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lisabon 2012 [21]
3 years ago
5

Find the equation of the tangent line. y=(9-x^2)^1/3 at (1,2)​

Mathematics
1 answer:
Schach [20]3 years ago
7 0

Get the derivative:

<em>y</em> = (9 - <em>x</em>²)¹ʹ³

d<em>y</em>/d<em>x</em> = 1/3 (9 - <em>x</em>²)⁻²ʹ³ d/d<em>x</em> [9 - <em>x</em>²]

d<em>y</em>/d<em>x</em> = 1/3 (9 - <em>x</em>²)⁻²ʹ³ (-2<em>x</em>)

d<em>y</em>/d<em>x</em> = -2/3 <em>x</em> (9 - <em>x</em>²)⁻²ʹ³

Evaluate it at <em>x</em> = 1 :

d<em>y</em>/d<em>x</em> (1) = -2/3 • 8⁻²ʹ³

Since 8 = 2³, we have

8⁻²ʹ³ = 1 / 8²ʹ³ = 1 / (2³)²ʹ³ = 1 / 2² = 1/4

Then the tangent line has equation

<em>y</em> - 2 = 1/4 (<em>x</em> - 1)   →   <em>y</em> = 1/4 <em>x</em> + 7/4

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