1. ## differentiable functions

The twice-differentiable function f is defined for all real numbers and satisies the following conditions:

f(0) = 2
f'(0) = -4
f''(0) = 3

The function h is given by h(x) = cos(kx)f(x) for all real numbers. where k is a constant. Find h'(x).

2. Use the product rule on h(x)

3. is the derivative:

cos(kx) * f'(x) - kf(x) * sin(kx)

4. yes

5. how can i find the equation for the line tangent to the graph of h at x = 0 ?

6. use y - y1 = m(x-x1)
x1 = 0, chuck x1=0 into your h equation to get y1.
then chuck x1 = 0 into your h' equation to get m.
sub everything in and rearrange.

7. do you get y = 2

8. no

9. for y1, i get 2

and for m, i get 0

what am i doing wrong?

11. h'(x) = cos(kx) * f'(x) - kf(x) * sin(kx)

cos(k*0) * f'(0) - kf(0) * sin(k*0)

cos(0) * f'(0) - kf(0) * sin(0)

1 * -4 - 2k * 0

-4 = m

y - 2 = -4(x-0)

y-2 = -4x

y=-4x+2

correct? and by the way thanks for everything

12. correct, good job.

Note: if your gradient is 0. you'll have a horizontal tangent & if your gradient is undefined/infinity then you'll have a vertical tangent. vertical tangents are of the form x = x1