   Chapter 3.1, Problem 39E

Chapter
Section
Textbook Problem

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.y = 3x2 – x3, (1, 2)

To determine

To find: The equation of the tangent line to the curve y=3x2x3 and to sketch the given curve and the tangent line at the point (1,2).

Explanation

Given:

The equation of the curve is, y=3x2x3 and the point is (1,2).

Derivative rules:

(1) Constant Multiple Rule: ddx[cf(x)]=cddxf(x)

(2) Power Rule: ddx(xn)=nxn1

(3) Difference Rule: ddx[f(x)g(x)]=ddx(f(x))ddx(g(x))

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

Where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1.

Calculation:

Obtain the derivative of y that is dydx as follows.

dydx=ddx(y) =ddx(3x2x3)

Apply the difference rule (3),

dydx=ddx(3x2)ddx(x3)

Apply the constant multiple rule (1),

dydx=3ddx(x2)ddx(x3

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