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Ifxy3-yx3=6is the equation of a curve, find the slope and the equation of the tangent line at the point(1,2). Computer plot the curve and the tangent line on the same axes.

Short Answer

Expert verified

The tangent has a slope of-211 .

The equation for the tangent is.....2x+11y-24=0.

Step by step solution

01

Explanation of Solution

The provide equation or curve isxy3-yx3=6.

02

Step 2: Implicit Differentiation

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

03

Calculation

Consider the equation of curve below:

xy3-yx3=6

Differentiate the provided above equation of curve,

x.3y3dydx+y3-y.3x2+x3.dydx=03xy3dydx+y3-3x2y-x3.dydx=0y3-3x2y+3xy2-x3dydx=0

Hence,

dydx=3x2y-y33xy2-x3

Put therole="math" localid="1658730361366" (1,2) for(x,y)

dydx(1,2)=3(1)2(2)-(2)33(1)(2)2-(1)3=6-812-1=-211

Then, the equation for tangent line,

y-2=-211(x-1)

Implies that,

2x+11y-24=0

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