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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 5.3.95b
Chapter 5, Problem 5.3.95b

Working with area functions Consider the function ƒ and the points a, b, and c.
(b) Graph ƒ and A.
ƒ(𝓍) = eˣ ; a = 0 , b = ln 2 , c = ln 4

Verified step by step guidance
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First, understand that the function given is \(f(x) = e^x\), which is an exponential function that grows as \(x\) increases.
Next, identify the points \(a = 0\), \(b = \ln 2\), and \(c = \ln 4\). These points will be important for defining intervals on the \(x\)-axis.
The area function \(A(x)\) typically represents the area under the curve \(f(t)\) from a fixed point \(a\) to a variable upper limit \(x\). So, define \(A(x)\) as the integral \(A(x) = \int_{0}^{x} e^t \, dt\).
To graph \(f(x) = e^x\), plot the exponential curve starting at \(f(0) = 1\) and increasing rapidly. Mark the points \(x = 0\), \(x = \ln 2\), and \(x = \ln 4\) on the \(x\)-axis to show the intervals of interest.
To graph the area function \(A(x)\), recognize that \(A(x)\) is the accumulation of the area under \(f(t)\) from \(0\) to \(x\). Since the integral of \(e^t\) is \(e^t\), \(A(x)\) can be expressed as \(A(x) = e^x - e^0 = e^x - 1\). Plot this function alongside \(f(x)\) to compare the original function and its area function.

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Key Concepts

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

Area Functions and the Definite Integral

An area function A(x) represents the accumulated area under the curve of a function ƒ from a fixed point a to a variable point x. It is defined as A(x) = ∫_a^x ƒ(t) dt, linking the concept of integration to area calculation under ƒ. Understanding this helps in interpreting A graphically and analytically.
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Definition of the Definite Integral

Exponential Functions and Their Properties

The function ƒ(x) = e^x is an exponential function with base e, characterized by its continuous growth and the unique property that its derivative equals itself. Knowing how to evaluate and graph e^x, especially at points like ln 2 and ln 4, is essential for plotting ƒ and understanding the behavior of the area function.
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Properties of Functions

Natural Logarithm and Its Relationship to Exponentials

The natural logarithm ln x is the inverse of the exponential function e^x, meaning ln(e^x) = x. Points like b = ln 2 and c = ln 4 correspond to x-values where e^x equals 2 and 4, respectively. This relationship is crucial for interpreting the given points and accurately graphing both ƒ and the area function A.
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Derivative of the Natural Logarithmic Function
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

Suppose ƒ is an even function and ∫⁸₋₈ ƒ(𝓍) d𝓍 = 18

(b) Evaluate ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 .

Textbook Question

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

(b) Verify that A'(𝓍) = ƒ(𝓍).

ƒ(t) = 3t + 1 , a = 2

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.


∫₀⁴ (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.


ƒ(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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Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(b) 4 + 5 + 6 + 7 + 8 + 9