Applications of Derivatives: Tangents, Normals, Approximations, Errors

Tangents and Normals to a Curve: Equations

The derivative of a function at a point provides the instantaneous rate of change and the slope of the curve at that specific point. This geometric interpretation is directly used to find the equations of the tangent line and the normal line to the curve at a given point.

Tangent Line

The tangent line is a straight line that passes through a point on a curve and has the same instantaneous slope as the curve at that point. It represents the best linear approximation of the curve near that point.

• Definition: The tangent line to the graph of a function $y=f(x)$ at a point $P(x_1, y_1)$ is the line that passes through $P$ with slope equal to the derivative of $f$ evaluated at $x_1$. For the tangent to exist, the function must be differentiable at $x_1$.
• Slope of the Tangent: If the function $f(x)$ is differentiable at $x_1$, the slope of the tangent line to the curve $y=f(x)$ at the point $P(x_1, y_1)$ is given by the value of the derivative $f'(x)$ evaluated at $x_1$.

  $m_{\text{tangent}} = f'(x_1) = \frac{dy}{dx}\bigg|_{x=x_1}$

  Here, $y_1 = f(x_1)$.

• Equation of the Tangent Line: Once we have the point $P(x_1, y_1)$ on the line and its slope $m_{\text{tangent}} = f'(x_1)$, we can use the point-slope form of the equation of a straight line, $y - y_1 = m(x - x_1)$. Substituting the slope of the tangent, the equation is:

  $y - y_1 = f'(x_1) (x - x_1)$

  This equation is valid provided $f'(x_1)$ is a finite real number.

Normal Line

The normal line to a curve at a point is the line perpendicular to the tangent line at that same point.

• Definition: The normal line to the graph of a function $y=f(x)$ at a point $P(x_1, y_1)$ is the straight line that passes through $P$ and is perpendicular to the tangent line at $P$.
• Slope of the Normal: The slopes of two perpendicular lines are negative reciprocals of each other (unless one is horizontal and the other is vertical). If the slope of the tangent line at $P(x_1, y_1)$ is $m_{\text{tangent}} = f'(x_1)$, then the slope of the normal line, $m_{\text{normal}}$, is:

  If $f'(x_1) \neq 0$, then $m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}} = -\frac{1}{f'(x_1)}$.

• Equation of the Normal Line: Using the point-slope form $y - y_1 = m(x - x_1)$ with the point $P(x_1, y_1)$ and the slope $m_{\text{normal}} = -\frac{1}{f'(x_1)}$, the equation of the normal line is:

  $y - y_1 = -\frac{1}{f'(x_1)} (x - x_1)$ (provided $f'(x_1) \neq 0$)

Special Cases for Tangents and Normals

The formulas for the normal line need careful consideration when the tangent line is horizontal or vertical.

• Horizontal Tangent: If the derivative at $x_1$ is zero, $f'(x_1) = 0$, the tangent line is horizontal.
  • The slope of the tangent is $m_{\text{tangent}} = 0$. The equation of the tangent line is $y - y_1 = 0(x - x_1)$, which simplifies to $y - y_1 = 0$ or $y = y_1$.
  • The normal line is perpendicular to a horizontal line, so it must be a vertical line. The slope of a vertical line is undefined.
  • The equation of the normal line passing through $(x_1, y_1)$ is $x = x_1$.
• Vertical Tangent: If the derivative at $x_1$ is undefined (e.g., $\lim\limits_{x \to x_1} \frac{f(x) - f(x_1)}{x - x_1} = \pm \infty$), the tangent line is vertical. This occurs at points like cusps or where there's a vertical slope.
  • The slope of the tangent is undefined. The equation of the tangent line passing through $(x_1, y_1)$ is $x = x_1$.
  • The normal line is perpendicular to a vertical line, so it must be a horizontal line. The slope of a horizontal line is 0.
  • The equation of the normal line passing through $(x_1, y_1)$ is $y = y_1$.

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Approximations using Differentials

Derivatives can be used to estimate how a function's value changes in response to a small change in its input. This application is based on the idea that, for a small interval, the function's graph is closely approximated by its tangent line. The change in the function's value ($\Delta y$) is approximated by the change in the y-value along the tangent line, which is given by the differential $dy$.

Differentials

Let $y = f(x)$ be a differentiable function. Consider a point $(x, f(x))$ on the graph.

• The differential of $x$, denoted by $dx$, is defined as an independent variable. It represents a small, arbitrary change in $x$. Often, we set $dx = \Delta x$, where $\Delta x$ is a small change in $x$.
• The differential of $y$, denoted by $dy$, is a dependent variable defined in terms of $x$ and $dx$. It is defined as the product of the derivative of $f$ at $x$ and the differential of $x$:

  $dy = f'(x) dx$

Geometrically, if we move along the tangent line at $(x, f(x))$ by an amount $dx$ horizontally, the corresponding vertical change along the tangent line is $dy$. Recall that the slope of the tangent is $f'(x)$, so $f'(x) = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{dy}{dx}$. This definition of $dy$ formalizes this relationship.

Now consider the actual change in the function value when $x$ changes by a small amount $\Delta x = dx$. The actual change in $y$ is $\Delta y = f(x + \Delta x) - f(x)$.

From the limit definition of the derivative, $f'(x) = \lim\limits_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$. For small $\Delta x$, the difference quotient is a good approximation of the derivative:

$\frac{f(x + \Delta x) - f(x)}{\Delta x} \approx f'(x)$

Multiplying by $\Delta x$:

$f(x + \Delta x) - f(x) \approx f'(x) \Delta x$

Since $\Delta y = f(x + \Delta x) - f(x)$ and $dy = f'(x) dx$, setting $dx = \Delta x$, we get the approximation:

$\Delta y \approx dy = f'(x) \Delta x$ (when $\Delta x$ is small)

This means that for a small change in $x$, the change in $y$ along the tangent line ($dy$) is approximately equal to the actual change in $y$ along the curve ($\Delta y$).

Approximation Formula using Differentials

The approximation $\Delta y \approx dy$ can be used to approximate the value of a function at a point near a known point. Since $\Delta y = f(x + \Delta x) - f(x)$, we have:

$f(x + \Delta x) - f(x) \approx dy$

$f(x + \Delta x) - f(x) \approx f'(x) \Delta x$

Adding $f(x)$ to both sides, we get the approximation formula:

$f(x + \Delta x) \approx f(x) + f'(x) \Delta x$

This formula is used to approximate the value of $f$ at a point $(x+\Delta x)$ that is close to a point $x$ where both $f(x)$ and $f'(x)$ are easy to calculate. The term $f(x) + f'(x) \Delta x$ is the value of the $y$-coordinate on the tangent line at $x + \Delta x$. Hence, this is also known as the linear approximation or tangent line approximation of $f(x + \Delta x)$.

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Errors and Differentials (Absolute, Relative, Percentage Error)

Differentials are not only useful for approximating function values but also for estimating the error that propagates through a calculation when there is a small error in the input measurement. If we measure a quantity $x$ with some error, and then calculate another quantity $y = f(x)$ based on that measurement, the error in $x$ will cause an error in $y$. Derivatives and differentials help us estimate this resulting error in $y$.

Definitions of Errors

Let $x$ be the measured value of a variable, and let $\Delta x$ be the error in the measurement of $x$. $\Delta x$ is the difference between the measured value and the true value. $|\Delta x|$ is the magnitude of the error.

Let $y = f(x)$ be the quantity calculated based on the measured value $x$. The actual change in $y$ due to the error $\Delta x$ is $\Delta y = f(x + \Delta x) - f(x)$. We want to estimate this error $\Delta y$.

• Absolute Error in $x$: The magnitude of the error in the measurement of $x$, denoted by $|\Delta x|$. This is the direct error amount. Often, $dx$ is used interchangeably with $\Delta x$ when referring to a small change or error in the independent variable.
• Propagated Absolute Error in $y$: The resulting error in $y$ due to the error in $x$. We approximate the actual error $\Delta y$ using the differential $dy$:

  Propagated Absolute Error in $y \approx |dy| = |f'(x) dx|$ or $|f'(x) \Delta x|$

  This estimates the magnitude of the error in the calculated value of $y$.

• Relative Error in $x$: The ratio of the absolute error in $x$ to the measured value of $x$ (assuming $x \neq 0$). It's often expressed as a fraction or decimal.

  Relative Error in $x = \frac{\Delta x}{x}$

  The magnitude of the relative error is $|\frac{\Delta x}{x}|$.

• Propagated Relative Error in $y$: The ratio of the propagated absolute error in $y$ to the calculated value of $y = f(x)$ (assuming $y \neq 0$).

  Propagated Relative Error in $y \approx \frac{dy}{y} = \frac{f'(x) dx}{f(x)}$ or $\frac{f'(x) \Delta x}{f(x)}$

  The magnitude is $|\frac{dy}{y}|$.

• Percentage Error in $x$: The relative error in $x$ expressed as a percentage.

  Percentage Error in $x = \left(\frac{\Delta x}{x}\right) \times 100\%$

  Magnitude: $|\frac{\Delta x}{x}| \times 100\%$.

• Propagated Percentage Error in $y$: The propagated relative error in $y$ expressed as a percentage.

  Propagated Percentage Error in $y \approx \left(\frac{dy}{y}\right) \times 100\%$ or $\left(\frac{f'(x) \Delta x}{f(x)}\right) \times 100\%$

  Magnitude: $|\frac{dy}{y}| \times 100\%$.

These formulas allow us to use the derivative to estimate the magnitude of the error in a computed quantity $y$ based on a known error in the measured quantity $x$.

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