   Chapter 12.2, Problem 39E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Each of Problems 41 and 42 has the form ∫ f ( x ) dx.(a) Evaluate each integral to obtain a family of functions.(b) Find and graph the family member that passes through the point (0,2). Call that function F ( x ) .(c) Find any x-values where f ( x ) is not defined but F ( x ) is.(d) At the x-values found in part (c), what kind of tangent line does F ( x ) have? ∫ 3 d x ( 2 x − 1 ) 3 / 5

(a)

To determine

To calculate: The value of integral 3dx(2x1)3/5.

Explanation

Given Information:

The provided integral is,

3dx(2x1)3/5

Formula used:

According to the power formula of integrals, if u=u(x), then:

undu=un+1n+1+C

Calculation:

Consider the provided integral:

3dx(2x1)3/5

Rewrite the provided integral by multiplying and dividing by 2 as,

322dx(2x1)3/5

Consider the power rule of integrals:

undu=un+1n+1+C

Now, to use the power rule, the integrand should have the function u(x) and its derivative u(x) and n1.

Here,

u=2x1

Then, on obtaining differentials,

du=2

Thus,

u=2x1n=35u=2

All required parts are present, so the integral is of the form,

322dx(2x1)3/

(b)

To determine

To graph: The family member of the solution to part (a) which passes through point (0,2).

(c)

To determine

The values of x where the function f(x) is not defined but F(x) is if f(x)=3(2x1)3/5 and F(x)=154(2x1)2/574.

(d)

To determine

The nature of tangent line of F(x) at the point obtained in part (c) which is x=12.

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