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Chapter 4: Q7P (page 203)

If y3-x2y=8is the equation of a curve, find the slope and the equation of the tangent line at the point(3,-1). Computer plot the curve and the tangent line on the same axes.

Short Answer

Expert verified

The slope of the tangent is 1 and the equation isx-y-4=0.

Step by step solution

01

Explanation of Solution

The provided equation or curve isy3-x2y=8

02

Step 2: Implicit Differentiation

To find the slopes of tangents to curves that aren't functions, we can utilize implicit differentiation (they fail the vertical line test). Parts of yare assumed to be functions that satisfy the supplied equation but yis not a function of x.

03

Calculation

Consider the equation of the curve below:

y3-x2y=8

Differentiate the provided above equation of curve with respect to x,

3y2dydx-x2dydx+y路2x=03y2dydx-x2路dydx-2xy=03y2-x2dydx-2xy=0

Hence,

dydx=2xy3y2-x2

Put the 3,-1for x,y

dydx3,-1=23-13-12-32=-6-6=1

Then, the equation for the tangent line,

y--1=1x-3y+1=x-3

Therefore,

x-y-4=0

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