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) = ∫₀ˣ sin² t dt
65
views
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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 guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
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.
Recommended video:
05:43
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.
Recommended video:
06:21Properties 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.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) Suppose ƒ is a negative increasing function, for 𝓍 > 0 . Then the area function A(𝓍) = ∫₀ˣ ƒ(t) dt is a decreasing function of 𝓍 .
60
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) = x² ― 1 on [2,5] ; n = 75
(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.
88
views
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].
81
views
Textbook Question
Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(b) Find the mass of the right half of the rod (5 ≤ x ≤ 10) .
46
views
Textbook Question
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
f(𝓍) = x³ on [-1,2]
(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.
49
views