   Chapter 2.1, Problem 27E

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# If f ( x ) = 3 x 2 − x 3 , find f ′ ( 1 ) and use it to find an equation of the tangent line to the curve y = 3 x 2 − x 3 at the point (1, 2).

To determine

To find:

The value of f'(1) and the equation of the tangent line to the curve at the point (1, 2).

Explanation

1) Concept:

First find f'x; then substitute x=1. Then, by using slope of line and point (1, 2), find equation of the tangent line.

2) Given:

fx=3x2-x3

3) Calculation:

Consider the derivative by using the limit formula.

f'x= limh0fx+h-f(x)h

Substituting for f(x+h) and f(x) we have

f'x= limh03x+h2-x+h3-(3x2-x3)h

Since fx+h=  3 x+h2-x+h3,

we shall use identities a+b2=a2+2ab+b2 and a+b3=a3+3a2b+3ab2+b3

to obtain,

f'x= limh03(x2+2xh+h2)-(x3+3x2h+3xh2+h3)-(3x2-x3)h

f'x= limh03x2+6xh+3h2-x3-3x2h-3xh2-h3-3x2+x3h

f'x= limh06xh+3h2-3x2h-

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