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At what point(s) is the tangent line to the curve 2 3 y x = 2 perpendicular to the line 4 3 10 x y − += ?

At what point(s) is the tangent line to the curve 2 3 y x = 2 perpendicular to the-example-1
User Lilie
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1 Answer

14 votes
14 votes

Answer:

Step-by-step explanation:

From the information,

equation of curve = y^2 = 2x^3

equation of line = 4x - 3y + 1 = 0

The first step is to find the slope of the line. Recall, the equation of a line in the slope intercept form is expressed a

y = mx + c

where

m represents slope

We would rearrange the given equation. We have

3y = 4x + 1

y = 4x/3 + 1/3

Thus,

m = 4/3

The slope of the perpendicular line = - 1/m = - 3/4

For the curve, we would find the derivative.

y = (2x^3)^1/2

y' = 2^3/2 * x^3/2

f'(x) = (3√x)/√2

We would equate the derivative to the slope of the perpendicular line. We have

(3√x)/√2 = - 3/4

3√x * 4 = - 3√2

12√x = - 3√2

√x = - 3√2/12

√x = -√2/4

Square both sides

x = (-√2/4)^2 = 2/16

x =

Substituting x = 1/8 into y = (2x^3)^1/2, we have

y = (2(1/8)^3)^1/2

y = √256

y =

User Grigor
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