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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 5.3.21b
Chapter 5, Problem 5.3.21b

Area functions for linear functions Consider the following functions Ζ’ and real numbers a (see figure).
(b) Verify that A'(𝓍) = Ζ’(𝓍).
fig
Ζ’(t) = 3t + 1 , a = 2

Verified step by step guidance
1
Step 1: Understand the problem. The goal is to verify that the derivative of the area function A(x) with respect to x is equal to the given function f(x). The area function A(x) represents the area under the curve y = f(t) from t = a to t = x.
Step 2: Recall the Fundamental Theorem of Calculus. It states that if A(x) is defined as the integral of f(t) from a to x, then the derivative of A(x) with respect to x is equal to f(x). Mathematically, this is expressed as: ddx A(x)=f(x)
Step 3: Define the area function A(x). Using the given function f(t) = 3t + 1 and the lower limit a = 2, the area function is: A(x)=∫ax(3t+1)dt
Step 4: Differentiate A(x) with respect to x. By the Fundamental Theorem of Calculus, the derivative of the integral with respect to its upper limit x is simply the integrand evaluated at x. Therefore, ddxA(x)=f(x)=3x+1
Step 5: Verify the result. The derivative of A(x) matches the given function f(x) = 3x + 1, confirming that A'(x) = f(x). This completes the verification.

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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 accumulated area under a curve from a fixed point 'a' to a variable point 'x'. In this context, it quantifies the area between the x-axis and the function f(t) = 3t + 1 over the interval [a, x]. Understanding this concept is crucial for analyzing how the area changes as 'x' varies.
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Finding Area When Bounds Are Not Given

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links 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 essential for verifying the relationship between the area function and the original function, as required in the question.
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Fundamental Theorem of Calculus Part 1

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(t) = mt + b, where m is the slope and b is the y-intercept. In this case, f(t) = 3t + 1 is a linear function with a slope of 3, indicating a constant rate of change. Understanding linear functions is vital for interpreting the graph and calculating the area under the curve.
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Linearization
Related Practice
Textbook Question

Displacement from a table of velocities The velocities (in mi/hr) of an automobile moving along a straight highway over a two-hour period are given in the following table.

(b) Find the midpoint Riemann sum approximation to the displacement on [0,2] with n = 2 and .n = 4 .

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Textbook Question

Working with area functions Consider the function Ζ’ and the points a, b, and c.

(b) Graph Ζ’ and A.

Ζ’(𝓍) = eΛ£ ; a = 0 , b = ln 2 , c = ln 4

Textbook Question

Suppose Ζ’ is an even function and βˆ«βΈβ‚‹β‚ˆ Ζ’(𝓍) d𝓍 = 18

(b) Evaluate βˆ«β‚‹β‚ˆβΈ 𝓍ƒ(𝓍) d𝓍 .

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£ sinΒ² t dt

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Textbook Question

{Use of Tech} Midpoint Riemann sums 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.


βˆ«β‚€β΄ (4𝓍― 𝓍²) d𝓍

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Textbook Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.


Ζ’(x) = 4 - 2x on [0,4]


(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

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