Derivative of Tabular Functions: Calculus I & II (AB and BC)
Assume f, g, h, and m are all continuous functions. Some of the values of f and g and their derivatives are given in the below table:
You also know that
.
(a) Approximate h'(1) using a secant line.
(b) If h'(-1) = 9, fill in the missing value on the chart above.
(c) Evaluate
for x = 0.
Difficulty:
Solution
(a) A good approximation would be the slope of the secant line to h passing through the points on h where x = 0 and x = 1. It is easy to calculuate secant slope, but first you need to evaluate h(1) and h(0)
You could also have used the x values 1 and 2 instead. Surprisingly, the answer there is -3. Who can be sure which is more accurate? We can't! All we are given is the information in the chart. Having x values closer to 1 would be helpful, but alas, you can't use what you don't have.
(b) Use the product rule to get h '(x) and then use the chart to fill in the resulting values. The missing component is easy to find by solving the simple equation.
(c) This is the tricky one that upped the difficulty level. No one thing is hard, but all together, it's a mini-marathon to solve the problem. First, you have to use the chain rule to get the requested derivative.
Time for that sinking feeling. This is gonna require some work. We'll need to find both m(x) and m '(x). Whereas the first is very easy to find, the second will require the quotient rule (it figures...)
Now, plug those values into the problem (don't lose focus on your original goal) to see what the next step will be.
All that's left is to evaluate h '(-2). Apply the product rule just like you did in part (b).
Great! Plug that into our problem and out pops the final answer.
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