Negative Questions MCQs for Sub-Topics of Topic 10: Calculus

Question 1. Which of the following statements about the concept of a limit is NOT true?

(A) The limit of a function as $x \to a$ describes the behavior of the function near $a$, not necessarily at $a.

(B) If $\lim\limits_{x \to a} f(x) = L$, then $f(a)$ must be defined and equal to $L$.

(C) The limit must be a finite value for it to exist in the standard sense.

(D) Limits can be used to understand the value a function approaches from either side of a point.

Question 2. For the limit $\lim\limits_{x \to a} f(x)$ to exist, which condition is NOT required?

(A) The left hand limit $\lim\limits_{x \to a^-} f(x)$ must exist.

(B) The right hand limit $\lim\limits_{x \to a^+} f(x)$ must exist.

(C) The left hand limit and the right hand limit must be equal.

(D) The function value $f(a)$ must be equal to the limit.

Question 3. Which of the following is NOT a condition under which direct substitution can be used to evaluate $\lim\limits_{x \to a} f(x)$?

(A) $f(x)$ is a polynomial function.

(B) $f(x)$ is a rational function and the denominator is non-zero at $x=a.

(C) $f(x)$ is a trigonometric function and is defined at $x=a.

(D) Evaluating $f(a)$ results in an indeterminate form like $\frac{0}{0}$.

Question 4. Which of the following is NOT an indeterminate form that requires special evaluation techniques?

(A) $\frac{0}{0}$

(B) $\frac{\infty}{\infty}$

(C) $1^\infty$

(D) $\frac{0}{1}$

Question 5. Which of the following limits is NOT evaluated correctly?

(A) $\lim\limits_{x \to 1} (2x + 3) = 5$

(B) $\lim\limits_{x \to 0} \frac{|x|}{x} = 0$

(C) $\lim\limits_{x \to \infty} \frac{1}{x^2} = 0$

(D) $\lim\limits_{x \to 2} \frac{x^2-4}{x-2} = 4$

Question 6. For the function $f(x) = \begin{cases} x+1 & , & x \leq 0 \\ x-1 & , & x > 0 \end{cases}$, which statement about its limit at $x=0$ is NOT true?

(A) The left hand limit exists and is 1.

(B) The right hand limit exists and is -1.

(C) The left hand limit and the right hand limit are equal.

(D) The limit as $x \to 0$ does not exist.

Question 7. Which method is NOT commonly used to evaluate indeterminate forms like $\frac{0}{0}$?

(A) Factorization and cancellation.

(B) Rationalization.

(C) Using standard limit formulas (like $\lim\limits_{x \to 0} \frac{\sin x}{x}$).

(D) Direct substitution.

Question 8. Which statement accurately describes a scenario where $\lim\limits_{x \to a} f(x)$ does NOT exist?

(A) $\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) = L$, where $L$ is a finite number.

(B) $\lim\limits_{x \to a^-} f(x) = -\infty$ and $\lim\limits_{x \to a^+} f(x) = +\infty$.

(C) The function $f(x)$ has a removable discontinuity at $x=a$.

(D) The function $f(x)$ is continuous at $x=a$.

Question 9. Consider the function $f(x) = \lfloor x \rfloor$ (floor function). Which statement about its limits is NOT true?

(A) $\lim\limits_{x \to 1^-} \lfloor x \rfloor = 0$

(B) $\lim\limits_{x \to 1^+} \lfloor x \rfloor = 1$

(C) $\lim\limits_{x \to 1} \lfloor x \rfloor$ exists.

(D) For any integer $n$, $\lim\limits_{x \to n} \lfloor x \rfloor$ does not exist.

Question 10. To evaluate $\lim\limits_{x \to 4} \frac{\sqrt{x}-2}{x-4}$, which method is NOT suitable among the common techniques for indeterminate forms?

(A) Rationalization of the numerator.

(B) Factorization of the denominator as $(\sqrt{x}-2)(\sqrt{x}+2)$.

(C) Direct substitution.

(D) L'Hopital's Rule (if introduced after derivatives).

Question 1. Which of the following is NOT a valid property from the algebra of limits, assuming $\lim\limits_{x \to a} f(x) = L$ and $\lim\limits_{x \to a} g(x) = M$ exist?

(A) $\lim\limits_{x \to a} (f(x) + g(x)) = L+M$

(B) $\lim\limits_{x \to a} (f(x) \cdot g(x)) = L \cdot M$

(C) $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, regardless of the value of $M$.

(D) $\lim\limits_{x \to a} c \cdot f(x) = c \cdot L$, where $c$ is a constant.

Question 2. Which of the following theorems is NOT primarily a theorem about limits?

(A) Squeeze Play Theorem (Sandwich Theorem)

(B) Intermediate Value Theorem

(C) Limit of a sum is the sum of the limits.

(D) $\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$ (considered a standard result, often proven using theorems)

Question 3. Which of the following is NOT a standard algebraic limit result?

(A) $\lim\limits_{x \to a} x^n = a^n$ for rational $n$ (where $a^n$ is defined).

(B) $\lim\limits_{x \to a} c = c$ for a constant $c$.

(C) $\lim\limits_{x \to \infty} x^n = \infty$ for $n > 0$.

(D) $\lim\limits_{x \to 0} \frac{e^x - 1}{x} = 1$

Question 4. Which of the following is NOT a standard trigonometric limit?

(A) $\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$

(B) $\lim\limits_{x \to 0} \frac{\tan x}{x} = 1$

(C) $\lim\limits_{x \to 0} \frac{1 - \cos x}{x} = 0$

(D) $\lim\limits_{x \to 0} \frac{\sin x}{\cos x} = 1$

Question 5. Which of the following is NOT a standard limit involving exponential or logarithmic functions?

(A) $\lim\limits_{x \to 0} \frac{e^x - 1}{x} = 1$

(B) $\lim\limits_{x \to 0} \frac{\ln(1+x)}{x} = 1$

(C) $\lim\limits_{x \to \infty} (1 + \frac{1}{x})^x = e$

(D) $\lim\limits_{x \to 0} (1+x)^x = e$

Question 6. If $\lim\limits_{x \to a} f(x) = L$ and $\lim\limits_{x \to a} g(x) = 0$, then $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ is NOT necessarily equal to which of the following?

(A) $\infty$

(B) $-\infty$

(C) A finite value

(D) $\frac{L}{0}$ (which is undefined in standard arithmetic)

(E) $L/0$ leading to an indeterminate form

Question 7. The Squeeze Play Theorem requires that for $x$ near $a$, which condition is NOT met?

(A) We have functions $g(x)$, $f(x)$, and $h(x)$ such that $g(x) \leq f(x) \leq h(x)$.

(B) $\lim\limits_{x \to a} g(x)$ exists.

(C) $\lim\limits_{x \to a} h(x)$ exists.

(D) $\lim\limits_{x \to a} g(x) < \lim\limits_{x \to a} h(x)$.

Question 8. Which of the following functions does NOT have a finite limit as $x \to \infty$?

(A) $f(x) = \frac{1}{x}$

(B) $f(x) = e^{-x}$

(C) $f(x) = \sin x$

(D) $f(x) = \frac{x}{x+1}$

Question 9. Which property is NOT part of the conclusion if $\lim\limits_{x \to a} f(x)$ exists and is equal to $L$?

(A) $\lim\limits_{x \to a^-} f(x) = L$

(B) $\lim\limits_{x \to a^+} f(x) = L$

(C) $f(a) = L$

(D) The limit $L$ is a finite real number.

Question 10. Which of the following limit calculations is NOT correct?

(A) $\lim\limits_{x \to 0} \frac{\sin(3x)}{5x} = \frac{3}{5}$

(B) $\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$

(C) $\lim\limits_{x \to 0} \frac{\tan x - x}{x} = 0$

(D) $\lim\limits_{x \to \infty} (1 + \frac{2}{x})^x = e^2$

Question 1. For a function $f(x)$ to be continuous at a point $x=a$, which condition is NOT required?

(A) $\lim\limits_{x \to a} f(x)$ exists.

(B) $f(a)$ is defined.

(C) The graph of $f(x)$ must be smooth at $x=a$ (no sharp corners).

(D) $\lim\limits_{x \to a} f(x) = f(a)$.

Question 2. Which statement about continuity on an interval is NOT true?

(A) A function is continuous on an open interval $(a,b)$ if it is continuous at every point in $(a,b)$.

(B) A function is continuous on a closed interval $[a,b]$ if it is continuous on $(a,b)$, continuous from the right at $a$, and continuous from the left at $b$.

(C) If a function is continuous on $[a,b]$, it must also be differentiable on $(a,b)$.

(D) Polynomial functions are continuous on any closed or open interval.

Question 3. Which of the following is NOT a standard type of discontinuity?

(A) Removable discontinuity

(B) Jump discontinuity

(C) Infinite discontinuity

(D) Oscillating discontinuity

(E) Vertical discontinuity

Question 4. If $f(x)$ and $g(x)$ are continuous functions, which of the following is NOT necessarily continuous?

(A) $f(x) + g(x)$

(B) $f(x) \cdot g(x)$

(C) $\frac{f(x)}{g(x)}$ where $g(x)$ can be zero.

(D) $(f \circ g)(x) = f(g(x))$

Question 5. Which function is NOT continuous at $x=0$?

(A) $f(x) = x^2$

(B) $f(x) = |x|$

(C) $f(x) = \sin x$

(D) $f(x) = \frac{1}{x}$

Question 6. Which statement about types of discontinuity is NOT correct?

(A) A removable discontinuity can be eliminated by redefining the function at a single point.

(B) A jump discontinuity occurs when the left and right limits are finite but unequal.

(C) An infinite discontinuity occurs when at least one of the one-sided limits is infinite.

(D) An oscillating discontinuity is a type of jump discontinuity.

Question 7. If $f(x)$ is continuous on a closed interval $[a,b]$, which property is NOT guaranteed by theorems about continuous functions?

(A) $f$ attains an absolute maximum value on $[a,b]$.

(B) $f$ attains an absolute minimum value on $[a,b]$.

(C) $f$ is differentiable on $(a,b)$.

(D) For any value $y_0$ between $f(a)$ and $f(b)$, there exists $c \in [a,b]$ such that $f(c) = y_0$.

Question 8. Which function is NOT continuous everywhere on its natural domain?

(A) $f(x) = x^3 - 2x + 1$

(B) $f(x) = \frac{1}{x^2 + 1}$

(C) $f(x) = \tan x$

(D) $f(x) = e^x$

Question 9. If $f$ is continuous at $g(a)$ and $g$ is continuous at $a$, the composite function $(f \circ g)(x) = f(g(x))$ is continuous at $a$. Which scenario does NOT fit this rule?

(A) $g(x) = x^2$, $f(u) = \sin u$, at $a=1$.

(B) $g(x) = x+1$, $f(u) = \frac{1}{u}$, at $a=-1$.

(C) $g(x) = \sin x$, $f(u) = e^u$, at $a=\pi/2$.

(D) $g(x) = |x|$, $f(u) = u^3$, at $a=0$.

Question 10. Which statement regarding the relationship between limits and continuity is NOT correct?

(A) If a function is continuous at a point, its limit at that point exists.

(B) If the limit of a function exists at a point, the function must be continuous at that point.

(C) A removable discontinuity occurs when the limit exists but does not equal the function value (or the function is undefined).

(D) Jump discontinuities occur when the one-sided limits exist but are unequal.

Question 1. Which is NOT a valid form for the definition of the derivative of $f(x)$ at $x=a$ from first principles?

(A) $\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$

(B) $\lim\limits_{x \to a} \frac{f(x) - f(a)}{x - a}$

(C) $\lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}$ (This gives $f'(x)$, not $f'(a)$)

(D) $\lim\limits_{h \to 0} \frac{f(a) - f(a-h)}{h}$

Question 2. For a function $f(x)$ to be differentiable at $x=a$, which condition is NOT necessary?

(A) $f(x)$ must be continuous at $x=a$.

(B) The left hand derivative at $x=a$ must exist and be finite.

(C) The right hand derivative at $x=a$ must exist and be finite.

(D) The graph of $f(x)$ must have a vertical tangent at $x=a$.

Question 3. Which statement about the relationship between differentiability and continuity is NOT true?

(A) If a function is differentiable at a point, it is also continuous at that point.

(B) If a function is continuous at a point, it is also differentiable at that point.

(C) Continuity is a necessary condition for differentiability.

(D) Differentiability implies continuity.

Question 4. Which function is NOT differentiable at $x=0$?

(A) $f(x) = x^2$

(B) $f(x) = e^x$

(C) $f(x) = \sin x$

(D) $f(x) = |x|$

Question 5. Which statement about differentiability in an interval is NOT correct?

(A) A function is differentiable on $(a,b)$ if it is differentiable at every point in $(a,b)$.

(B) A function is differentiable on $[a,b]$ if it is differentiable on $(a,b)$, and the right hand derivative at $a$ and the left hand derivative at $b$ exist and are finite.

(C) If a function is differentiable on $[a,b]$, its graph is smooth with no breaks, corners, or cusps in that interval.

(D) If a function is continuous on $[a,b]$, it is guaranteed to be differentiable on $(a,b)$.

Question 6. Which graphical feature at $x=a$ does NOT necessarily imply that a continuous function $f(x)$ is not differentiable at $x=a$?

(A) A sharp corner.

(B) A cusp.

(C) A vertical tangent.

(D) A horizontal tangent.

Question 7. If a function is differentiable on an interval, it does NOT necessarily possess which property on that interval?

(A) It is continuous on the interval.

(B) Its derivative is also differentiable on the interval.

(C) It has a well-defined tangent line at every point in the interval.

(D) The left and right derivatives are equal at every point in the interval.

Question 8. Which statement regarding the process of differentiation from an applied perspective is NOT accurate?

(A) Differentiation provides the instantaneous rate of change.

(B) Differentiation is used to find the slope of the tangent line.

(C) Differentiation of a position function gives velocity.

(D) Differentiation gives the total accumulation of a quantity over an interval.

Question 9. The derivative from first principles $f'(a) = \lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists if and only if the left-hand limit and the right-hand limit exist and are equal. Which pair of limits is NOT relevant here?

(A) $\lim\limits_{h \to 0^-} \frac{f(a+h) - f(a)}{h}$ (Left Hand Derivative)

(B) $\lim\limits_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$ (Right Hand Derivative)

(C) $\lim\limits_{x \to a^-} f(x)$ (Left Hand Limit of the function)

(D) $\lim\limits_{x \to a^+} f(x)$ (Right Hand Limit of the function)

Question 10. Which function is continuous at a point but NOT differentiable at that same point?

(A) $f(x) = x^3$ at $x=0$.

(B) $f(x) = \sqrt{x}$ at $x=0$.

(C) $f(x) = |x-1|$ at $x=1$.

(D) $f(x) = \lfloor x \rfloor$ at $x=0.5$.

Question 1. Which of the following is NOT a correct basic rule of differentiation?

(A) $\frac{d}{dx}(c) = 0$ for a constant $c$.

(B) $\frac{d}{dx}(x^n) = nx^{n-1}$ for real $n$.

(C) $\frac{d}{dx}(u+v) = \frac{du}{dx} + \frac{dv}{dx}$.

(D) $\frac{d}{dx}(uv) = \frac{du}{dx} \frac{dv}{dx}$.

Question 2. Which is NOT a standard derivative formula?

(A) $\frac{d}{dx}(\sin x) = \cos x$

(B) $\frac{d}{dx}(\cos x) = \sin x$

(C) $\frac{d}{dx}(\tan x) = \sec^2 x$

(D) $\frac{d}{dx}(\sec x) = \sec x \tan x$

Question 3. Which statement about the derivative of a constant function $f(x)=c$ is NOT true?

(A) The derivative is 0.

(B) The rate of change of the function is zero.

(C) The graph of the function is a horizontal line.

(D) The derivative is equal to the constant $c$ itself.

Question 4. Which of the following is NOT a formula from the algebra of derivatives?

(A) Sum Rule: $(u+v)' = u' + v'$

(B) Difference Rule: $(u-v)' = u' - v'$

(C) Product Rule: $(uv)' = u'v + uv'$

(D) Power Rule: $(x^n)' = nx^{n-1}$

Question 5. Which function is NOT differentiable everywhere on its natural domain?

(A) $f(x) = x^5$

(B) $f(x) = \cos x$

(C) $f(x) = e^x$

(D) $f(x) = \frac{1}{x}$

Question 6. Which derivative formula is incorrect?

(A) $\frac{d}{dx}(a^x) = a^x \ln a$

(B) $\frac{d}{dx}(\ln x) = \frac{1}{x}$

(C) $\frac{d}{dx}(\log_a x) = \frac{1}{x \log_{10} a}$

(D) $\frac{d}{dx}(e^x) = e^x$

Question 7. Which function's derivative is NOT negative for all $x$ in its domain?

(A) $f(x) = -x^3$

(B) $f(x) = e^{-x}$

(C) $f(x) = \cos x$ (on $(0, \pi)$ it is negative, but not always)

(D) $f(x) = -\ln x$ (on $(0, \infty)$ it is negative)

Question 8. Which function is NOT its own derivative?

(A) $f(x) = e^x$

(B) $f(x) = e^x + 5$

(C) $f(x) = \sin x$

(D) $f(x) = Ce^x$ for any constant $C$.

Question 9. Which differentiation rule is NOT directly derived from limit properties using the definition of the derivative?

(A) Sum Rule

(B) Constant Multiple Rule

(C) Product Rule

(D) Power Rule for integer exponents

(E) Chain Rule

Question 10. Which of the following expressions is NOT a polynomial function?

(A) $3x^4 - 2x + 1$

(B) $x^{100}$

(C) $\sqrt{x} + 5x$

(D) $7$

Question 1. Which of the following is NOT a composite function?

(A) $f(x) = \sin(x^2)$

(B) $f(x) = \sqrt{x+1}$

(C) $f(x) = e^{3x}$

(D) $f(x) = x^2 + 5x - 2$

Question 2. Which of the following is NOT a correct representation of the chain rule?

(A) If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

(B) $(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$.

(C) $\frac{d}{dx} f(g(x)) = f'(g(x))$.

(D) If $y = f(u)$, $u=g(v)$, $v=h(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}$.

Question 3. Which function's derivative CANNOT be found using only the power rule and constant multiple rule and requires the chain rule?

(A) $f(x) = (x+5)^2$

(B) $f(x) = x^2 + 5^2$

(C) $f(x) = 7x^3$

(D) $f(x) = x^{-1/2}$

Question 4. Which function is NOT a composite function of two simpler functions?

(A) $y = \sin(3x+1)$

(B) $y = \ln(x^2)$

(C) $y = e^x + x^2$

(D) $y = (x^3-1)^5

Question 5. Which statement about the application of the chain rule is NOT true?

(A) When differentiating $f(g(x))$, we differentiate the outer function $f$ first, keeping the inner function $g(x)$ unchanged inside $f'$.

(B) We then multiply the result by the derivative of the inner function $g'(x)$.

(C) The chain rule is used when the argument of a function is not simply the independent variable $x$.

(D) The chain rule applies only when the inner function is a linear function.

Question 6. If $y=f(u)$ and $u=g(x)$, which relationship is NOT correct?

(A) $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$

(B) $y'(x) = f'(u) g'(x)$

(C) $\frac{dy}{dx} = f'(g(x)) g'(x)$

(D) $\frac{dx}{dy} = \frac{dx}{du} \frac{du}{dy}$

Question 7. Which derivative calculation using the chain rule is incorrect?

(A) $\frac{d}{dx}(\sin(2x)) = 2 \cos(2x)$

(B) $\frac{d}{dx}(e^{x^2}) = 2x e^{x^2}$

(C) $\frac{d}{dx}(\ln(\cos x)) = \frac{1}{\cos x}$

(D) $\frac{d}{dx}((x^2+x)^3) = 3(x^2+x)^2 (2x+1)$

Question 8. Which type of function is NOT typically differentiated using the chain rule as the primary method?

(A) A function of the form $f(g(x))$ where $f$ and $g$ are non-linear.

(B) A simple polynomial like $f(x) = x^4 + 2x$.

(C) A function like $y = \sin(\ln x)$.

(D) A function like $y = e^{\sqrt{x}}$.

Question 9. When applying the chain rule to differentiate $y = f(g(h(x)))$, which order of differentiation is NOT followed?

(A) Differentiate the outermost function $f$ with respect to its argument $g(h(x))$.

(B) Multiply by the derivative of the middle function $g$ with respect to its argument $h(x)$.

(C) Multiply by the derivative of the innermost function $h$ with respect to $x$.

(D) Differentiate the innermost function $h$ with respect to $x$, then the middle function $g$ with respect to its result, then the outermost function $f$ with respect to its result.

Question 10. Which statement about composite functions is NOT accurate?

(A) A composite function is formed by applying one function to the result of another.

(B) For functions $f$ and $g$, $(f \circ g)(x) = f(g(x))$.

(C) The domain of $f \circ g$ is the set of $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

(D) The composition of two functions is always commutative, i.e., $(f \circ g)(x) = (g \circ f)(x)$.

Question 1. Which of the following equations does NOT define $y$ implicitly as a function of $x$?

(A) $x^2 + y^2 = 25$

(B) $\sin(xy) = x$

(C) $e^{x+y} = xy$

(D) $y = x^3 - 4x + 1$

Question 2. When performing implicit differentiation on an equation involving $x$ and $y$, which step is NOT part of the process?

(A) Differentiate both sides of the equation with respect to $x$.

(B) Treat $y$ as a function of $x$ and use the chain rule for terms involving $y$.

(C) Treat $x$ as a function of $y$ and use the chain rule for terms involving $x$.

(D) Solve the resulting equation algebraically for $\frac{dy}{dx}$.

Question 3. If $y = f(x)$ and $x = g(y)$ is its inverse function, then $\frac{dy}{dx}$ is NOT necessarily equal to which expression?

(A) $\frac{1}{dx/dy}$

(B) $\frac{1}{g'(y)}$

(C) $g'(y)$

(D) $\frac{1}{g'(f(x))}$

Question 4. Which of the following is NOT a standard derivative formula for inverse trigonometric functions?

(A) $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$

(B) $\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$

(C) $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{\sqrt{1+x^2}}$

(D) $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}$

Question 5. Implicit differentiation is NOT needed to find $\frac{dy}{dx}$ for which type of equation?

(A) An equation where $y$ is explicitly expressed as a function of $x$ (e.g., $y = x^2$).

(B) An equation where $x$ and $y$ are mixed together (e.g., $x^2y + xy^2 = 1$).

(C) An equation defining a relation that is not a single-valued function of $x$ (e.g., $x^2+y^2=1$).

(D) An equation like $y^3 + \sin y = x$.

Question 6. Which statement about implicit differentiation is NOT true?

(A) It relies on the chain rule.

(B) We differentiate every term with respect to the independent variable (usually $x$).

(C) Terms involving only the independent variable are differentiated as usual.

(D) Terms involving the dependent variable are differentiated as if it were the independent variable, without multiplying by $\frac{dy}{dx}$.

Question 7. If $y = \text{cot}^{-1} x$, then $\frac{dy}{dx}$ is NOT equal to which expression?

(A) $-\frac{1}{1+x^2}$

(B) $\frac{d}{dx}(\tan^{-1}(1/x))$ for $x>0$

(C) $\frac{1}{1+x^2}$

(D) $-\frac{1}{1+x^2}$ (This is the correct derivative, so the question is asking which is NOT equal to it)

Question 8. The derivative of which inverse trigonometric function involves $\sqrt{1-x^2}$ in the denominator?

(A) $\sin^{-1} x$

(B) $\cos^{-1} x$

(C) $\tan^{-1} x$

(D) Both $\sin^{-1} x$ and $\cos^{-1} x$ involve $\sqrt{1-x^2}$. Which ONE of the options does NOT involve it?

Question 9. Which range of $x$ is NOT generally applicable for the standard derivative formula $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$?

(A) $|x| < 1$

(B) $-1 < x < 1$

(C) $x = 1$

(D) $x = 0$

Question 10. If $y=f(x)$ and $x=g(y)$ is its inverse, and $f'(x) \neq 0$, the derivative $\frac{dx}{dy}$ is NOT equal to which expression?

(A) $g'(y)$

(B) $\frac{1}{dy/dx}$

(C) $1/f'(x)$

(D) $f'(x)$

Question 1. Logarithmic differentiation is NOT typically the most efficient method for differentiating which type of function?

(A) $y = x^x$

(B) $y = \frac{(x+1)\sqrt{x-2}}{x^3}$

(C) $y = \sin(x^2 + e^x)$

(D) $y = (\sin x)^{\cos x}$

Question 2. Which is NOT a primary reason for using logarithmic differentiation?

(A) To handle functions where both the base and the exponent are variables.

(B) To simplify differentiation of complicated products and quotients.

(C) To make addition and subtraction operations easier to differentiate.

(D) To convert exponents into coefficients by taking the logarithm.

Question 3. Which statement about functions in parametric form $x=f(t), y=g(t)$ is NOT true?

(A) Both $x$ and $y$ are expressed in terms of a third variable, the parameter $t$.

(B) The parameter $t$ must always represent time.

(C) This form is useful for describing curves where $y$ is not a single-valued function of $x$.

(D) To find points on the curve, we plug in values of $t$ into the equations for $x$ and $y$.

Question 4. Which formula for finding the derivative $\frac{dy}{dx}$ of a parametric function $x=f(t), y=g(t)$ is NOT correct?

(A) $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, provided $\frac{dx}{dt} \neq 0$.

(B) $\frac{dy}{dx} = \frac{g'(t)}{f'(t)}$, provided $f'(t) \neq 0$.

(C) $\frac{dy}{dx} = \frac{d}{dt}(\frac{y}{x})$.

(D) $\frac{dy}{dx} = \frac{\dot{y}}{\dot{x}}$, using dot notation for differentiation with respect to $t$.

Question 5. To find $\frac{dy}{dx}$ for $x=f(t), y=g(t)$, which derivative is NOT explicitly required in the primary formula $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$?

(A) $\frac{dy}{dt}$

(B) $\frac{dx}{dt}$

(C) $\frac{d^2 y}{dt^2}$

(D) Both $\frac{dy}{dt}$ and $\frac{dx}{dt}$ are required.

Question 6. Which function form does NOT directly benefit from simplifying differentiation by using logarithmic differentiation?

(A) $y = f(x)^{g(x)}$

(B) $y = f(x) \cdot g(x) \cdot h(x)$

(C) $y = \frac{f(x) g(x)}{h(x)}$

(D) $y = f(x) + g(x) + h(x)$

Question 7. Which property of logarithms is NOT typically useful in the process of logarithmic differentiation?

(A) $\ln(ab) = \ln a + \ln b$

(B) $\ln(a/b) = \ln a - \ln b$

(C) $\ln(a^b) = b \ln a$

(D) $\ln(a+b) = \ln a + \ln b$

Question 8. A parametric representation is NOT suitable for describing which type of curve?

(A) A circle.

(B) A parabola.

(C) A line segment.

(D) A curve that cannot be represented by any equation in $x$ and $y$.

Question 9. Which derivative calculation using parametric equations is incorrect?

(A) If $x=t^2, y=t^3$, then $\frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3}{2}t$ (for $t \neq 0$).

(B) If $x=\cos\theta, y=\sin\theta$, then $\frac{dy}{dx} = \frac{\cos\theta}{-\sin\theta} = -\cot\theta$ (for $\sin\theta \neq 0$).

(C) If $x=e^t, y=e^{2t}$, then $\frac{dy}{dx} = \frac{2e^{2t}}{e^t} = 2e^t$.

(D) If $x=\ln t, y=t^2$, then $\frac{dy}{dx} = \frac{1/t}{2t} = \frac{1}{2t^2}$.

Question 10. Logarithmic differentiation CANNOT simplify the differentiation of which function type?

(A) Products of many functions.

(B) Quotients of many functions.

(C) Functions raised to variable powers.

(D) Sums or differences of functions.

Question 1. Which notation does NOT represent a second order derivative of $y$ with respect to $x$?

(A) $\frac{d^2 y}{dx^2}$

(B) $f''(x)$

(C) $y''$

(D) $(y')^2$

Question 2. Which statement about higher order derivatives is NOT true?

(A) The second derivative is the derivative of the first derivative.

(B) Higher order derivatives can be calculated by successive differentiation.

(C) The third derivative is the rate of change of acceleration.

(D) If the first derivative is zero at a point, all higher order derivatives must also be zero at that point.

Question 3. The second derivative does NOT provide information about which property of the function's graph?

(A) Concavity.

(B) Points of inflection.

(C) Whether a critical point is a local maximum or minimum (using the Second Derivative Test).

(D) The instantaneous slope of the tangent line.

Question 4. If $y=f(x)$, the third derivative $\frac{d^3 y}{dx^3}$ is NOT the derivative of which expression with respect to $x$?

(A) $\frac{d^2 y}{dx^2}$

(B) $y''$

(C) $\frac{dy}{dx}$

(D) $f''(x)$

Question 5. Which function does NOT have derivatives of all orders for all real numbers $x$?

(A) $f(x) = x^5$

(B) $f(x) = \sin x$

(C) $f(x) = e^x$

(D) $f(x) = |x|$

Question 6. If the position function of an object is $s(t)$, which of the following does NOT represent the jerk?

(A) $s'''(t)$

(B) $\frac{d^3 s}{dt^3}$

(C) The derivative of acceleration with respect to time.

(D) The integral of acceleration with respect to time.

Question 7. Which relationship involving higher order derivatives is NOT correct?

(A) If $y = \sin x$, then $y'' = -y$.

(B) If $y = e^{2x}$, then $y'' = 4y$.

(C) If $y = x^3$, then $y''' = 6$.

(D) If $y = \cos x$, then $y''' = \sin x$.

Question 8. The process of finding higher order derivatives does NOT involve which operation?

(A) Differentiation.

(B) Finding the derivative of the previous derivative.

(C) Successive application of differentiation rules.

(D) Integration.

Question 9. Which function's second derivative is NOT always positive for all real $x$?

(A) $f(x) = x^2$

(B) $f(x) = e^x$

(C) $f(x) = x^4$

(D) $f(x) = \sin x$

Question 10. If $f'(c)=0$ and $f''(c)=0$, the Second Derivative Test does NOT tell us which of the following?

(A) Whether $c$ is a local maximum.

(B) Whether $c$ is a local minimum.

(C) Whether $c$ is a point of inflection (definitively).

(D) Whether $f'(x)$ changes sign at $c$.

Question 1. Which of the following is NOT a necessary condition for Rolle's Theorem to apply to a function $f(x)$ on the interval $[a,b]$?

(A) $f(x)$ must be continuous on $[a,b]$.

(B) $f(x)$ must be differentiable on $(a,b)$.

(C) $f(a)$ must be equal to $f(b)$.

(D) $f'(x)$ must be zero at the endpoints $a$ and $b$.

Question 2. If the conditions of Rolle's Theorem are met for $f(x)$ on $[a,b]$, which statement about the conclusion is NOT true?

(A) There exists at least one $c$ in the open interval $(a,b)$.

(B) At this point $c$, the derivative $f'(c) = 0$.

(C) Geometrically, there is a horizontal tangent to the curve at $x=c$.

(D) There exists a unique $c$ in $(a,b)$ such that $f'(c)=0$.

Question 3. Which of the following is NOT a necessary condition for Lagrange's Mean Value Theorem to apply to a function $f(x)$ on the interval $[a,b]$?

(A) $f(x)$ must be continuous on the closed interval $[a,b]$.

(B) $f(x)$ must be differentiable on the open interval $(a,b)$.

(C) $f(a)$ must be equal to $f(b)$.

(D) The interval $[a,b]$ must be finite.

Question 4. If the conditions of the Mean Value Theorem are met for $f(x)$ on $[a,b]$, which statement is NOT a valid geometric interpretation of the conclusion?

(A) There exists a point $c \in (a,b)$ where the tangent line is parallel to the secant line connecting $(a, f(a))$ and $(b, f(b))$.

(B) The instantaneous rate of change at some point equals the average rate of change over the interval.

(C) The slope of the tangent at $x=c$ is equal to $\frac{f(b) - f(a)}{b - a}$.

(D) The tangent line at $x=c$ passes through the points $(a, f(a))$ and $(b, f(b))$.

Question 5. Which statement comparing Rolle's Theorem and the Mean Value Theorem is NOT correct?

(A) Rolle's Theorem is a special case of the Mean Value Theorem.

(B) The conditions for MVT are less restrictive than those for Rolle's Theorem.

(C) The conclusion of MVT is $f'(c) = \frac{f(b) - f(a)}{b - a}$, while for Rolle's Theorem it's $f'(c) = 0$.

(D) If a function satisfies the conditions of MVT, it must also satisfy the conditions of Rolle's Theorem.

Question 6. Which function does NOT satisfy the conditions of Rolle's Theorem on the interval $[-1, 1]$?

(A) $f(x) = x^2$

(B) $f(x) = \sin(\pi x)$

(C) $f(x) = |x|$

(D) $f(x) = x^3 - x$

Question 7. Which application CANNOT be directly proven or demonstrated using the Mean Value Theorem?

(A) Proving that if $f'(x) = 0$ on an interval, then $f(x)$ is constant on that interval.

(B) Estimating the value of $f(b)$ given $f(a)$ and bounds on $f'(x)$.

(C) Proving that if $f'(x) > 0$ on an interval, then $f(x)$ is strictly increasing on that interval.

(D) Proving that a continuous function on $[a,b]$ attains its maximum value.

Question 8. For the function $f(x) = 1/x$ on the interval $[-1, 1]$, which statement regarding the Mean Value Theorem is NOT true?

(A) The function is defined on $[-1, 1]$.

(B) The function is not continuous on $[-1, 1]$.

(C) The conditions of the Mean Value Theorem are not satisfied on $[-1, 1]$.

(D) There exists a value $c \in (-1, 1)$ such that $f'(c) = \frac{f(1) - f(-1)}{1 - (-1)}$.

Question 9. If a function is discontinuous on the closed interval $[a,b]$, which of the following theorems CANNOT be applied to the function on this interval?

(A) Intermediate Value Theorem

(B) Extreme Value Theorem

(C) Rolle's Theorem

(D) Lagrange's Mean Value Theorem

(E) All theorems from (A) to (D) require continuity, so none can be applied directly.

Question 10. For the function $f(x) = x^2$ on $[0, 2]$, we found $c=1$ such that $f'(c) = \frac{f(2)-f(0)}{2-0}$. Which statement about this result is NOT true?

(A) $c=1$ is in the interval $(0, 2)$.

(B) $f'(1) = 2$.

(C) $\frac{f(2)-f(0)}{2-0} = 2$.

(D) The theorem guarantees that $c$ is the midpoint of the interval.

Question 1. Which quantity is NOT typically represented as a rate of change using the first derivative with respect to time $t$?

(A) Velocity (rate of change of position).

(B) Acceleration (rate of change of velocity).

(C) Marginal cost (rate of change of total cost with respect to quantity).

(D) Total revenue.

Question 2. When solving a related rates problem, which step is NOT typically involved?

(A) Identify the variables and the given and unknown rates of change.

(B) Find an equation that relates the variables.

(C) Differentiate the equation with respect to time implicitly.

(D) Integrate the equation with respect to time implicitly.

Question 3. In economics, which of the following is NOT a marginal concept calculated using derivatives?

(A) Marginal Cost.

(B) Marginal Revenue.

(C) Marginal Profit.

(D) Average Cost.

Question 4. The rate of change of the volume $V$ of a sphere with respect to its radius $r$ is $\frac{dV}{dr} = 4\pi r^2$. This rate is NOT equal to which expression?

(A) The surface area of the sphere.

(B) $4\pi r^2$.

(C) The derivative of $V = \frac{4}{3}\pi r^3$ with respect to $r$.

(D) The rate at which the radius is changing.

Question 5. If the radius of a circle is decreasing at a rate of $2$ cm/s, its area is NOT necessarily decreasing at which rate?

(A) $4\pi r$ cm$^2$/s

(B) The rate is negative since the area is decreasing.

(C) $2 \cdot (2\pi r)$ cm$^2$/s, where $2\pi r$ is the circumference.

(D) The rate depends on the current radius $r$.

Question 6. Which scenario is NOT an example of a related rates problem?

(A) A ladder sliding down a wall.

(B) Water draining from a conical tank.

(C) The distance between two cars moving in different directions.

(D) Finding the average velocity over a time interval.

Question 7. Marginal revenue is NOT the derivative of which function?

(A) Total Revenue Function.

(B) Average Revenue Function.

(C) $R(x)$ where $R$ is total revenue.

(D) The function representing the total income from sales.

Question 8. Marginal cost is NOT the rate of change of total cost with respect to which variable?

(A) Quantity produced.

(B) Time.

(C) Number of units.

(D) $x$ where $x$ is the number of units.

Question 9. If the quantity of production $x$ (in units) is changing with time $t$ (in hours) such that $x(t)$ is a differentiable function, and the total cost is $C(x)$, the rate of change of total cost with respect to time is NOT given by which expression?

(A) $\frac{dC}{dt}$

(B) $C'(x) \cdot \frac{dx}{dt}$

(C) $\frac{dC}{dx} \cdot \frac{dx}{dt}$ (by chain rule)

(D) $C'(t)$

Question 10. Which statement about the units of a rate of change like $\frac{dy}{dx}$ is NOT correct?

(A) The units are units of $y$ per unit of $x$.

(B) If $y$ is in meters and $x$ is in seconds, $\frac{dy}{dx}$ is in meters/second.

(C) If $y$ is in $\textsf{₹}$ and $x$ is in units, $\frac{dy}{dx}$ is in $\textsf{₹}$/unit.

(D) The units are dimensionless if $y$ and $x$ have the same units.

Question 1. Which formula is NOT the equation of the tangent line to the curve $y=f(x)$ at the point $(x_0, y_0)$?

(A) $y - y_0 = f'(x_0)(x - x_0)$

(B) $y = f'(x_0)x + y_0 - f'(x_0)x_0$

(C) $y = m(x - x_0) + y_0$ where $m = f'(x_0)$.

(D) $x - x_0 = f'(x_0)(y - y_0)$

Question 2. Which formula is NOT the equation of the normal line to the curve $y=f(x)$ at the point $(x_0, y_0)$, assuming $f'(x_0) \neq 0$?

(A) $y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)$

(B) $(y - y_0) f'(x_0) = -(x - x_0)$

(C) $y = -\frac{1}{f'(x_0)}x + y_0 + \frac{x_0}{f'(x_0)}$

(D) $y - y_0 = f'(x_0)(x - x_0)$

Question 3. Differentials are NOT used to calculate which of the following?

(A) Approximate change in a function's value ($\Delta y$).

(B) Linear approximation of a function near a point.

(C) Estimation of errors in calculated quantities based on measurement errors.

(D) The exact value of a function at a specific point away from the known point.

Question 4. Which statement about using differentials for approximations is NOT true?

(A) $f(x + \Delta x) \approx f(x) + f'(x) \Delta x$ for small $\Delta x$.

(B) $dy = f'(x) dx$ is the differential of $y$.

(C) $\Delta y \approx dy$ for small $\Delta x$.

(D) The approximation is exact for any size of $\Delta x$.

Question 5. If $\Delta x$ is the error in measuring $x$, and $y=f(x)$, which formula for the related error in $y$ using differentials is NOT correctly defined?

(A) Absolute error in $y \approx dy = f'(x) dx$ where $dx = \Delta x$.

(B) Relative error in $y \approx \frac{dy}{y} = \frac{f'(x) dx}{f(x)}$.

(C) Percentage error in $y \approx \frac{dy}{y} \times 100\%$.

(D) Absolute error in $y = f(x+\Delta x) - f(x)$.

(E) Approximate error in $y = f(x) + f'(x) \Delta x$.

Question 6. The slope of the normal line to a curve at a point is NOT related to the slope of the tangent line at that point in which way?

(A) The product of their slopes is -1 (if the tangent is not horizontal or vertical).

(B) They are perpendicular to each other.

(C) If the tangent has slope $m$, the normal has slope $-1/m$ (if $m \neq 0$).

(D) The slopes are always equal in magnitude but opposite in sign.

Question 7. Which approximate value CANNOT be directly estimated using differentials based on a simple nearby value?

(A) $\sqrt{101}$ (using $\sqrt{100}$)

(B) $(2.01)^3$ (using $2^3$)

(C) $\sin(46^\circ)$ (using $\sin(45^\circ)$)

(D) The exact value of $f(a+\Delta x)$ where $f'(a)=0$.

Question 8. If a measurement $x$ has a percentage error, the calculated quantity $y=x^n$ has an approximate percentage error that is NOT necessarily related in which way?

(A) The percentage error in $y$ is approximately $n$ times the percentage error in $x$.

(B) The relative error $\frac{dy}{y} \approx n \frac{dx}{x}$.

(C) The relationship is exact, not approximate.

(D) This method applies to non-integer values of $n$ as well.

Question 9. Which statement about the linear approximation using tangents and differentials is NOT correct?

(A) The tangent line provides a good approximation of the function near the point of tangency.

(B) The error in the approximation $|\Delta y - dy|$ is small for small $\Delta x$.

(C) The approximation is usually less accurate as $\Delta x$ increases.

(D) The approximation is always an overestimate of the true value.

Question 10. Which formula relating errors and differentials is NOT correct?

(A) $\Delta x = dx$ (when $dx$ is considered as a small change in $x$)

(B) $\Delta y = f(x+dx) - f(x)$

(C) $dy = f'(x) dx$

(D) $\Delta y = dy$ for any function $f(x)$ and any value of $dx$.

Question 1. Which of the following is NOT a definition of an increasing function $f(x)$ on an interval?

(A) For any $x_1, x_2$ in the interval, if $x_1 < x_2$, then $f(x_1) \leq f(x_2)$.

(B) The graph of the function rises or stays constant as you move from left to right.

(C) The first derivative $f'(x) \geq 0$ for all $x$ in the interval.

(D) For any $x_1, x_2$ in the interval, if $x_1 < x_2$, then $f(x_1) < f(x_2)$.

Question 2. If $f'(x) < 0$ for all $x$ in an interval, the function is NOT necessarily which of the following on that interval?

(A) Decreasing.

(B) Strictly decreasing.

(C) Monotonic.

(D) Having a local minimum in the interior of the interval.

Question 3. Which test is NOT used to determine the intervals where a function is increasing or decreasing?

(A) First Derivative Test.

(B) Checking the sign of $f'(x)$.

(C) Analyzing the graph of $f'(x)$.

(D) Second Derivative Test.

Question 4. Which statement about finding intervals of monotonicity is NOT true?

(A) We find the points where $f'(x) = 0$ or $f'(x)$ is undefined.

(B) These points divide the domain into intervals.

(C) We test the sign of $f'(x)$ in each interval.

(D) If $f'(x) > 0$ in an interval, the function is decreasing in that interval.

Question 5. Which function is NOT strictly increasing on its entire domain?

(A) $f(x) = e^x$

(B) $f(x) = x^3$

(C) $f(x) = \ln x$ (for $x>0$)

(D) $f(x) = x^2$

Question 6. If $f(x)$ is decreasing on an interval, its derivative $f'(x)$ is NOT necessarily which of the following on that interval?

(A) Less than or equal to zero.

(B) Strictly less than zero.

(C) Negative or zero.

(D) Indicating a non-rising tangent.

Question 7. Which function is NOT monotonic on the interval $(0, \pi)$?

(A) $f(x) = \sin x$

(B) $f(x) = \cos x$

(C) $f(x) = x^2$

(D) $f(x) = e^x$

Question 8. A function that is not monotonic on an interval must necessarily NOT satisfy which condition over that interval?

(A) $f'(x) \geq 0$ for all $x$ in the interval.

(B) $f'(x) \leq 0$ for all $x$ in the interval.

(C) $f'(x)$ having a constant sign (either $\geq 0$ or $\leq 0$) throughout the interval.

(D) $f(x)$ being continuous on the interval.

Question 9. If a function has a local extremum in the interior of an interval, which statement is TRUE?

(A) The function is monotonic on that interval.

(B) The function is not monotonic on that interval.

(C) The derivative is always positive or always negative on that interval.

(D) The function is strictly increasing on the interval.

Question 10. Which application does NOT directly involve finding intervals of increasing or decreasing functions?

(A) Finding where profit is maximized (often occurs at the boundary of increasing/decreasing intervals).

(B) Analyzing the behavior of population growth over time.

(C) Determining where a function is concave up or concave down.

(D) Finding the production level where marginal cost is minimized.

Question 1. Which of the following is NOT a type of extremum?

(A) Local maximum.

(B) Local minimum.

(C) Absolute maximum.

(D) Absolute minimum.

(E) Point of inflection.

Question 2. A local extremum of a differentiable function $f(x)$ can occur only at a critical point where $f'(x)=0$. It CANNOT occur at which type of point?

(A) A point where $f'(x) > 0$.

(B) A point where $f'(x) < 0$.

(C) A point where $f'(x)$ is undefined (if considering local extrema generally).

(D) Both (A) and (B).

Question 3. Which is NOT a test commonly used to classify critical points as local maxima, minima, or neither?

(A) First Derivative Test.

(B) Second Derivative Test.

(C) Checking if $f'(c)=0$ at the critical point $c$.

(D) L'Hopital's Rule.

Question 4. The absolute maximum of a continuous function on a closed interval $[a,b]$ CANNOT occur at which point?

(A) At a critical point in the open interval $(a,b)$.

(B) At the left endpoint $a$.

(C) At the right endpoint $b$.

(D) At a point in the open interval $(a,b)$ that is not a critical point.

Question 5. When solving optimization word problems using calculus, which step is NOT typically involved?

(A) Setting up a function to be maximized or minimized.

(B) Finding the derivative of the function.

(C) Finding the critical points by setting the second derivative to zero.

(D) Using derivative tests or endpoint evaluation to find the optimal value.

Question 6. If $f'(c)=0$ at a point $c$ in the domain, which conclusion is NOT necessarily true?

(A) $c$ is a critical point.

(B) The tangent line at $x=c$ is horizontal (if differentiable).

(C) $c$ is a local extremum.

(D) The function might have a local maximum, minimum, or inflection point at $c$.

Question 7. Which statement about local vs. absolute extrema is NOT correct?

(A) A local maximum is the highest point in its immediate neighborhood.

(B) An absolute maximum is the highest point over the entire domain or interval.

(C) A function can have multiple local maxima but only one absolute maximum (if it exists).

(D) Every local extremum is also an absolute extremum.

Question 8. A continuous function on an open interval $(a,b)$ does NOT necessarily have which property?

(A) It is bounded on $(a,b)$.

(B) It attains an absolute maximum value on $(a,b)$.

(C) It satisfies the Intermediate Value Theorem on $(a,b)$.

(D) It has a derivative at every point in $(a,b)$.

Question 9. The Second Derivative Test is NOT conclusive in which case?

(A) $f'(c) = 0$ and $f''(c) > 0$.

(B) $f'(c) = 0$ and $f''(c) < 0$.

(C) $f'(c) = 0$ and $f''(c) = 0$.

(D) $f'(c)$ is undefined.

Question 10. Which application does NOT involve finding extrema?

(A) Minimizing the surface area of a cylinder with a fixed volume.

(B) Finding the maximum height of a projectile.

(C) Determining the intervals where a function is increasing or decreasing.

(D) Finding the point on a curve closest to a given point.

Question 1. Integration is NOT correctly described as which of the following?

(A) The reverse process of differentiation.

(B) Finding the antiderivative of a function.

(C) A process of summation (in the context of definite integrals).

(D) The process of finding the instantaneous slope of a function.

Question 2. Which of the following is NOT a property of indefinite integrals?

(A) $\int c f(x) dx = c \int f(x) dx$ for a constant $c$.

(B) $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$.

(C) $\int (f(x) - g(x)) dx = \int f(x) dx - \int g(x) dx$.

(D) $\int f(x)g(x) dx = \int f(x) dx \int g(x) dx$.

Question 3. Which of the following is NOT a standard formula for indefinite integrals?

(A) $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

(B) $\int \sin x dx = \cos x + C$.

(C) $\int e^x dx = e^x + C$.

(D) $\int \frac{1}{x} dx = \ln |x| + C$ for $x \neq 0$.

Question 4. The indefinite integral of a function is NOT which of the following?

(A) The collection of all its antiderivatives.

(B) Represented by the symbol $\int f(x) dx + C$.

(C) A single, unique function.

(D) A family of curves.

Question 5. The constant of integration $C$ in the indefinite integral $\int f(x) dx = F(x) + C$ does NOT represent which of the following?

(A) An arbitrary constant.

(B) The difference between any two antiderivatives of $f(x)$.

(C) A specific fixed value determined by initial conditions.

(D) The vertical shift of the graph of the antiderivative.

Question 6. Which notation is NOT commonly used for the indefinite integral of $f(x)$ with respect to $x$?

(A) $\int f(x) dx$

(B) $\int f(x)$

(C) $\int f(x) \text{ } dx$

(D) $\int f(x)$ (without the differential $dx$)

Question 7. Finding the antiderivative of a function $f(x)$ means finding a function $F(x)$ whose derivative $F'(x)$ is NOT equal to which of the following?

(A) $f(x)$

(B) $\frac{d}{dx} (\int f(x) dx)$

(C) $f'(x)$

(D) The integrand of $\int f(x) dx$

Question 8. Geometrically, the indefinite integral represents a family of curves. Which statement about this family is TRUE?

(A) All curves in the family are identical.

(B) All curves have the same slope at points with the same $x$-coordinate.

(C) The curves intersect each other at various points.

(D) The curves are horizontal translations of each other.

Question 9. Which function's indefinite integral does NOT involve a logarithm?

(A) $f(x) = \frac{1}{x}$

(B) $f(x) = \frac{1}{x+5}$

(C) $f(x) = x^2$

(D) $f(x) = \frac{1}{2x-1}$

Question 10. Which property does NOT hold for indefinite integrals?

(A) $\int_a^b f(x) dx = F(b) - F(a)$ where $F$ is an antiderivative (This is a property of definite integrals).

(B) $\int k f(x) dx = k \int f(x) dx$

(C) $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$

(D) $\frac{d}{dx} \left( \int f(x) dx \right) = f(x)$

Question 1. Which integral is NOT suitable for evaluation using a simple substitution $u=g(x)$ where $g'(x)$ (or a constant multiple) is present in the integrand?

(A) $\int 2x (x^2+1)^4 dx$

(B) $\int \cos x \sin x dx$

(C) $\int e^x \sqrt{e^x+3} dx$

(D) $\int x \cos x dx$

Question 2. Which formula is NOT the correct integration by parts formula?

(A) $\int u dv = uv - \int v du$

(B) $\int uv' dx = uv - \int u'v dx$

(C) $\int f(x) g'(x) dx = f(x)g(x) - \int f'(x)g(x) dx$

(D) $\int uv dx = u \int v dx - \int (u' \int v dx) dx$

Question 3. The method of integration by substitution is NOT based on which differentiation rule?

(A) Chain Rule.

(B) Product Rule.

(C) Power Rule applied after substitution.

(D) Derivative of a composite function.

Question 4. When using integration by parts $\int u dv$, which choice of $u$ and $dv$ is NOT likely to simplify the resulting integral $\int v du$ compared to the original integral?

(A) Choosing $u$ as a function that simplifies when differentiated, and $dv$ as a function that is easily integrable.

(B) Choosing $u = \ln x$ and $dv = x dx$ for $\int x \ln x dx$.

(C) Choosing $u = e^x$ and $dv = \sin x dx$ for $\int e^x \sin x dx$.

(D) Choosing $u = x^2$ and $dv = e^x dx$ for $\int x^2 e^x dx$.

(E) Choosing $u = \sin x$ and $dv = x dx$ for $\int x \sin x dx$ (instead of $u=x, dv=\sin x dx$).

Question 5. Which integral is NOT a standard form that can often be solved using integration by parts?

(A) $\int x^n e^{ax} dx$

(B) $\int x^n \sin(ax) dx$ or $\int x^n \cos(ax) dx$

(C) $\int e^{ax} \sin(bx) dx$ or $\int e^{ax} \cos(bx) dx$

(D) $\int \frac{1}{x^2+a^2} dx$

Question 6. If we use the substitution $u=g(x)$ to evaluate $\int f(g(x)) g'(x) dx$, the differential $dx$ is NOT transformed into which expression?

(A) $\frac{du}{g'(x)}$

(B) $du$ if $g'(x) = 1$

(C) $\frac{du}{g'(x)}$ is the general transformation.

(D) $du \cdot g'(x)$

Question 7. Integration by parts is NOT the standard or most efficient technique for which type of integral?

(A) $\int x \ln x dx$

(B) $\int \tan x dx$

(C) $\int \sin^{-1} x dx$

(D) $\int e^{x} \cos x dx$

Question 8. Which integral CANNOT be evaluated by a simple substitution or using the standard form $\int \frac{f'(x)}{f(x)} dx$?

(A) $\int \frac{2x}{x^2+1} dx$

(B) $\int \frac{\ln x}{x} dx$

(C) $\int \frac{e^x}{e^x+1} dx$

(D) $\int \frac{1}{x^2+1} dx$

Question 9. The acronym LIATE (Logarithmic, Inverse Trig, Algebraic, Trigonometric, Exponential) is NOT used for which purpose in integration?

(A) To suggest the order of preference for choosing the function $u$ in integration by parts.

(B) To help decide which part of the integrand becomes $u$ and which becomes $dv$ in $\int u dv$.

(C) To find the antiderivative of a function directly.

(D) As a heuristic rule, not a strict mathematical requirement.

Question 10. Which statement about the integration by parts formula $\int u dv = uv - \int v du$ is NOT true?

(A) The integral $\int v du$ must be easier to evaluate than $\int u dv$ for the method to be useful.

(B) The choice of $u$ and $dv$ is crucial for simplifying the integration process.

(C) The formula can be derived from the product rule of differentiation.

(D) The formula can be used to integrate any product of two functions.

Question 1. Partial fraction decomposition is NOT applicable to which type of rational function?

(A) Proper rational functions (degree of numerator < degree of denominator).

(B) Improper rational functions (degree of numerator $\geq$ degree of denominator) after polynomial long division.

(C) Functions with an irreducible quadratic factor in the denominator.

(D) Functions that are not rational functions (e.g., $\frac{\sin x}{x}$).

Question 2. For partial fraction decomposition, which type of factor in the denominator does NOT correspond to a constant term in the numerator of the partial fraction?

(A) A non-repeated linear factor $(x-a)$.

(B) A repeated linear factor $(x-a)^n$ for $n>1$.

(C) An irreducible quadratic factor $(ax^2+bx+c)$.

(D) All linear factors (whether repeated or not) correspond to constant numerators.

Question 3. The substitution $t = \tan(x/2)$ is NOT commonly used to integrate which type of function?

(A) Rational functions of $\sin x$ and $\cos x$.

(B) Integrals of the form $\int \frac{dx}{a+b\cos x}$.

(C) Integrals of the form $\int \frac{dx}{a+b\sin x}$.

(D) Functions that are not rational functions of $\sin x$ or $\cos x$, like $\int x \sin x dx$.

Question 4. Which is NOT a standard integral formula involving square roots of quadratic expressions?

(A) $\int \frac{dx}{\sqrt{a^2 - x^2}}$

(B) $\int \frac{dx}{\sqrt{x^2 + a^2}}$

(C) $\int \frac{dx}{\sqrt{x^2 - a^2}}$

(D) $\int \frac{dx}{\sqrt{a^2 + x^2}}$ (same as B)

(E) $\int \sqrt{a^2 - x^2} dx$

(F) $\int \frac{dx}{\sqrt{ax+b}}$

Question 5. Which rational function requires polynomial long division before partial fraction decomposition?

(A) $\frac{x}{x^2+1}$

(B) $\frac{x^3}{x^2+1}$

(C) $\frac{1}{(x-1)(x+2)}$

(D) $\frac{x^2+1}{(x-1)(x+2)}$

(E) Which ONE of the options does NOT require polynomial long division before partial fractions?

Question 6. Which integral CANNOT be evaluated using standard formulas from the section on special forms (like $\int \frac{dx}{\sqrt{a^2 \pm x^2}}$, $\int \frac{dx}{x^2 \pm a^2}$, $\int \frac{dx}{\sqrt{x^2 \pm a^2}}$ etc.) after completing the square or simple manipulation?

(A) $\int \frac{dx}{\sqrt{x^2 - 4x + 5}}$

(B) $\int \frac{dx}{x^2 + 6x + 10}$

(C) $\int \frac{dx}{\sqrt{4 - x^2}}$

(D) $\int \frac{x}{x^2+4} dx$

(E) $\int \frac{dx}{\sqrt{x^3+1}}$

Question 7. For a repeated quadratic factor $(ax^2+bx+c)^n$ in the denominator of a rational function, the partial fraction decomposition terms are of the form $\frac{A_1 x + B_1}{ax^2+bx+c} + \frac{A_2 x + B_2}{(ax^2+bx+c)^2} + \dots + \frac{A_n x + B_n}{(ax^2+bx+c)^n}$. Which statement about the numerators is NOT correct?

(A) Each term has a numerator of the form $Ax+B$.

(B) The numerator coefficients $A_i$ and $B_i$ are constants to be determined.

(C) The degree of the numerator is one less than the degree of the quadratic factor.

(D) The numerators are just constants, $A_i$.

Question 8. The substitution $t=\tan(x/2)$ transforms $dx$ into $\frac{2dt}{1+t^2}$. Which expression is NOT the correct transformation for $\sin x$ or $\cos x$ under this substitution?

(A) $\sin x = \frac{2t}{1+t^2}$

(B) $\cos x = \frac{1-t^2}{1+t^2}$

(C) $\sin x = \frac{1-t^2}{1+t^2}$

(D) Both (A) and (B) are correct transformations.

Question 9. Which integral is NOT a rational function of $\sin x$ and $\cos x$ that can be potentially simplified by the substitution $t = \tan(x/2)$?

(A) $\int \frac{\sin x}{1+\cos x} dx$

(B) $\int \frac{1}{2 + \sin x} dx$

(C) $\int \frac{\cos x}{3 + \sin x} dx$

(D) $\int \sec x dx$

(E) $\int x \sin x dx$

Question 10. Which of the following is NOT a standard integral result or directly derived from one?

(A) $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln|\frac{x-a}{x+a}| + C$

(B) $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln|\frac{a+x}{a-x}| + C$

(C) $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} - \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + C$

(D) $\int \frac{dx}{x \sqrt{x^2 - a^2}} = \frac{1}{a} \sec^{-1}|\frac{x}{a}| + C$

Question 1. The definite integral $\int_{a}^{b} f(x) dx$ is NOT defined as the limit of which type of sum, as the number of subintervals increases indefinitely and the width of the largest subinterval approaches zero?

(A) Riemann sum.

(B) Sum of areas of rectangles under the curve.

(C) $\lim\limits_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x_i$.

(D) An arithmetic series or geometric series.

Question 2. Which is NOT a statement from the Fundamental Theorems of Integral Calculus?

(A) If $F'(x) = f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$.

(B) If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$.

(C) Differentiation and integration are inverse processes.

(D) The value of a definite integral is always positive.

Question 3. The definite integral $\int_{a}^{b} f(x) dx$ does NOT necessarily represent the area under the curve $y=f(x)$ from $x=a$ to $x=b$ if which condition holds?

(A) The function $f(x)$ is continuous on $[a,b]$.

(B) The function $f(x)$ is negative over part or all of the interval $[a,b]$.

(C) The function $f(x)$ is piecewise continuous on $[a,b]$.

(D) The limits of integration $a$ and $b$ are finite.

Question 4. Which property does NOT hold for definite integrals?

(A) $\int_{a}^{b} c f(x) dx = c \int_{a}^{b} f(x) dx$ for a constant $c$.

(B) $\int_{a}^{b} (f(x) + g(x)) dx = \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx$.

(C) $\int_{a}^{b} f(x)g(x) dx = \int_{a}^{b} f(x) dx \int_{a}^{b} g(x) dx$.

(D) $\int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx$.

Question 5. The value of $\int_{a}^{a} f(x) dx$ is NOT equal to which value?

(A) 0.

(B) $F(a) - F(a)$ where $F$ is an antiderivative.

(C) The area under the curve from $a$ to $a$.

(D) A non-zero constant if $f(a) \neq 0$.

Question 6. The Fundamental Theorem of Integral Calculus part 1 states that if $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$. This theorem is NOT used to find which of the following?

(A) The derivative of an integral with a variable upper limit.

(B) The relationship between differentiation and integration.

(C) The value of a definite integral with constant limits of integration.

(D) The integrand of a function defined as an integral with a variable limit.

Question 7. The Fundamental Theorem of Integral Calculus part 2 ($\int_a^b f(x) dx = F(b) - F(a)$) is NOT applicable if which condition is met?

(A) $f(x)$ is continuous on $[a,b]$.

(B) $F(x)$ is an antiderivative of $f(x)$.

(C) $f(x)$ has an infinite discontinuity within $(a,b)$.

(D) The limits $a$ and $b$ are finite.

Question 8. The definite integral $\int_{a}^{b} f(x) dx$ for an integrable function $f(x)$ is NOT sensitive to which factor?

(A) The values of the limits of integration, $a$ and $b$.

(B) The function $f(x)$ itself.

(C) The constant of integration $C$ in the antiderivative.

(D) Whether $f(x)$ is positive or negative on the interval.

Question 9. Which statement about the geometric interpretation of $\int_{a}^{b} f(x) dx$ is NOT correct?

(A) It represents the net signed area between the curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$.

(B) Area above the x-axis is counted as positive, and area below is counted as negative.

(C) If $f(x) \geq 0$ on $[a,b]$, it represents the total area under the curve.

(D) It represents the total area bounded by the curve, regardless of whether $f(x)$ is positive or negative.

Question 10. The process of evaluating $\int_{a}^{b} f(x) dx$ using the Fundamental Theorem does NOT involve which step?

(A) Finding an antiderivative $F(x)$ of $f(x)$.

(B) Evaluating the antiderivative at the upper limit $b$, i.e., $F(b)$.

(C) Evaluating the antiderivative at the lower limit $a$, i.e., $F(a)$.

(D) Adding an arbitrary constant $C$ to the result.

Question 1. Which of the following is NOT a property of definite integrals?

(A) $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(t) dt$ (Dummy variable property).

(B) $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$ (Interval splitting property).

(C) $\int_{a}^{b} f(x) dx \leq \int_{a}^{b} g(x) dx$ if $f(x) \leq g(x)$ on $[a,b]$ (Comparison property).

(D) $\int_{a}^{b} f(x)g(x) dx = (\int_{a}^{b} f(x) dx)(\int_{a}^{b} g(x) dx)$.

Question 2. When evaluating a definite integral $\int_{a}^{b} f(x) dx$ using substitution $u=g(x)$, which change is NOT required?

(A) Replacing $dx$ with $du$ according to the substitution.

(B) Replacing $f(x)$ with an expression in terms of $u$.

(C) Changing the limits of integration from $a, b$ (for $x$) to $g(a), g(b)$ (for $u$).

(D) Adding the constant of integration $C$ at the end.

Question 3. Which integral CANNOT be simplified using properties of definite integrals like symmetry ($\int_{-a}^a$) or $\int_0^a f(x) dx = \int_0^a f(a-x) dx$?

(A) $\int_{-\pi/2}^{\pi/2} \sin x dx$

(B) $\int_{-1}^{1} x^2 dx$

(C) $\int_{0}^{1} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{1-x}} dx$

(D) $\int_{1}^{2} \frac{1}{x} dx$

Question 4. Which statement about the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ is NOT true?

(A) It holds for any integrable function $f(x)$.

(B) It is particularly useful for integrals involving trigonometric functions like $\sin x$ and $\cos x$ over $[0, \pi/2]$.

(C) It allows changing the variable of integration from $x$ to $a-x$ without changing the limits.

(D) It implies that $\int_0^a f(x) dx = 0$ for any function $f(x)$.

Question 5. For an odd function $f(x)$ (i.e., $f(-x) = -f(x)$), the definite integral $\int_{-a}^{a} f(x) dx$ is NOT equal to which value or expression?

(A) 0.

(B) $\int_{-a}^{0} f(x) dx + \int_{0}^{a} f(x) dx$.

(C) $2 \int_{0}^{a} f(x) dx$.

(D) The net signed area from $-a$ to $a$ is zero.

Question 6. The value of $\int_{-a}^{a} f(x) dx$ for an even function $f(x)$ (i.e., $f(-x) = f(x)$) is NOT equal to which expression?

(A) $2 \int_{0}^{a} f(x) dx$.

(B) $\int_{-a}^{0} f(x) dx + \int_{0}^{a} f(x) dx$.

(C) 0.

(D) Twice the area under the curve from $0$ to $a$.

Question 7. Which method is NOT a standard technique for evaluating definite integrals?

(A) Using the Fundamental Theorem of Calculus.

(B) Substitution method (with change of limits).

(C) Integration by parts (applied to the definite integral).

(D) Partial differentiation with respect to the limits.

Question 8. If $\int_{a}^{b} f(x) dx = 0$, which conclusion is NOT necessarily true?

(A) $f(x)$ is identically zero on $[a,b]$.

(B) The net signed area between the curve and the x-axis is zero.

(C) If $f(x) \geq 0$ on $[a,b]$, then $f(x)$ must be zero everywhere on $[a,b]$.

(D) The positive area is equal to the absolute value of the negative area.

Question 9. Which property of definite integrals relates to the sum or difference of functions?

(A) $\int_{a}^{b} (f(x) + g(x)) dx = \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx$.

(B) $\int_{a}^{b} c f(x) dx = c \int_{a}^{b} f(x) dx$.

(C) $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$.

(D) $\int_{a}^{b} f(x)g(x) dx \neq (\int_{a}^{b} f(x) dx)(\int_{a}^{b} g(x) dx)$ in general.

(E) Which ONE of the options does NOT relate to the sum or difference of functions?

Question 10. The property $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$ is NOT valid if which condition holds?

(A) The function $f(x)$ is integrable on the interval containing $a, b, c$.

(B) $c$ is not between $a$ and $b$ (e.g., $c < a < b$ or $a < b < c$).

(C) $a < c < b$.

(D) The property is valid even if $c$ is outside the interval $[a,b]$.

(E) The property is NOT valid if $f(x)$ is discontinuous at $c$ (within the interval of integration).

Question 1. The area bounded by the curve $y=f(x)$, the x-axis, and the lines $x=a$, $x=b$ is NOT given by $\int_{a}^{b} f(x) dx$ if which condition holds?

(A) The function $f(x) \geq 0$ on $[a,b]$.

(B) The function $f(x) \leq 0$ on $[a,b]$.

(C) The function $f(x)$ changes sign on the interval $[a,b]$.

(D) The function $f(x)$ is constant on $[a,b]$.

Question 2. Which formula is NOT correct for finding the area between two curves $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$, assuming $f(x) \geq g(x)$ on $[a,b]$?

(A) $\int_{a}^{b} (f(x) - g(x)) dx$.

(B) $\int_{a}^{b} (\text{upper curve} - \text{lower curve}) dx$.

(C) $\int_{a}^{b} f(x) dx - \int_{a}^{b} g(x) dx$.

(D) $\int_{a}^{b} (f(x) + g(x)) dx$.

Question 3. To find the total area bounded by the curve $y=f(x)$, the x-axis, $x=a$, and $x=b$ when $f(x)$ crosses the x-axis in the interval, we CANNOT simply evaluate $\int_{a}^{b} f(x) dx$. Which statement is TRUE about finding the total area in this case?

(A) We must integrate the absolute value of the function: $\int_{a}^{b} |f(x)| dx$.

(B) We find the points where the curve crosses the x-axis, split the integral, and sum the absolute values of the integrals over the subintervals.

(C) The integral $\int_{a}^{b} f(x) dx$ gives the net signed area, not the total area.

(D) All of the above statements are true.

Question 4. The area bounded by a curve $x=f(y)$, the y-axis, and horizontal lines $y=c$, $y=d$ (where $c

(A) $\int_{c}^{d} f(y) dy$

(B) $\int_{c}^{d} x dy$

(C) $\int_{c}^{d} |f(y)| dy$

(D) $\int_{c}^{d} f(x) dx$

Question 5. The area between two curves $x=f(y)$ and $x=g(y)$ from $y=c$ to $y=d$ is NOT given by which formula, assuming $f(y) \geq g(y)$ on $[c,d]$?

(A) $\int_{c}^{d} (f(y) - g(y)) dy$

(B) $\int_{c}^{d} (\text{right curve} - \text{left curve}) dy$

(C) $\int_{c}^{d} f(y) dy - \int_{c}^{d} g(y) dy$

(D) $\int_{c}^{d} (f(x) - g(x)) dx$

Question 6. Integration with respect to $x$ (using vertical strips) is NOT suitable as the primary method for finding the area of which region?

(A) Area under $y = x^2$ from $x=0$ to $x=1$.

(B) Area between $y=x$ and $y=x^2$.

(C) Area bounded by $x=y^2$ and $x=4$.

(D) Area bounded by $y = \sin x$ from $x=0$ to $x=\pi/2$.

Question 7. Integration with respect to $y$ (using horizontal strips) is NOT suitable as the primary method for finding the area of which region?

(A) Area bounded by $x = y^2$ and the y-axis from $y=0$ to $y=2$.

(B) Area bounded by $x = y$ and $x = y^2$.

(C) Area bounded by $y = x^3$, the x-axis, and $x=2$.

(D) Area bounded by $x = e^y$, the y-axis, $y=0$, and $y=1$.

Question 8. Which statement about finding the area between curves is NOT correct?

(A) We need to find the points of intersection of the curves to determine the limits of integration.

(B) We need to determine which function is the upper/right boundary and which is the lower/left boundary in the region.

(C) The formula $\int_{a}^{b} (\text{upper} - \text{lower}) dx$ or $\int_{c}^{d} (\text{right} - \text{left}) dy$ always gives the total area.

(D) If the upper/lower or right/left function changes within the interval, we need to split the integral.

Question 9. If $f(x)$ is always below $g(x)$ on the interval $[a,b]$, the area between the curves $y=f(x)$ and $y=g(x)$ is NOT given by which integral?

(A) $\int_{a}^{b} (g(x) - f(x)) dx$.

(B) $\int_{a}^{b} |f(x) - g(x)| dx$.

(C) $\int_{a}^{b} f(x) dx - \int_{a}^{b} g(x) dx$.

(D) $\int_{a}^{b} (g(x) - f(x)) dx$ (This is the correct formula, so the question asks which is NOT equal to it).

Question 10. The area calculation using definite integrals is NOT directly applicable to which type of region?

(A) A region bounded by a curve and an axis over a finite interval.

(B) A region bounded between two curves over a finite interval.

(C) A region bounded by a curve and an axis over an infinite interval (an improper integral).

(D) An unbounded region in the plane (e.g., the area under $y=1/x^2$ from $x=1$ to $\infty$).

Question 1. Which of the following is NOT a necessary component of a differential equation?

(A) An independent variable.

(B) A dependent variable.

(C) Derivatives of the dependent variable with respect to the independent variable.

(D) An arbitrary constant $C$ appearing explicitly in the equation.

Question 2. Which statement about the order of a differential equation is NOT true?

(A) The order is the highest order of the derivative present in the equation.

(B) The equation $y''' + (y'')^2 + y = x$ has order 3.

(C) The equation $\frac{dy}{dx} = x+y$ has order 1.

(D) The order is determined by the highest power of the highest order derivative.

Question 3. The degree of a differential equation is NOT defined if which condition holds?

(A) The equation can be written as a polynomial in terms of the derivatives.

(B) The equation contains radicals involving derivatives that cannot be removed by raising both sides to an integer power.

(C) The equation is linear in the dependent variable and its derivatives.

(D) The highest order derivative has an integer power.

Question 4. A particular solution of a differential equation does NOT contain which elements?

(A) The independent variable.

(B) The dependent variable.

(C) Arbitrary constants.

(D) Specific values for the arbitrary constants derived from initial/boundary conditions.

Question 5. Which statement about the general solution of a differential equation is NOT correct?

(A) It contains arbitrary constants.

(B) The number of arbitrary constants is usually equal to the order of the equation.

(C) It represents a family of solutions.

(D) It is a unique function that satisfies the differential equation.

Question 6. Which type of solution is NOT always obtainable from the general solution by specializing the arbitrary constants?

(A) A particular solution derived from initial conditions.

(B) The trivial solution $y=0$ (if it satisfies the equation).

(C) A singular solution.

(D) Any specific solution belonging to the family represented by the general solution.

Question 7. Formulating a differential equation representing a family of curves involves eliminating the arbitrary constants by differentiation. This process does NOT involve which step?

(A) Differentiating the equation of the family of curves with respect to the independent variable.

(B) Obtaining a system of equations including the original equation and its derivatives.

(C) Using integration to find the relationship between variables and derivatives.

(D) Eliminating the arbitrary constants from the system of equations.

Question 8. The number of arbitrary constants in the general solution of a differential equation of order $n$ is NOT necessarily equal to which value?

(A) The order $n$ of the equation.

(B) The number of independent variables.

(C) The number of integrations performed to obtain the general solution.

(D) The number of parameters in the family of curves it represents.

Question 9. Which of the following equations is NOT a differential equation?

(A) $\frac{dy}{dx} = 5x + 2$

(B) $\frac{d^2 y}{dx^2} + y = 0$

(C) $y = x^2 + 3x - 5$

(D) $(\frac{dy}{dx})^2 + y^2 = 1$

Question 10. Which statement about the definition of a differential equation from an applied mathematics perspective is NOT correct?

(A) They describe relationships involving rates of change.

(B) They arise naturally in models of dynamic systems.

(C) They always involve derivatives with respect to time.

(D) They connect a function with its derivatives.

Question 1. Which is NOT a standard method for solving first-order differential equations?

(A) Variable separable method.

(B) Homogeneous method.

(C) Integrating Factor method (for linear equations).

(D) Partial fraction decomposition method.

Question 2. The variable separable method is NOT directly applicable to which type of first-order differential equation?

(A) $\frac{dy}{dx} = f(x) g(y)$

(B) $\frac{dy}{dx} = x+y$

(C) $\frac{dy}{dx} = \frac{x^2}{y^2}$

(D) $\frac{dy}{dx} = e^{x+y}$

Question 3. A homogeneous differential equation $\frac{dy}{dx} = f(x,y)$ does NOT satisfy which property?

(A) $f(tx, ty) = f(x,y)$ for any non-zero scalar $t$.

(B) It can be written in the form $\frac{dy}{dx} = F(\frac{y}{x})$.

(C) The terms in $f(x,y)$ involving $x$ and $y$ have the same degree.

(D) It is always a linear differential equation.

Question 4. The substitution $y=vx$ is NOT suitable for solving which type of first-order differential equation?

(A) Homogeneous differential equations.

(B) Equations reducible to homogeneous form.

(C) Linear differential equations of the form $\frac{dy}{dx} + P(x)y = Q(x)$ where $Q(x) \neq 0$.

(D) Variable separable equations (though it might work, it's not the standard method).

Question 5. Which differential equation CANNOT be solved by the variable separable method?

(A) $\frac{dy}{dx} = \frac{x}{y}$

(B) $\frac{dy}{dx} = xy$

(C) $\frac{dy}{dx} = x \sin y$

(D) $\frac{dy}{dx} = x + y$

Question 6. Which differential equation is NOT a homogeneous equation?

(A) $\frac{dy}{dx} = \frac{x+y}{x}$

(B) $\frac{dy}{dx} = \frac{x^2+y^2}{xy}$

(C) $(x^2+y^2)dx - 2xy dy = 0$

(D) $\frac{dy}{dx} = \frac{x+y+1}{x+y-1}$

Question 7. Equations of the form $\frac{dy}{dx} = \frac{ax+by+c}{Ax+By+C}$ where $aB-Ab=0$ (parallel lines) are NOT solved by which substitution?

(A) $v = ax+by$ (or $v=Ax+By$).

(B) $\frac{dv}{dx} = a + b \frac{dy}{dx}$.

(C) Reducing it to a variable separable form in $v$ and $x$.

(D) $x = X+h, y = Y+k$ and choosing $h, k$ to make the linear terms zero.

Question 8. Which step is NOT typically involved in solving a homogeneous differential equation using the substitution $y=vx$?

(A) Substitute $y=vx$ and $\frac{dy}{dx} = v + x \frac{dv}{dx}$ into the original equation.

(B) Simplify the equation to a form where variables $v$ and $x$ are separable.

(C) Integrate the separated equation with respect to $x$ and $v$.

(D) Substitute $v = y/x$ back into the solution to get the general solution in terms of $x$ and $y$.

(E) Add arbitrary constants to both sides during integration.

Question 9. The general approach to solving a first-order differential equation does NOT involve which step?

(A) Identifying the type of the differential equation (separable, homogeneous, linear, etc.).

(B) Applying the appropriate solution method for that type.

(C) Obtaining a general solution containing arbitrary constants.

(D) Finding higher order derivatives of the solution.

Question 10. Which solution method is NOT applicable to the differential equation $\frac{dy}{dx} = \sin(x+y)$?

(A) Using the substitution $v = x+y$ to reduce it to a variable separable form.

(B) Treating it as a linear differential equation.

(C) Treating it as a homogeneous differential equation.

(D) Both (B) and (C).

Question 1. Which of the following is NOT the standard form of a first-order linear differential equation in $y$?

(A) $\frac{dy}{dx} + P(x)y = Q(x)$

(B) $y' + P(x)y = Q(x)$

(C) $\frac{dy}{dx} = Q(x) - P(x)y$

(D) $\frac{dy}{dx} + P(x)y + Q(x) = 0$

(E) $y' = P(x)y + Q(x)$

Question 2. The integrating factor (IF) for the linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is NOT given by which formula?

(A) $e^{\int P(x) dx}$

(B) $e^{\int Q(x) dx}$

(C) A function that makes the LHS the derivative of a product.

(D) $e^{\int P(x) dx}$ (This is the correct formula, so the question asks which is NOT equal to it).

Question 3. Which differential equation is NOT a first-order linear differential equation?

(A) $\frac{dy}{dx} + xy = x^2$

(B) $y' + y = \sin x$

(C) $\frac{dy}{dx} + y^2 = x$

(D) $(1+x^2)\frac{dy}{dx} + 2xy = 4x$

Question 4. The method of integrating factor is NOT used to solve which type of differential equation?

(A) First-order linear differential equations.

(B) Differential equations of the form $\frac{dy}{dx} + P(x)y = Q(x)$.

(C) Bernoulli's equation, which is reducible to a linear form.

(D) Second-order linear differential equations directly.

Question 5. The general solution of a linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is NOT given by which formula?

(A) $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) dx + C$, where IF is the integrating factor.

(B) $y = \frac{1}{\text{IF}} \left( \int Q(x) \cdot (\text{IF}) dx + C \right)$.

(C) $\frac{d}{dx}(y \cdot \text{IF}) = Q(x) \cdot \text{IF}$.

(D) $y = \int Q(x) dx / \int P(x) dx + C$.

Question 6. Which step is NOT part of the standard method for solving a first-order linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$?

(A) Calculate the integrating factor IF $= e^{\int P(x) dx}$.

(B) Multiply the entire equation by the integrating factor.

(C) Recognize the LHS as the derivative of the product $y \cdot \text{IF}$.

(D) Differentiate both sides with respect to $x$ to find $y$.

Question 7. The integrating factor for a linear differential equation of the form $\frac{dx}{dy} + P(y)x = Q(y)$ is NOT given by which formula?

(A) $e^{\int P(y) dy}$

(B) A function that makes the LHS the derivative of a product of $x$ and the IF.

(C) $e^{\int P(x) dx}$

(D) $e^{\int P(y) dy}$ (This is the correct formula, so the question asks which is NOT equal to it).

Question 8. In the standard form of a linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$, the functions $P(x)$ and $Q(x)$ MUST NOT contain which variable?

(A) The independent variable $x$.

(B) The dependent variable $y$.

(C) Arbitrary constants (although they may be included in the expression for $P(x)$ or $Q(x)$).

(D) Transcendental functions.

Question 9. Which function CANNOT be the integrating factor for a first-order linear differential equation in $y$?

(A) $e^{x^2}$

(B) $\sin x$

(C) $x^3$

(D) $y$

Question 10. The general solution of a first-order linear differential equation contains how many arbitrary constants?

(A) 0

(B) 1

(C) 2

(D) The number of arbitrary constants is NOT necessarily 0.

Question 1. Differential equations are NOT suitable for modeling phenomena involving which characteristic?

(A) Rates of change that depend on the current state.

(B) Processes that evolve continuously over time.

(C) Instantaneous events without continuous change.

(D) Relationships between a quantity and how fast it is changing.

Question 2. Which field does NOT typically use differential equations for modeling?

(A) Physics (e.g., motion, heat transfer).

(B) Biology (e.g., population growth, disease spread).

(C) Economics (e.g., market dynamics, investment growth).

(D) Pure algebra (solving algebraic equations).

Question 3. Formulating a differential equation from a real-world problem involves translating the description of the rate of change into mathematical terms. This does NOT require translating which aspect into mathematical terms?

(A) "The rate of change of quantity $Y$ with respect to $X$".

(B) "Is proportional to".

(C) "The current value of quantity $Z$".

(D) "The total amount of quantity $Y$ at a specific fixed instant".

Question 4. The solution to a differential equation modeling a physical system (e.g., the position of a particle) does NOT necessarily provide which of the following without additional information?

(A) A general solution describing a family of possible states.

(B) A unique particular solution for the specific scenario.

(C) The relationship between the variables.

(D) How the system evolves over time.

Question 5. Which real-world process is NOT typically modeled by a simple first-order ordinary differential equation?

(A) Exponential population growth.

(B) Radioactive decay.

(C) Newton's Law of Cooling.

(D) Wave motion in a string (often modeled by partial differential equations).

Question 6. Applied problems modeled by differential equations CANNOT be solved by finding which type of solution?

(A) General solution.

(B) Particular solution.

(C) Singular solution (in some cases).

(D) Only singular solutions, excluding general or particular solutions.

Question 7. The concept of "rate of change" in a word problem does NOT correspond to which mathematical representation?

(A) The first derivative of a quantity with respect to an independent variable.

(B) The instantaneous rate of change.

(C) The slope of the tangent line.

(D) The value of the function itself at a given point.

Question 8. Which factor is NOT typically used to determine the specific solution (particular solution) in an applied problem modeled by a differential equation?

(A) Initial conditions (values of variables at a starting point).

(B) Boundary conditions (values of variables at different points).

(C) The order of the differential equation.

(D) Specific constraints given in the problem statement.

Question 9. Differential equations are NOT applied to model which phenomenon?

(A) The motion of objects under forces.

(B) The flow of heat in a material.

(C) The spread of information in a social network over time.

(D) The calculation of the exact derivative of a function at a point.

Question 10. Solving applied problems using differential equations does NOT necessarily involve which mathematical technique *after* obtaining the general solution?

(A) Using initial or boundary conditions.

(B) Solving algebraic equations to find the constant(s).

(C) Interpreting the particular solution in the context of the original problem.

(D) Finding the degree of the solution function.