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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.10.63c
Chapter 3, Problem 3.10.63c

62–65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x)=(x²−1)sin^−1 x on [−1,1]

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Step 1: Understand the problem. We need to verify that the zeros of the derivative of the function f(x) = (x^2 - 1)sin^{-1}(x) correspond to points where the original function f(x) has a horizontal tangent line.
Step 2: Find the derivative f'(x). Use the product rule for differentiation, since f(x) is a product of two functions: u(x) = x^2 - 1 and v(x) = sin^{-1}(x). The product rule states that (uv)' = u'v + uv'.
Step 3: Differentiate u(x) and v(x) separately. For u(x) = x^2 - 1, the derivative u'(x) = 2x. For v(x) = sin^{-1}(x), the derivative v'(x) = 1/√(1-x^2).
Step 4: Apply the product rule. Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula to find f'(x) = (x^2 - 1)(1/√(1-x^2)) + (2x)(sin^{-1}(x)).
Step 5: Set f'(x) = 0 to find the zeros of the derivative. Solve the equation (x^2 - 1)(1/√(1-x^2)) + (2x)(sin^{-1}(x)) = 0 to find the x-values where f'(x) is zero. These x-values correspond to points where f(x) has a horizontal tangent line.

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Key Concepts

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

Derivative and Critical Points

The derivative of a function, denoted as f', represents the rate of change of the function f. Critical points occur where the derivative is zero or undefined, indicating potential locations for horizontal tangent lines. In this context, finding the zeros of f' helps identify where the slope of f is zero, which corresponds to horizontal tangents.
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Critical Points

Horizontal Tangent Lines

A horizontal tangent line occurs at points on the graph of a function where the slope is zero. This means that the derivative of the function at those points is equal to zero. Understanding this concept is crucial for verifying the relationship between the zeros of the derivative and the behavior of the original function.
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Slopes of Tangent Lines

Graphing Functions

Graphing a function involves plotting its values on a coordinate plane to visualize its behavior. By graphing both f(x) and its derivative f'(x), one can observe how the zeros of f' correspond to points where f has horizontal tangents. This visual representation aids in understanding the relationship between a function and its derivative.
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Graph of Sine and Cosine Function
Related Practice
Textbook Question

Tangents and inverses Suppose L(x)=ax+b (with a≠0) is the equation of the line tangent to the graph of a one-to-one function f at (x0,y0). Also, suppose M(x)=cx+d is the equation of the line tangent to the graph of f^−1 at (y0,x0).


c. Prove that L^−1(x)=M(x).

Textbook Question

Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>

d. Determine the velocity of the airliner at noon (t = 6) and explain why the velocity is negative.

Textbook Question

Deriving trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Textbook Question

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

c. What is the height of the stone at the highest point?

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Textbook Question

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y=0y=0 when the mass hangs at rest. Suppose you push the mass to a position y0y_0 units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is y=y0cos(tkm)y=y_0\(\cos\)\(\left\)(t\(\sqrt{\frac{k}{m}\)}\(\right\)), where k>0k>0 is a constant measuring the stiffness of the spring (the larger the value of kk, the stiffer the spring) and yy is positive in the upward direction.

Use equation (4) to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness (kk is increased by a factor of 44)?

Textbook Question

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>


c. d/dx ((f(x)g(x)) |x=3

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