Question

How do I find x?

Answer 1

f(x) = sin(x) - tan(x)
when x = pi/4, find f(x).
f(pi/4) = ?
(x, ?) is a point on the curve and also on the tangent. ---- (1)

What is the slope of the curve at x = pi/4 ?
The derivative gives the slope. Find f'(x) and then f'(pi/4)
f'(pi/4) = ? ---- (2)

Equation of tangent is: y = mx + b
m is the slope. Sub from (2)

(1) gives a point on the line. Sub x and y values and solve for b.

Answer 2

\[
f(x) = \sin(x) - \tan(x) \\
f(\pi/2) = \sin(\pi/2) - \tan(\pi/2) = \frac{\sqrt{2}}{2} - 1 = \frac{\sqrt{2}-2}{2} \\
f'(x) = \frac{d}{dx}\sin(x) - \frac{d}{dx}\tan(x) = \cos(x) - \sec^2(x) \\
f'(\pi/2) = \cos(\pi/2) - \sec^2(\pi/2) = \frac{\sqrt{2}}{2} -
\left(\frac{2}{\sqrt{2}}\right)^2 = \frac{\sqrt{2}}{2} - \frac{4}{2} =
\frac{\sqrt{2}-4}{2}\\
\]You can find the equation of the tangent knowing the slope and a point on the line.

Answer 3

Thank you!

Answer 4

You are welcome.