Last two CALC questions. :)
posted by Samantha .
8. If f(x) = sin(7 − 5x), find f′(π), which is the derivative at π:
−0.754
−0.657
0
* 0.657
3.770
9. Given the function:
g(x)={x+b, x<0
{cos(x), x≥0
Find the value of b, if any, that will make the function differentiable at x = 0.
0
1
* 2
No such value exists.
There is not enough information to determine the value.

8.
f'(x) = 5cos(75x)
f'(π) = 5cos(75π)
= 3.769
or your last choice of 3.770 
#9.first, you need
0+b = cos(0)
so b=1
Now f is continuous
But you also need the derivative to be defined at x=0, so
1 = sin(0)
But it does not. What that means is the left limit and right limit do not agree, so g' has a break at x=0.
If you plot both parts of g(x), you can see that the slope changes abruptly at x=0:
http://www.wolframalpha.com/input/?i=plot+y%3Dx%2B1+and+y%3Dcos%28x%29