Problem Six: 2001-2002

Approximating Derivatives: Calculus I (AB)

Given the functions f and g as defined below, continuous on x > 0, approximate h '(.85) if h(x) = f(g(x)).

Difficulty:

Solution:

Use the chain rule to find the derivative of h(x): h'(x) = f '(g(x)) g'(x). To evaluate this derivative for x = .85, you first need to calculate g(.85). That is pretty easy using a calculator; the answer is .792797. We don't need to be too exact with decimals, since this is only an approximation anyway, so I will just round off to .8. Therefore, your answer may vary slightly from mine. You'll see why this makes sense when we try to find the derivative of f. Next, we need to find g'(.85). The quotient rule is the way to go there.

Let's fill in what we know.

How do you calculate f '? You have no formula...only a graph. The best way is to use a secant line containing the x value in question. Because we want an approximation for f at x = .8, I'll use the point (.8, .9) which appears to be on the graph of f (remember we're approximating). Another point close to .9 on the graph of f appears to be (.6, .95). To approximate f '(.8), I'll find the slope of the secant line through those points. It is -1/4. Now we have all we need to answer the question:

Rates of Change: Calculus II (BC)

Difficulty:

Solution:

For simplicity's sake, I'll call the four things above A, B, C, and D in that order. By far, it's easiest to find A with a simple substitution.

Before we can find B, we need to find C. Remember that the instantaneous rate of change is the same thing as the derivtaive, so C really is asking us to calculate f '. Use the product rule and substitute in 7/ 6

Now, take the derivative of this expression to get the second derivative and plug in 2/ 3 to find B

Surprise! They have the exact same value. You can't be sure of that using the calculator, as only a decimal results. However, when you work it out by hand, you get the exact same answer. All that's left is to find D. Remember that the average rate of change is given by the slope of a secant line (whereas C was given by the slope of a tangent line).

The correct order is A, then D, then both B and C, since the last two values are equal.

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