Calculus Review



Calculus Review

AGEC 317

Material from MATH 142

The ability to manipulate equations using calculus is critical in economic analysis. The following problems review single-variable calculus. In AGEC 317, the single-variable calculus will be extended to include more than a single variable.

1. Informally, the concept of a limit is what value does one variable (say y) approach as another variable (say x) approaches a specific value (say n). For a limit to make sense, y must be a function of x. The variable x can approach n from values either greater than or less than n. The idea of a limit will be used in AGEC 317 will only to illustrate the concept of a derivative.

(a) Graph the following linear line, y = 1 + 3x, in x, y space. What is the limit of y = f(x) when x approaches 2?

(b) Graph the following function, y = 5 if x is less than 3 and y = 10 if x is greater than or equal to 3. What is the limit of this function as x approaches 3?

2. The following rules of differentiation must be memorized for AGEC 317. A derivative finds the slope of the function at a point. The first order derivative of y with respect to x is denoted as either dg/dx or g’(x), where the prime denotes the derivative with respect to x.

Constant function rule if g(x) = c, dg/dx = 0. Derivative of a constant is zero.

Power function rule if g(x) = cxn, dg/dx = cnxn-1.

Sum rule if y = f(x) + g(x), [pic].

Difference rule if y = f(x) - g(x), [pic].

Product rule if y = f(x) g(x),[pic].

Quotient rule if [pic]

Chain rule if [pic]

Log rule (natural or base 10) if y = ln f(x), [pic]

Exponential rule if [pic]

Find dy/dx (first derivative) for the following functions.

[pic]

(j) What does the first derivative measure?

3. Because the first order derivative is, in general, a function of the variable x, the first derivative is itself differentiable with respect to x, if it is continuous and smooth. Higher order derivatives are the derivative of the derivatives. For example, the second derivative is the derivative of the first derivative with respect to x: [pic] Note, higher order derivatives are denoted by a superscript number, which is the same as the order of the derivative or by the number of primes, which denote the order. The same rules of differentiation used in question 2 apply to higher order derivatives.

Find [pic]for the following functions:

[pic]

(e) Find the first six derivatives of the function f(x) = a + 2x2 + 3x5.

(f) What does the second derivative measure?

4. Examples of the use of calculus in economics include comparative statics, elasticities, marginal analysis, and optimization. MATH 142 introduced some of these concepts. AGEC 317 will introduce, review, reinforce and use these concepts, whereas higher-level AGEC courses will expand and assume knowledge of these basic concepts and the use of calculus.

Elasticities, in general, can be calculated as [pic], where f(x) is the function of interest and x is the variable of interest.

Marginal analysis is accomplished by taking the derivative of the function, for example the marginal product of a production function is the first derivative with respect to x.

To find the relative extreme points of a function, the first derivative is set equal to zero and the resulting equation solved. The second derivative test indicates whether the point is a maximum or minimum. If the first derivative,[pic]at x = x0, then the value of the function at the point x0 will be a

(i) relative maximum if the second derivative is [pic],

(ii) relative minimum if the second derivative is [pic].

An inflection point is on the function where the rate of change of the slope changes. At this point, the second derivative equals zero, that is, [pic]

For the following functions, find the elasticity with respect to x, marginal function with respect to x, and if the function contains a maximum or minimum.

[pic]

(d) What is the elasticity for the function in (a) at x = 2 and x = 3?

(e) For the function, y = x3 - 3x + 5, find the stationary values (maximum, minimum, and inflection points).

5. Show that the function y = x + 1/x (with x not equal to zero) has two relative extrema, one being a maximum and the other being a minimum. Is the y value associated with the “minimum” larger or smaller than the maximum. How is this paradoxical result possible? Hint: plot the function and the graph will help to explain.

6. What is the difference between relative and global extreme points? Does calculus find relative or global extreme points? How must global points be found?

Answers

1. (a) When graphed it is clear that the function approaches 7 when x approaches 2 from either side of 2.

(b) No limit exists, in this case as the function f(x) does not approach the same number from both sides of x → 3.

2.

[pic]

(j) The first derivative measures the instantaneous rate of change of the function or the slope of the function.

3.

[pic]

(f) The second derivative measures the instantaneous rate of change of the instantaneous rate of change of the original function.

4.

[pic]

(d) [pic]

(e) y’ = 3x2 - 3 = 3(x2 - 1) set this equal to zero and solve for x gives two extremes at x = 1 and x = -1.

y’’ = 6x. At x = 1, y’’(x) = 6(1) = 6 > 0 so have minimum at x = 1

y’’ = 6x. At x = -1, y’’(x) = 6(-1) = -6 < 0 so have maximum at x = -1

y’’(x) equals zero at x = 0, inflection point at x = 0, y = 5.

Graph of the function

[pic]

5. y’ = x + 1/x2 set this equal to zero and solve for x gives two extremes at x = 1 and x = -1.

y’’ = 2/x3. At x = 1, y’’(x) = 2/(1) = 2 > 0 so have minimum at x = 1.

At x = 1, y(1) = 2.

y’’ = 2/x3. At x = -1, y’’(x) = 2/(-1) = -2 < 0 so have maximum at x = -1

At x = -1, y(-1) = -2.

At the minimum y = 2, but at the maximum y = -2. This seems paradoxical, but a graph of the function reveals why this result is true. Graph of the function

[pic]

6. Relative or local maximums or minimums are extreme points, in the sense they represent an extreme in the immediate neighborhood of the point. Global extremes are the absolute maximum or minimum of the function. Calculus finds all relative extremes. To find the global, all relative extremes must be evaluated to find the absolute maximum or minimum.

-----------------------

Approach x=2 from either side, y =7

2

7

Approach x = 3 from greater than three, y = 10. Approach x = 3 from less than three, y = 5.

3

y = 10

y = 5

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download