   Chapter 2.6, Problem 34E

Chapter
Section
Textbook Problem

# (a) The curve with equation y 2 = x 2 + 3 x 2 is called the Tschimhausen cubic. Find an equation of the tangent line to this curve at the point ( 1 , − 2 ) .(b) At what points does this curve have horizontal tangents?(c )Illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen.

To determine

(a)

To find:

An equation of tangent line to the curve at the point (1, -2)

Explanation

1) Concept:

Slope of tangent line is derivative of curve at that point. Find the derivative by using implicit differentiation and substitute the given point in it. This will be the slope of tangent line. Use the equation of point slope form for the given point and the calculated slope to find the equation of tangent line.

2) Formula:

i. Product rule

ddxf*g=fddxg+gddxf

ii. The Power rule

ddxxn=nxn-1

3) Given :

y2=x3+3x2

4) Calculation:

y2=x3+3x2

Differentiate with respect to x

ddxy2=ddxx3+3x2)

Use Product rule for differentiation

ddxy2=ddxx3+3ddxx2

2ydydx=3x2+3*2x

2ydydx

To determine

(b)

To find:

The point where the curve has horizontal tangent.

To determine

(c)

To illustrate:

Part a and part b by graphing on a same graph

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