   Chapter 2.7, Problem 9E

Chapter
Section
Textbook Problem

(a) Find the slope of the tangent to the curve y = 3 + 4x2 – 2x3 at the point where x = a.(b) Find equations of the tangent lines at the points (1, 5) and (2, 3).(c) Graph the curve and both tangents on a common screen.

(a)

To determine

To find: The slope of the tangent line to the curve.

Explanation

Given:

The equation of the curve is y=3+4x22x3.

The curve passing through the points (1, 5) and (2, 3).

Formula used:

The slope of the tangent curve y=f(x) at the point P(a,f(a)) is,

m=limh0f(a+h)f(a)h (1)

Calculation:

Obtain the slope of the tangent to the curve at the point x=a.

Since f(x)=3+4x22x3, substitute f(a)=3+4a22a3 and f(a+h)=3+4(a+h)22(a+h)3 in equation (1),

m=limh0f(a+h)f(a)h=limh0(3+4(a+h)22(a+h)3)(3+4a22a3)h=limh0(3+4(a2+h2+2ah)2(a3+3a2h+3ah2+h3))(3+4a22a3)h=limh0(3+4a2+4h2+42ah2a323a2h23ah22h3)(3+4a22a3)h

Perform the mathematical operations as shown below

(b)

To determine

To find: The equation of the tangent lines to the curve y=3+4x22x3 at the points (1,5) and (2, 3).

(c)

To determine

To sketch: The graph of the curve and the tangent lines.

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