`f(x) = tan(x), (pi/4,1)` (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) = tan(x), (pi/4,1)` (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, 2.3 - Problem 67 - 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) = tan x` , t the point `((pi)/4, 1), ` using the following formula, such that:

`f(x) - f((pi)/4) = f'((pi)/4)(x - (pi)/4)`

Notice that `f((pi)/4) = 1.`

You need to evaluate f'(x) and then `f'((pi)/4):`

`f'(x) = (tan x)'`

`f'(x) = 1/(cos^2 x) => f'((pi)/4) = 1/(cos^2((pi)/4))`

Since `cos ((pi)/4) = sqrt2/2 => (cos^2((pi)/4)) = 2/4 = 1/2`

`f'((pi)/4) = 1/(1/2) => f'((pi)/4) = 2`

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

`f(x) - 1 = 2(x - (pi)/4)`

Hence, evaluating the equation of the tangent line to teĀ given curve , at the given point, yields `f(x) = 2x - (pi)/2 + 1.`

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