   Chapter 3.1, Problem 33E

Chapter
Section
Textbook Problem

Find an equation of the tangent line to the curve at the given point.y = 2x3 – x2 + 2, (1, 3)

To determine

To find: The equation of the tangent line to the curve at the point.

Explanation

Given:

The curve is y=2x3x2+2.

The point is (1,3).

Derivative rules:

(1) Derivative of a Constant Function: ddq(c)=0

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

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

(4) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))

(5) 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:

The derivative of y is dydx, which is obtained as follows.

dydx=ddx(y)=ddx(2x3x2+2)

Apply the Sum rule (4) and Difference Rule (5),

dydx=ddx(2x3x2) +ddx(2)=ddx

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