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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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 = 3x2x3, (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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