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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 5.3.16b
Chapter 5, Problem 5.3.16b

Area functions for constant functions Consider the following functions Ζ’ and real numbers a (see figure).
(b) Verify that .A'(𝓍) = Ζ’(𝓍)
  fig                                                                                                                                                          
Ζ’(t) = 5 , a = -5

Verified step by step guidance
1
Step 1: Understand the problem. The function Ζ’(t) = 5 is a constant function, and the area function A(x) represents the area under the curve of Ζ’(t) from t = a to t = x. We are tasked with verifying that the derivative of A(x), denoted A'(x), equals Ζ’(x).
Step 2: Recall the definition of the area function. A(x) is defined as the integral of Ζ’(t) from t = a to t = x: A(x) = ∫[a, x] Ζ’(t) dt. Substituting Ζ’(t) = 5, we have A(x) = ∫[a, x] 5 dt.
Step 3: Compute the integral. The integral of a constant function c over an interval [a, x] is given by c * (x - a). Therefore, A(x) = 5 * (x - a).
Step 4: Differentiate A(x) with respect to x. Using the derivative rules, differentiate A(x) = 5 * (x - a). Since a is a constant, its derivative is 0, and the derivative of x is 1. Thus, A'(x) = 5.
Step 5: Verify the result. The derivative A'(x) = 5 matches the original function Ζ’(x) = 5, confirming that A'(x) = Ζ’(x). This verifies the relationship between the area function and the original function.

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Key Concepts

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

Area Function

An area function, denoted as A(x), represents the area under a curve from a fixed point a to a variable point x on the x-axis. In this context, if f(t) is a constant function, the area A(x) can be calculated as the product of the height f(t) and the width (x - a). This concept is fundamental in understanding how the area changes as x varies.
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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if A(x) is the area function defined as the integral of f(t) from a to x, then the derivative A'(x) equals f(x). This theorem is crucial for verifying relationships between area functions and their corresponding functions, particularly in the context of constant functions.
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Fundamental Theorem of Calculus Part 1

Constant Functions

A constant function is a function that always returns the same value, regardless of the input. In this case, f(t) = 5 is a constant function, meaning the height of the rectangle representing the area under the curve remains unchanged as x varies. Understanding constant functions is essential for analyzing the area function and its derivative, as it simplifies the calculations involved.
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Exponential Functions
Related Practice
Textbook Question

{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.


(b) Calculate g'(𝓍)


g(𝓍) = βˆ«β‚€Λ£ sin (Ο€tΒ² ) dt ( a Fresnel integral) 

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Textbook Question

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

                                                                                                                                                                    

(b) βˆ« (Ζ’(𝓍))ⁿ Ζ’'(𝓍) d𝓍 = 1/(n + 1) (Ζ’(𝓍))ⁿ⁺¹ + C , n β‰  ―1 .

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Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = βˆ«β‚€^Ο€/2 (cos ΞΈ ― 2 sin ΞΈ) dΞΈ = ―1

(b) βˆ«β‚€^Ο€/2 (4 cos ΞΈ ― 8 sin ΞΈ) dΞΈ

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Textbook Question

Working with area functions Consider the function Ζ’ and its graph.

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


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Textbook Question

Working with area functions Consider the function Ζ’ and its graph.

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


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Textbook Question

Generalizing the Mean Value Theorem for Integrals Suppose Ζ’ and g are continuous on [a, b] and let h(𝓍) = (𝓍―b) βˆ«β‚Λ£ Ζ’(t) dt + (𝓍―a) βˆ«β‚“α΅‡g(t)dt.                                                                                                                                                                                                                                                                                                                                

(b) Show that there is a number c in (a, b) such that βˆ«β‚αΆœ Ζ’(t) dt = Ζ’(c) (b ― c)                                                                                                              

                                                                                                                                                                                

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

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