   Chapter 3.5, Problem 34E

Chapter
Section
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.

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Convert the expressions in Exercises 31-36 to positive exponent form. 3x4

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 73-80, find the indicated limits, if they exist. 76. limx2x2+3x+1x4x2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Multiply. (1cos)(1+cos)

Trigonometry (MindTap Course List)

True or False: The degree of f (x) = 4x3 + 7x6 + 1 is 7.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: is a convergent series.

Study Guide for Stewart's Multivariable Calculus, 8th 