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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.1.89a
Chapter 4, Problem 4.1.89a

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.


a. Evaluate g(2), h(2), g'(2), and h'(2).

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First, evaluate g(2) by substituting x = 2 into the function g(x) = xf(x) + 1. Since f(2) = 0, g(2) = 2 * f(2) + 1 = 2 * 0 + 1 = 1.
Next, evaluate h(2) by substituting x = 2 into the function h(x) = xf(x) + x + 1. Again, since f(2) = 0, h(2) = 2 * f(2) + 2 + 1 = 0 + 2 + 1 = 3.
To find g'(x), use the product rule for differentiation. The function g(x) = xf(x) + 1 can be differentiated as g'(x) = f(x) + x * f'(x). Substitute x = 2 to find g'(2).
For h'(x), differentiate h(x) = xf(x) + x + 1. Using the product rule and the sum rule, h'(x) = f(x) + x * f'(x) + 1. Substitute x = 2 to find h'(2).
Since f has a local extreme value at x = 2, f'(2) = 0. Use this information to simplify g'(2) and h'(2) further by substituting f'(2) = 0 into the expressions obtained in the previous steps.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Differentiability and Local Extrema

A function is differentiable at a point if it has a defined derivative there, which implies it is smooth and has no sharp corners. A local extreme value occurs at a point where the function reaches a maximum or minimum compared to nearby points. For a differentiable function, local extrema can be identified using the first derivative test, where the derivative changes sign around the point.
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Finding Extrema Graphically

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For example, to evaluate g(2) or h(2), we substitute x = 2 into the respective function definitions. This process is essential for finding specific values of functions at given points, which is a fundamental skill in calculus.
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Derivative of a Product

The derivative of a product of two functions can be found using the product rule, which states that if u(x) and v(x) are functions, then the derivative of their product u(x)v(x) is u'(x)v(x) + u(x)v'(x). This rule is crucial when differentiating functions like g(x) and h(x), which are defined as products of x and f(x), allowing us to compute g'(2) and h'(2) effectively.
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Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The function f(x) = √x has a local maximum on the interval [0,∞).

Textbook Question

{Use of Tech} Basketball shot A basketball is shot with an initial velocity of v ft/s at an angle of 45° to the floor. The center of the basketball is 8 ft above the floor at a horizontal distance of 18 feet from the center of the basketball hoop when it is released. The height h (in feet) of the center of the basketball after it has traveled a horizontal distance of x feet is modeled by the function h(x) = 32x² / v² + x + 8 (see figure). <IMAGE>



b. During the flight of the basketball, show that the distance s from the center of the basketball to the front of the hoop is s = √ (x - 17.25)² + ( -(4x² / 81) + x - 2)² (Hint: The diameter of the basketball hoop is 18 inches.) 

Textbook Question

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.


b. Does either g or h have a local extreme value at x = 2? Explain.

Textbook Question

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b. Find the time and the displacement when the object reaches its lowest point.

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Textbook Question

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.


a. Plot a graph of the curve when a = 3.