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:

• Example: If , , :

• 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: 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 is an absolute maximum of on domain if for all in .

• Absolute Minimum: The value is an absolute minimum of on domain if for all in .

• Local Maximum: is a local maximum if when is near .

• Local Minimum: is a local minimum if when is near .

• Example: For , there are no absolute maxima or minima on ; for , there is an absolute minimum at .

The Extreme Value Theorem

If is continuous on a closed interval , then attains an absolute maximum value and an absolute minimum value at some numbers and in .

• 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 in the domain of for which is zero or undefined is called a critical point for .

• Finding Critical Points:

  1. Solve for .

  2. Check where is undefined (and is in the domain of ).

• Example: Find all critical points of : Set : So, is a critical point.

• Example: Find the critical numbers of : Simplify numerator: Set So, and are critical numbers.

Fermat's Theorem

If a function has a local minimum or maximum at , and if is defined, then .

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

Summary Table: Types of Extrema

Key Steps for Finding Extrema

• Find the derivative .

• Solve for critical points.

• Check where is undefined within the domain.

• Evaluate 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.