Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.4.82

Chapter 4, Problem 4.4.82

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
82. y' = sin t, for 0 ≤ t ≤ 2π

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Step 1: To find the second derivative y'', differentiate y' = sin(t) with respect to t. Use the derivative rule for sine: d/dt[sin(t)] = cos(t). This gives y'' = cos(t).

Step 2: Analyze the critical points of y' = sin(t) by finding where y' = 0. Solve sin(t) = 0 for t in the interval [0, 2π]. The solutions are t = 0, π, and 2π.

Step 3: Determine the intervals where y' is positive or negative. Since sin(t) is positive on (0, π) and negative on (π, 2π), this indicates where the function f(x) is increasing or decreasing.

Step 4: Analyze the concavity of f(x) using y'' = cos(t). Find where y'' = 0 by solving cos(t) = 0. The solutions are t = π/2 and 3π/2. Use test intervals to determine where y'' > 0 (concave up) and y'' < 0 (concave down).

Step 5: Combine the information from Steps 2–4 to sketch the general shape of the graph of f(x). Mark the critical points, intervals of increase/decrease, and concavity changes to create a rough sketch of the function.

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Key Concepts

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

First Derivative

The first derivative of a function, denoted as f'(x) or y', represents the rate of change of the function with respect to its variable. It provides information about the slope of the tangent line to the graph of the function at any given point. In this case, y' = sin(t) indicates how the function f(t) changes as t varies from 0 to 2π.

Second Derivative

The second derivative, denoted as f''(x) or y'', is the derivative of the first derivative. It measures the rate of change of the first derivative, providing insights into the concavity of the function. A positive second derivative indicates that the function is concave up, while a negative second derivative indicates concave down. For the given problem, finding y'' involves differentiating y' = sin(t).

Graphing Procedure

The graphing procedure involves analyzing the first and second derivatives to sketch the general shape of the function. Steps typically include identifying critical points where the first derivative is zero or undefined, determining intervals of increase or decrease, and using the second derivative to assess concavity. This systematic approach helps in visualizing the behavior of the function f(t) based on its derivatives.

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Graphing The Derivative

Related Practice

Textbook Question

Root Finding

2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

Textbook Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(2x³ − 5x + 7) dx

Textbook Question

109. Suppose the derivative of the function y = f(x) is

y'=(x-1)^2(x-2).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection? (Hint: Draw the sign pattern for y'.)

Textbook Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(1 − cot²x) dx

Textbook Question

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

f(x) = x⁴ + 3x + 1, [−2, −1]

Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.82. y' = sin t, for 0 ≤ t ≤ 2π

Verified video answer for a similar problem:

Key Concepts

First Derivative

Second Derivative

Graphing Procedure

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
82. y' = sin t, for 0 ≤ t ≤ 2π