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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.3.16b
Capitolo 5, Problema 5.3.16b

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

Guida verificata passo dopo passo
1
Step 1: Understand the problem. The function ƒ(t) = 5 is a constant function, and the area function A(x) represents the area under the curve of ƒ(t) from t = a to t = x. We are tasked with verifying that the derivative of A(x), denoted A'(x), equals ƒ(x).
Step 2: Recall the definition of the area function. A(x) is defined as the integral of ƒ(t) from t = a to t = x: A(x) = ∫[a, x] ƒ(t) dt. Substituting ƒ(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 ƒ(x) = 5, confirming that A'(x) = ƒ(x). This verifies the relationship between the area function and the original function.

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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
Pratica correlata
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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 .

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Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       

(b) ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍

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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.


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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n. 

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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.


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Domanda del libro di testo

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)

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