Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.3.91b
Chapter 5, Problem 5.3.91b

Working with area functions Consider the function ƒ and its graph.
(b) Estimate the points (if any) at which A has a local maximum or minimum.


<IMAGE>

Verified step by step guidance
1
Step 1: Understand the problem. The function A(t) represents the area under the curve of ƒ(t) from a fixed point (usually t = 0) to t. To find local maxima or minima of A(t), we need to analyze the behavior of ƒ(t), as the derivative of A(t) is ƒ(t).
Step 2: Recall that A(t) has a local maximum or minimum where its derivative, ƒ(t), changes sign. Specifically, A(t) has a local maximum where ƒ(t) transitions from positive to negative, and a local minimum where ƒ(t) transitions from negative to positive.
Step 3: Examine the graph of ƒ(t). The graph starts positive near t = 0, decreases, and crosses the t-axis (becomes zero) at approximately t = 6. After t = 6, ƒ(t) becomes negative and continues to decrease.
Step 4: Based on the graph, A(t) will have a local maximum at t = 6 because ƒ(t) transitions from positive to negative at this point. There are no points where ƒ(t) transitions from negative to positive, so A(t) does not have a local minimum.
Step 5: Summarize the findings. The local maximum of A(t) occurs at t = 6, and there are no local minima for A(t) based on the given graph of ƒ(t).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Local Maximum and Minimum

A local maximum of a function occurs at a point where the function value is higher than the values of the function at nearby points, while a local minimum occurs where the function value is lower. These points are critical for understanding the behavior of the function and can be identified using the first derivative test, where the derivative changes sign.
Recommended video:
06:02
The Second Derivative Test: Finding Local Extrema

Derivative and Critical Points

The derivative of a function measures the rate of change of the function with respect to its variable. Critical points occur where the derivative is zero or undefined, indicating potential locations for local maxima or minima. Analyzing these points helps in determining the function's increasing or decreasing behavior.
Recommended video:
04:50
Critical Points

Area Function

An area function typically represents the accumulation of quantities, such as the area under a curve. In the context of calculus, it can be related to the integral of a function. Understanding how the area function behaves can provide insights into the local extrema of the original function, especially when considering the relationship between the area and the function's values.
Recommended video:
05:06
Finding Area When Bounds Are Not Given
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           

                                                                                                                                                                    

(b) ∫ (ƒ(𝓍))ⁿ ƒ'(𝓍) d𝓍 = 1/(n + 1) (ƒ(𝓍))ⁿ⁺¹ + C , n ≠ ―1 .

45
views
Textbook Question

{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.


∫₀¹ (𝓍² + 1) d𝓍

80
views
Textbook Question

{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.


ƒ(𝓍) = 3 √x on [0,4] ; n = 40

(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.

66
views
Textbook Question

Working with area functions Consider the function ƒ and its graph.

(b) Estimate the points (if any) at which A has a local maximum or minimum.


64
views
Textbook Question

Area functions for constant functions Consider the following functions ƒ and real numbers a (see figure).

(b) Verify that .A'(𝓍) = ƒ(𝓍)

                                                                                                                                                            

ƒ(t) = 5 , a = -5

62
views
Textbook Question

Generalizing the Mean Value Theorem for Integrals Suppose ƒ and g are continuous on [a, b] and let h(𝓍) = (𝓍―b) ∫ₐˣ ƒ(t) dt + (𝓍―a) ∫ₓᵇg(t)dt.                                                                                                                                                                                                                                                                                                                                

(b) Show that there is a number c in (a, b) such that ∫ₐᶜ ƒ(t) dt = ƒ(c) (b ― c)                                                                                                              

                                                                                                                                                                                

(Source: The College Mathematics Journal, 33, 5, Nov 2002)

53
views