# `f(x) = sec(x), (pi/3,2)` (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...

`f(x) = sec(x), (pi/3,2)` (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.

### Textbook Question

Chapter 2, 2.3 - Problem 68 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the equation of the tangent line to the curve` f(x) = sec x` , t the point `((pi)/3, 2), ` using the following formula, such that:

`f(x) - f((pi)/3) = f'((pi)/3)(x - (pi)/3)`

Notice that `f((pi)/3) = 2.`

You need to evaluate f'(x) and then `f'((pi)/3):`

`f'(x) = (sec x)' f'(x) = sec x*tan x => f'((pi)/3) = sec ((pi)/3) *tan ((pi)/3)`

`sec ((pi)/3) = 1/(cos((pi)/3)) = 2`

`tan ((pi)/3)  = sqrt 3`

` f'((pi)/3) = 2sqrt 3`

You need to replace the values into the equation of tangent line:

`f(x) - 2 = 2sqrt 3*(x - (pi)/3)`

Hence, evaluating the equation of the tangent line to te given curve , at the given point, yields `f(x) = 2*(sqrt 3)*x - 2*(sqrt 3)*(pi)/3 + 2.`