(Solved by Humans)-MA3310 Project Project Topic List Topic: Numerical

Paper Details

?It goes with Topic 2 Question 8 on the attachment?Graph the gasoline demand elasticity function for 0 ? p 6.

MA3310
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
f (a h) f (a)
implies that
h
h 0

Assuming the limit exists, the definition of the derivative f (a) lim
f (a)

f (a h) f (a)
h

(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

MA3310
Project
Project Topic List

b. By equation (1), f (4)
calculating
and describe

4h 2
. Therefore we estimate f '(4) by
h

4h 2
for values of h near 0. Complete columns 2 and 5 of table given below1
h

4h 2
behaves as h approaches 0.
h
4h 2
h

h

f (4 h) f (4)

h

Error

h

0.1

?0.1

0.01

?0.01

0.001

?0.001

0.0001

?0.0001

4h 2
h

Error

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
f (a) lim

difference formula (CDF)
4. Again consider f (x) =

h0

f (a h) f (a h)
2h

(2)

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

MA3310
Project
Project Topic List

h

Approximation

Error

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)

f(t) (feet)

0

0

0.5

4

1.0

15

1.5

33

2.0

55

2.5

81

3.0

109

3.5

138

4.0

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).

3

MA3310
Project
Project Topic List

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

. The calculator rounds f(1 + 10?14) to 1 and therefore the forward difference formula becomes

f (1 1014) f (1)
14

10

, which is estimated to equal

11
1014

or 0.

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

MA3310
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

MA3310
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, E

D / D
. Compute the elasticity for the changes in Steps 3 and 4 (it should be
p / p

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
dp D .
p0 p / p
p0 p D

E lim

Now the elasticity is a function of p. Show that for the gasoline demand function the elasticity is
E(p)

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

MA3310
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



Solution details: