   Chapter 9.6, Problem 27E Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

Chapter
Section Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

In Problems 25-28, write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct. 27 .  y  =  3 x 2 − 2  at  x  = 3

To determine

To calculate: The equation of line tangent to the function y=3x22 at x=3 and check by plotting the graph of the tangent line and the function.

Explanation

Given Information:

The function is y=3x22 and the line is tangent at x=3.

Formula used:

Power rule:

If y=un and u is differentiable function of x, then,

dydx=nun1dudx

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

f(x)=u(x)+v(x)

According to the product rule, if f(x)=u(x)v(x), then,

f(x)=u(x)v(x)+v(x)u(x)

According to the point slope form, the equation of a line with slope m and point (x1,y1) is,

yy1=m(xx1)

Calculation:

Consider the provided function,

y=3x22

Rewrite the function,

y=(3x22)12

The equation of line tangent to the equation is found using the point-slope form.

The slope of tangent is found by differentiating the function.

Now, consider (3x22) to be u,

y=u12

Differentiate both sides with respect to x,

dydx=ddx(u12)

Simplify using the power rule,

dydx=12u121dudx=12u12dudx=12u12dudx

Substitute (3x22) for u,

dydx=12(3x22)12(ddx(3x22))=12(3x22)12(ddx(3x2)ddx(2))=12(3x22)12(3ddx(x2)ddx

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