Module 1 Functions and Limits Project Project Topic List Topic: Numerical differentiation Topics and skills: Derivatives, calculator While the rules of differentiation allow us to compute the derivative of just about any function, there are practical situations in which these rules cannot be used. For example, in some applications, a relationship between two variables may be given as a set of data points, but not as a formula. In situations like this, the rate of change of one variable with respect to the other (that is, the derivative) might be needed, but the rules do not apply to sets of data. This project focuses on methods for approximating the derivative of a function at a particular point. Backward and Forward Difference Formulas Assuming the limit exists, the definition of the derivative f ( a h) f ( a) f ( a) h f ( a h) f ( a) f ( a) lim h h 0 implies that (1) for h near 0. If h > 0, then (1) is referred to as a forward difference formula and if h < 0, (1) is a backward difference formula. The geometry of these formulas is shown in Figure 1. 1. Why do you think (1) is called the forward difference formula if h > 0 and a backward difference formula if h < 0? 2. Let f (x) = x a. Find the exact value of f '(4). 1 Module 1 Functions and Limits Project Project Topic List b. By equation (1), calculating and describe h f (4 h) f (4) f (4) h 4h 2 h 4h 2 . h Therefore we estimate f '(4) by for values of h near 0. Complete columns 2 and 5 of table gicen below1 4 h 2 h behaves as h approaches 0. 4 h 2 h Error h 0.1 0.01 0.001 0.0001 4h 2 h Error -0.1 -0.01 -0.001 -0.0001 Table 1 3. The accuracy of an approximation is given by Error = |exact value - approximate value|. Use the exact value of f'(4) in part (a) to complete columns 3 and 6 in Table 1. Describe the behavior of the errors as h approaches 0. Centered Difference Formulas Another formula that is used to approximate the derivative of a function at a point is the centered difference formula (CDF) (2) f ( a h) f ( a h) f ( a) lim 2h h 0 4. Again consider f (x) = . x a. Graph f near the point (4, 2) and let h = in the centered difference formula. Show the line whose slope is computed by the centered difference formula and explain why the formula approximates f '(4). b. Use the centered difference formula to approximate f '(4) by completing Table 2. h 0.1 0.01 0.001 2 Approximation Error Module 1 Functions and Limits Project Project Topic List 0.0001 Table 2 5. Use the CDF (2) and a table similar to Table 2 to find a good approximation to f'(0) for f(x) = (1 + x)-1. 6. Use the CDF (2) and a table similar to Table 2 to find a good approximation to f'(/6) for f(x) = sin x. 7. Table 3 gives the distance f(t) fallen by a smokejumper t seconds after she opens her chute. a. Use the forward difference formula (1) with h = 0.5 to estimate the velocity of the skydiver at t = 2 s. b. Repeat part (a) using the centered difference formula (2). t (seconds) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 f(t) (feet) 0 4 15 33 55 81 109 138 169 Table 2 Computer Rounding Error Using difference approximations to approximate derivatives with a computer or calculator is prone to rounding errors. These errors occur when a calculator rounds a number before using it in an arithmetic calculation. Such rounding may lead to remarkably inaccurate results. 8. Consider the function f (x) = x10. a. Use analytical methods to find the exact value of f'(1). b. Use the forward difference formula to approximate f'(1) using values of h = 10-2, 10-3, and 10-4. What do you observe? c. Compute approximations to f'(1) using h = 10-n for n = 5, 6, 7, ..., 15 What do you observe? In Step 8c, you should find that for small enough values of h, the approximations to f'(1) eventually are 0, which is clearly a bad estimate. Here is why this error occurs. Suppose h = 10-14. 3 Module 1 Functions and Limits Project Project Topic List -14 The calculator rounds f(1 + 10 ) to 1 and therefore the forward difference formula becomes f 1014 ) (1) (1 1014 , which is estimated to equal 1 1 or 0. 1014 d. The remedy to rounding errors in this situation is to use small - but not too small -values of h. Based on the approximations computed in parts (b) and (c), what is a good approximation to f'(1)? 4 Module 1 Functions and Limits Project Project Topic List Topic: Elasticity in economics Topics and skills: Derivatives Economists apply the term elasticity to supply, demand, income, capital, labor, and many other variables in systems with input and output. In a few words, elasticity describes how changes in the input to a system are related to changes in the output. And because elasticity involves change, it also involves derivatives. In this project we investigate the idea of elasticity as it applies to demand functions. It's a common experience that as the price of an item increases, the number of sales of that item generally decreases. This relationship is expressed in a demand function. 1. Suppose a gas station has the linear demand function D(p) = 1200 - 200p (Figure 1). According to this function, how many gallons of gas can the gas station owners expect to sell per month if the price is set at $4 per gallon? 2. Evaluate D'(p) and show that the demand function is decreasing. Explain why demand functions are usually decreasing functions. 3. Suppose the price of a gallon of gasoline (Steps 1 and 2) increases from $3.50 to $4.00 per gallon; call this change p. What is the resulting change in the number of gallons sold, call it D? (Note that the change is a decrease, so it should be negative.) 4. Now express the answer to Step 3 in terms of percentages: What is the percent change in price, p/p, when it increases from $3.50 to $4.00 per gallon? What is the resulting percent change in the number of gallons sold, D/D? (Note that the percent change is negative.) 5 Module 1 Functions and Limits Project Project Topic List 5. The elasticity in the demand is the ratio of the percent change in demand to the percent change in price; that is, D / D E p / p . Compute the elasticity for the changes in Steps 3 and 4 (it should be negative). 6. The elasticity is simplified by considering small changes in p and D. In this case we use the definition of the derivative and write D / D D p dD p lim D dp D . p 0 p / p p 0 p E lim Now the elasticity is a function of p. Show that for the gasoline demand function the elasticity is p E( p) 6p . 7. The elasticity may be interpreted as the percent change in the demand that results for every one percent change in the price. For example if E(p) = -2, a one-percent increase in price produces a two-percent decrease in demand. If the price of gasoline is p = $4.50 and there is a 3.5% increase in the price, what is the elasticity and the corresponding percent change in the number of gallons sold? 8. Graph the gasoline demand elasticity function for 0 p < 6. 9. When - < E < -1, the demand is said to be elastic. When -1 < E < 0, the demand is said to be inelastic. When E = -, the demand is perfectly elastic and when E = 0 the demand is perfectly inelastic. Essential goods such as basic foods tend to have inelastic demands; discretionary items, such as electronic equipment have elastic demands. Explain the meaning of these terms in this context. 10. For what prices is the gasoline demand function elastic and inelastic? 11. The demand for processed pork in Canada is described by the function D(p) = 286 - 20p1. Graph the demand function, compute the elasticity, and graph the elasticity. For what prices is the demand function elastic and inelastic? 12. Show that the general linear demand function D(p) = a - bp, where a and b are positive real numbers, has a decreasing elasticity for 0 p < a/b. Show that for the general linear demand function, E = -1, when p = a/2b. 6 Module 1 Functions and Limits Project Project Topic List 13. Not all demand functions are linear. Compute the elasticity for the exponential demand function D(p) = ae-bp, where a and b are positive real numbers. 14. Show that the demand function D(p) = a/pb, where a and b are positive real numbers, has a constant elasticity for all positive prices. 7