Ch. 4 - Applications of Derivatives

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

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.3.25a

Chapter 4, Problem 4.3.25a

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

f(r) = 3r³ + 16r

Verified step by step guidance

First, find the derivative of the function f(r) = 3r³ + 16r. The derivative, f'(r), will help us determine where the function is increasing or decreasing.

Calculate the derivative: f'(r) = d/dr (3r³ + 16r). Use the power rule for differentiation, which states that d/dr (r^n) = n*r^(n-1).

Apply the power rule: f'(r) = 3 * 3r² + 16 * 1 = 9r² + 16.

Set the derivative f'(r) = 9r² + 16 equal to zero to find critical points, which are potential points where the function changes from increasing to decreasing or vice versa.

Solve the equation 9r² + 16 = 0 for r. These critical points will help determine the intervals of increase and decrease. Analyze the sign of f'(r) around these points to identify the intervals where the function is increasing or decreasing.

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Key Concepts

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

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. For the function f(r) = 3r³ + 16r, finding the derivative and setting it to zero will help identify these critical points.

First Derivative Test

The First Derivative Test is used to determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign of the derivative before and after each critical point, we can ascertain the intervals where the function is increasing or decreasing. This test is crucial for understanding the behavior of f(r) = 3r³ + 16r around its critical points.

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if its derivative is negative. By applying the First Derivative Test to f(r) = 3r³ + 16r, we can identify these intervals, which are key to understanding the overall shape and behavior of the function.

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Related Practice

Textbook Question

51. Frictionless cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 10 cos πt.

a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then?

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Textbook Question

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = sec²x − 2tan x, −π/2 < x < π/2

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Textbook Question

20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?

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Textbook Question

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(x) = (x + 1)², −∞ < x ≤ 0

156

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Textbook Question

Identifying Extrema