`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.

Expert Answers
gsarora17 eNotes educator| Certified Educator

`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.