   Chapter 9.5, Problem 24E ### 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

# In Problems 23-26, write the equation of the tangent line to the graph of the function at the indicated point. Check the reasonableness of your answer by graphing both the function and the tangent line. y = ( 4 x 2 + 4 x + 1 ) ( 7 − 2 x ) at x = 2

To determine

To calculate: The equation of the tangent line to the graph of the function y=(4x2+4x+1)(72x) at point x=2. Also, check the answer by graphing function and the tangent line.

Explanation

Given Information:

The provided function is y=(4x2+4x+1)(72x).

Formula Used:

As per the quotient rule, if two functions are given in the form f(x)g(x), then the derivative is given as:

ddx(fg)=fggfg2

The slope (m) of the function is the first derivative of the function.

The equation of the tangent at any point (x1,y1) is given by

yy1=y(xx1)

Calculation:

Consider the provided function y=(4x2+4x+1)(72x).

For the derivative of y=(4x2+4x+1)(72x), follow the steps:

Consider f(x)=4x2+4x+1 and g(x)=72x.

Apply the product rule of the expression,

ddx(f.g)=ddx((4x2+4x+1)(72x))

Evaluate the expression further,

y=(4x2+4x+1)ddx(72x)+(72x)ddx(4x2+4x+1)=(4x2+4x+1)(ddx(7)2ddx(x))+(72x)(4ddx(x2)+4ddx(x)+ddx(1))=(4x2+4x+1).(02)+(72x).(8x+4+0)=(4x2+4x+1).(2)+(72x).(8x+4)

The slope of the tangent at x=2 can be found by the first derivative of the expression,

So, the value of the slope is,

f(2)=(4(2)2+4(2)+1).(2)+(72(2)).(8(2)+4)=(25).(2)+(3).(20)=50+60=10

Hence, the slope of the tangent line to the graph of the function y=(4x2+4x+1)(72x) at x=2 is 10.

Put x=2 in the function in order to find the value of y as,

y=(4(2)2+4(2)+1)

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