Four functions (f, g, h, and j) are continuous and differentiable for all real numbers, and some of their values (and the values of their derivatives) are given by the below table:
If you know that
and
, fill in the correct numbers for each blank value in the table.
Difficulty:
Woo, Nelly, this is a bunch of work, and there are a lot of ways to figure out the answer:
Here's how I did it.
1. If h(1) = 0, then either f(1) or g(1) must equal 0, since the two are being multiplied. We know that g(1) = 3, so therefore f(1) = 0.
2. Use the fact that h'(1) = 6 to figure out f '(1). Since h is a product, use the Product Rule to find its derivative :
3. Since j(1) = -1, you can find an unexpected value, g(0), thanks to the fact that f(1) = 0:
4. Now you can figure out j'(1), since g'(0) = 1; use the Chain Rule to differentiate j(x):
5. Let's calculate j(0); we have all the necessary pieces:
6. While we're doing functions evaluated at 0, let's calculate h(0):
7. Since we know what the values of j'(0) and f '(0) are, we can find g'(1):
8. We only have h'(0) left, so return to the Product Rule:
You don't have to do these steps in the order I listed; in fact, you can do other calculations, but either way, you'll end up with the same results.
