`f(x) = sqrt(2x^2 - 7), (4,5)` (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) = sqrt(2x^2 - 7), (4,5)` (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 the graphing utility to confirm your results.

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Textbook Question

Chapter 2, 2.4 - Problem 73 - Calculus of a Single Variable (10th Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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`f(x)=sqrt(2x^2-7)`

Slope of the tangent line is the derivative of the function at that point.

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

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

Slope of the tangent line (m) at (4,5) can be found by plugging in the value of x in f'(x).

Slope (m)= (2*4)/`sqrt(2*(4^2)-7)`

Slope (m)= 8/5

Equation of the tangent line can be found by using point slope form of the line.

y-y_1=m(x-x_1)

y-5=8/5(x-4)

5y-25=8x-32

5y=8x-7

y=8/5x-7/5

`y=8/5x-7/5`

Red line represents the tangent line and black represents the function.

 

 

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