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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding an Equation of a Function In Exercises 47-50, find an equation of the function f that has the given derivative and whose graph passes through the given point.

f ( x )   =   3 x 2 ( 2     x 3 ) 4 ;   ( 0 ,   7 )

To determine

To calculate: The equation of the function f that has given derivative f(x)=3x2(2x3)4

and given point (0,7).

Explanation

Given Information:

The provided derivative is f(x)=3x2(2x3)4 and the provided point is (0,7).

Formula Used:

According to the general power rule for integration,

If u is a differentiable function of x, then

undu=un+1n+1+C,

where n1

Calculation:

Consider the derivative f(x)=3x2(2x3)4.

It can be written as,

df(x)dx=3x2(2x3)4

Integrate df(x)dx to obtain f(x).

f(x)=3x2(2x3)4dx …… (1)

Let 2x3=u.

Differentiate with respect to x,

ddx(2x3)=dudxddx(2)ddx(x3)=dudx3x2=dudx3x2dx=du

Substitute the value of 3x2dx=du and 2x3

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