Question

Which of the following is the best linear approximation for f(x) = cos(x) near x equals pi divided by 2?

Answer 1

y equals x minus pi over 2
y equals negative x plus pi over 2
y equals negative x plus pi over 2 plus 1
y equals x minus pi over 2 plus 1

Answer 2

The derivative of cos(x) is -sin(x), but idk where to go from there.

Answer 3

We start with this:\[
dy = \frac {dy}{dx}dx\\
\Delta y\approx \frac{dy}{dx}\Delta x
\]

Answer 4

so we have the tangent line at x=pi/2 is
\[y-f( \frac{\pi}{2})=f'( \frac{\pi}{2})(x- \frac{\pi}{2})\]

Answer 5

oh and that is actually what @wio wrote in fancy notation :p

Answer 6

\[y=f'(\frac{\pi}{2})(x-\frac{\pi}{2})+f(\frac{\pi}{2}) \\ f(x) \approx f'(\frac{\pi}{2})(x-\frac{\pi}{2})+f(\frac{\pi}{2} ) \text{ for values of } x \text{ near } \frac{\pi}{2}\]

Answer 7

\[
y-y_0 \approx \frac{dy}{dx}(x-x_0)
\]

Answer 8

so you found f'(x)
already now plug pi/2 into f' to find the slope of your line

Answer 9

-sin(pi/2) ?

Answer 10

yah
do you know what that equals

Answer 11

-1?

Answer 12

\[y=f'(\frac{\pi}{2})(x-\frac{\pi}{2})+f(\frac{\pi}{2}) \\ y=-1(x-\frac{\pi}{2})+f(\frac{\pi}{2})\]

Answer 13

cool

Answer 14

now one step really left
find f(pi/2)

Answer 15

plug pi/2 into your original function that is

Answer 16

-x+pi/2 is the answer? correct?

Answer 17

:)

Answer 18

thank you i made that way harder than it was lol

Answer 19

no problem