Applications of Differentiation: Error Propagation, Maximum and Minimum Values, and Critical Points

Applications of Differentiation

Error Propagation Using Differentials

When measurements are used to calculate other quantities, small errors in the measurements can propagate and affect the final result. Differentials provide a way to estimate these errors.

• Measured Quantity: Let x be the exact value and a the measured value. The measurement error is dx, so the measured value is a + dx.

• Error Estimation: If f(x) is a function of the measured quantity, the error in f(x) can be estimated by: $df = f'(a) dx$

• Example: If $f(x) = x^3 + x^2$, $a = 2$, $dx = 0.01$: $f'(x) = 3x^2 + 2x$ $f'(2) = 3(2)^2 + 2(2) = 12 + 4 = 16$ $df = 16 imes 0.01 = 0.16$

• Application Example: Suppose the side length of a cube is measured to be 6 cm with an error of 0.2 cm. Estimate the error in the computed volume: $V = x^3$ $dV = 3x^2 dx$ $dV = 3(6)^2(0.2) = 3 imes 36 imes 0.2 = 21.6$ So, the estimated error in volume is 21.6 cm3.

Maximum and Minimum Values

Absolute and Local Extreme Values

Understanding the maximum and minimum values of a function is essential in calculus, especially for optimization problems.

• Absolute Maximum: The value $f(c)$ is an absolute maximum of $f$ on domain $D$ if $f(c) \\geq f(x)$ for all $x$ in $D$.

• Absolute Minimum: The value $f(c)$ is an absolute minimum of $f$ on domain $D$ if $f(c) \\leq f(x)$ for all $x$ in $D$.

• Local Maximum: $f(c)$ is a local maximum if $f(c) \\geq f(x)$ when $x$ is near $c$.

• Local Minimum: $f(c)$ is a local minimum if $f(c) \\leq f(x)$ when $x$ is near $c$.

• Example: For $y = x^3$, there are no absolute maxima or minima on $(-\infty, \infty)$; for $y = x^2$, there is an absolute minimum at $x = 0$.

The Extreme Value Theorem

If $f$ is continuous on a closed interval $[a, b]$, then $f$ attains an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in $[a, b]$.

• Application: This theorem guarantees the existence of extreme values for continuous functions on closed intervals.

Critical Points and Fermat's Theorem

Critical Points

Critical points are where the derivative of a function is zero or undefined. These points are candidates for local maxima or minima.

• Definition: Any value of $x$ in the domain of $f$ for which $f'(x)$ is zero or undefined is called a critical point for $f$.

• Finding Critical Points:

  1. Solve $f'(x) = 0$ for $x$.

  2. Check where $f'(x)$ is undefined (and $x$ is in the domain of $f$).

• Example: Find all critical points of $y = x^2 - x$: $y' = 2x - 1$ Set $y' = 0$: $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$ So, $x = \frac{1}{2}$ is a critical point.

• Example: Find the critical numbers of $g(y) = \frac{y^2 - y + 1}{y^2 - y + 1}$: $g'(y) = \frac{(1)(y^2 - y + 1) - (2y - 1)(y^2 - y + 1)}{(y^2 - y + 1)^2}$ Simplify numerator: $-y^2 + 2y$ Set $-y^2 + 2y = 0 \Rightarrow y(y - 2) = 0 \Rightarrow y = 0, y = 2$ So, $y = 0$ and $y = 2$ are critical numbers.

Fermat's Theorem

If a function $f(x)$ has a local minimum or maximum at $x = c$, and if $f$ is defined, then $f'(c) = 0$.

• Application: This theorem helps identify where local extrema can occur.

Summary Table: Types of Extrema

Type	Definition	How to Find
Absolute Maximum	$f(c) \geq f(x)$ for all $x$ in $D$	Check endpoints and critical points
Absolute Minimum	$f(c) \leq f(x)$ for all $x$ in $D$	Check endpoints and critical points
Local Maximum	$f(c) \geq f(x)$ near $c$	Check critical points
Local Minimum	$f(c) \leq f(x)$ near $c$	Check critical points

Key Steps for Finding Extrema

• Find the derivative $f'(x)$.

• Solve $f'(x) = 0$ for critical points.

• Check where $f'(x)$ is undefined within the domain.

• Evaluate $f(x)$ at critical points and endpoints (if on a closed interval).

• Compare values to determine absolute and local maxima/minima.

Additional info: These notes cover key concepts from Chapter 5 - Applications of Differentiation, including error estimation, extreme values, and critical points, with examples and step-by-step procedures for finding extrema.