`f(x) = 2/(root(4)(x^3)), (1,2)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the...

`f(x) = 2/(root(4)(x^3)), (1,2)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.

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Textbook Question

Chapter 2, 2.2 - Problem 55 - Calculus of a Single Variable (10th Edition, Ron Larson).
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kalau | (Level 2) Adjunct Educator

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Eliminate the radical by rewriting it as a fraction.

The function becomes:

`f(x) = 2/ (x^3)^(1/4) = 2/x^(3/4) = 2x^(-3/4)`

Take the derivative by using the power rule.

`f'(x) = -3/4 (2)(x^(-3/4-1))`

`f'(x) = -3/2 (x^(-7/4))` 

`f'(x) = -3/(2x^(7/4))`

Substitute x=1.

`f'(1) = -3/(2(1)^(7/4)) = -3/2`

With the slope of the point, and the given point (1,2), use the slope intercept form to find the equation.

`y=mx+b`

`2=(-3/2)(1)+b`

`2+3/2=b`

`b=7/2`

The equation of the tangent line is:  

`y= -3/2 x +7/2`

Graph both the equation of the tangent line with the original function.  They should intersect at (1,2).

See the image attached.

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