| This tutorial: Part A: Numerical Approach |
| Next tutorial: Part B: Graphical Viewpoint |
| Following tutorial: Part C: The Derivative Function |
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Calculating the Instantaneous Rate of Change
In this tutorial, we continue with the topic of average rate of change over an interval discussed in the previous tutorial, but this time we look at shorter and shorter intervals.
You are based in Indonesia, and you monitor the value of the US Dollar on the foreign exchange market very closely during a rather active five-day period. Suppose you find that the value of one US Dollar can be well approximated by the function
(The rupiah is the Indonesian currency), where t is time in days. (t = 0 represents the value of the Dollar at noon on Monday.)
Let us begin by considering the average rate of change of the Dollar's value over various intervals. (Press the "Prev" button or here to go to a tutorial on average rate of change.)
Q What was the value of the Dollar at noon on Tuesday?
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Q According to the graph, when was the value of the Dollar rising most rapidly?
| | Monday at noon |
| Tuesday at noon |
| | Wednesday night midnight (t = 2.5) |  | Saturday at noon (t = 5) |
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From above, the value of the Dollar was given by
Q Now compute the average rate od change of R over the intervals [1, 1+h] for h = 1, 0.01, 0.001, and 0.0001 (see the last example in the previous tutorial). Note There are several ways to do this:
Now complete the following table.
In terms of the value of the Dollar, this suggests that, as measured over a small time interval around noon on Tuesday (t = 1), the Dollar was rising at a rate of close to 300 rupiahs per day.
To put it another way, the Dollar was increasing at an instantaneous rate of 300 rupiahs per day at noon on Tuesday. And that is what much of calculus is concerned with: studying the instantaneous rate of change of a function.
The process of letting h get smaller and smaller is called taking the limit as h approaches 0. We write h
0 as shorthand for "h approaches 0." Taking the limit of the average rates of change gives us the instantaneous rate of change. Here is the notation for this limit.
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Instantaneous Rate of Change of f(x) at x = a: The Derivative
The instantaneous rate of change of f(x) at x = a is defined by taking the limit of the average rates of change of f over the intervals [a, a+h], as h approaches 0. We write:
Units: The units of f' are units of f per unit of x. Quick Example If f(x) = 7,500 + 500x - 100x2, then the calculation you did above above suggests (correctly) that
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The cost (in dollars) of producing x dumbbell sets per day at the Taft Sports Company is calculated by its marketing staff to be given by the formula
Now make a table showing the values of the averge rate of change of C over the interval [100, 100+h] for h = 1, 0.1, 0.01, 0.001, and 0.0001. Use your table to estimate the instantaneous rate of change of cost that results from an increase in production level from the current level of 100 dumbbell sets.
Here again is that little utility that computes the average rate of change over any interval. Enter the technology formula for C(x) in the formula box below, and the values for the end-points a = 100, and b = 100+h using the various values of h.
Q Do we always need to make tables of difference quotients as above in order to calculate an approximate value for the derivative?
A We can usually approximate the value of the derivative by using a single, small, value of h. In the example above, the value h = 0.0001 would have given a pretty good approximation. The problem with using a fixed value of h is that (1) we do not get an exact answer, only an approximationof the derivative, and (2) how good an approximation it is depends on the function we're differentiating. With many of the functions you encounter, considering, it is a good enough approximation.
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Calculating a Quick Approximation of the Derivative
We can calculate an approximate value of f'(a) by using the formula
with a small value of h (recall that the symbol " Alternative Formula: the "Balanced Difference Quotient" The following alternative formula often gives a more accurate result, and is the one used in many calculators (the nDeriv function of the TI-83 does this; by default it uses h = 0.001, but this may be changed via an optional argument).
Example Let f(x) = x2 - x-0.4 again, and use the balanced approximation with h = 0.0001 to estimate f'(5). You must round the answer to at least 5 decimal places!
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The following is similar to Example 4 in Section 3.2 of Applied Calculus.
If I throw a ball upward at a speed of 80 ft/s, its height t seconds later will be
How fast will the ball be rising exactly 2 seconds after I throw it (t = 2)?
| | 16 ft/sec |
| 0 ft/sec (not moving) | |
| | undefined | |
96 ft/sec | |
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You could now try some of the exercises in Section 3.5 in Applied Calculus or Section 10.5 in Finite Mathematics and Applied Calculus), but you will need the material in the next tutorial to answer the questions about graphs.
