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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

(a) The curve with equation y2 = x3 + 3x2 is called the Tschirnhausen 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.

(a)

To determine

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

Explanation

Given:

The curve with equation y2=x3+3x2.

The point is (1,2).

Derivative rules:

(1) Chain rule: If y=f(u) and u=g(x)  are both differentiable function, then

dydx=dydududx.

(2) Product rule: If y=f(u) and u=g(x)  are both differentiable function, then

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)

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

Calculation:

Obtain the equation of tangent line to the given point.

y2=x3+3x2

Differentiate the above equation implicitly with respect to x,

ddx(y2)=ddx(x3+3x2)ddx(y2)=ddx(x3)+3ddx(x2)ddx(y2)=3x2

(b)

To determine

To find: The points if the curve has horizontal tangent.

(c)

To determine

To sketch: The curve and tangent line to the given point.

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