Question

Find the equation of the line tangent to the graph of f(x)=−2cos(x) at x=π4

Give your answer in point-slope form y−y0=m(x−x0). You should leave your answer in terms of exact values, not decimal approximations.

Answer 1

We are given the function,

$\small f(x)=-2\cos x$

Firstly, let us find the value of the function at x=π/4

$\small f\left ({\pi\over 4}\right)=-2\cos {\pi\over 4 \right }=-2\times {1\over \sqrt{2}}=-\sqrt{2}$

Now, we differentiating the function with respect to x, we get

$\small f'(x)=-2(-\sin x)=2\sin x$

Then, the value of the derivative at x=π/4 gives us the slope of the tangent at that point. So, we have

$\small f'\left ( {\pi \over 4} \right )=2\sin {\pi \over 4}=2\times {1\over \sqrt {2}}=\sqrt{2}$

Thus, the tangent to the graph f(x) at x=π/4 passes through the point
$\small \left ( {\pi\over 4},-\sqrt{2} \right )$ and has slope $\small \sqrt 2$

Thus the equation of the tangent, in point-slope form is

$\small y-(-\sqrt{2})=\sqrt{2}\left (x-{\pi\over 4} \right )$

Simplifying we get

$\small y+\sqrt{2}=\sqrt{2}\left (x-{\pi\over 4} \right )$

which is our required answer.