Microeconomic Theory: Basic Math Concepts

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Microeconomic Theory: Basic Math Concepts"

Transcription

1 Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66

2 Basic Math Concepts In this lecture we will review some basic mathematical concepts employed throughout the course: 1 Single Variable Functions 2 Limit of a Function and Continuity 3 Derivative, Higher Order Derivatives, Differential 4 Multi-Variable Functions 5 Partial Derivatives of a Multi Variable Functions 6 Total Differential, Chain Rule, Second Partial Derivatives 7 Simple Integration Van Essen (U of A) Basic Math Concepts 2 / 66

3 What is a Function? Definition y is a function of x if there is a relationship between y and x that defines, for each value of x, a corresponding value of y. Van Essen (U of A) Basic Math Concepts 3 / 66

4 Useful Single Variable Functions and their Graphs We describe a few special families of single variable functions: linear functions, general polynomial functions, the exponential function, and the natural logarithm function. Van Essen (U of A) Basic Math Concepts 4 / 66

5 Linear Functions A linear function is a function of the form f (x) = ax + b, where a and b are parameters. The parameter a is the slope of the function and the parameter b is the vertical intercept i.e., the value the function takes at x = 0. The main property of a linear function is that the dependent variable changes at a constant rate with respect to the independent variable i.e., their graph is just a straight line. Van Essen (U of A) Basic Math Concepts 5 / 66

6 Linear Functions Example The graph of the linear function f (x) = 2x + 1 is given below: y x 4 Figure: Linear Function f (x) = 2x + 1 Van Essen (U of A) Basic Math Concepts 6 / 66

7 Polynomial Functions The linear function is a special case of a polynomial function. An n-th degree polynomial takes the form f (x) = a 0 + a 1 x + a 2 x a n x n. The quadratic function i.e., a function of the form f (x) = a 0 + a 1 x + a 2 x 2. Van Essen (U of A) Basic Math Concepts 7 / 66

8 Polynomial Functions The graph of a quadratic is easy to remember it is either a mountain or a valley. It looks like a mountain if a 2 < 0, and it looks like a valley if a 2 > 0. A quadratic with a 2 > 0 is given below. y x 4 Figure: Example of Quadratic with a 2 > 0. Van Essen (U of A) Basic Math Concepts 8 / 66

9 Inverse Functions Definition If y = f (x), then x = h(y) is the inverse function of f if f (h(y)) = y. Example If y = f (x) = 2x, then the function x = h(y) = 1 2 y is the inverse function since f (h(y)) = y = y The exponential and logarithm functions are inverses of one another. Van Essen (U of A) Basic Math Concepts 9 / 66

10 Exponential and Natural Log Functions The exponential and logarithm functions are two useful functions for a variety of reasons. The exponential function is a function of the form f (x) = ae x and the natural logarithm function is of the form f (x) = ln x. y x 4 Figure: The exponential and natural logartihm functions Van Essen (U of A) Basic Math Concepts 10 / 66

11 Exponential and Natural Log Functions Logarithms and exponential functions are inverses of one another. In other words, ln e x = x e ln x = x Van Essen (U of A) Basic Math Concepts 11 / 66

12 Exponential and Natural Log Functions In addition to the two inverse relationships, the following four properties of these functions are also useful ones to keep in mind. 1 ln(xy) = ln x + ln y 2 ln( x y ) = ln x ln y 3 ln x n = n ln x 4 e x +y = e x e y Example Suppose f (x, y) = x 2 y 3 where x, y > 0, then by properties (1) and (3) we have that ln f (x, y) = 2 ln x + 3 ln y. Van Essen (U of A) Basic Math Concepts 12 / 66

13 Derivative of a Single Variable Function In economics, we are often interested in measuring the rate of change of a function. For instance, a firm is interested in knowing how much profit will change if it produces an additional unit of output. A manager may care about how production is changing when he adds an additional unit of labor to the production line. A buyer wants to know how much happier they will be if the consume another unit of a certain product. All of these questions are about marginal changes and it turns out that the set of tools provided by calculus are well suited for answering these types of questions. Van Essen (U of A) Basic Math Concepts 13 / 66

14 Derivative of a Single Variable Function A derivative tells us the instantaneous rate of change of a function. We first need to first define rate of change. Suppose we have a function f : R R and want to know how this function would change if we go from input x to a new input x + x, where x is the change in x. Van Essen (U of A) Basic Math Concepts 14 / 66

15 Derivative of a Single Variable Function In order to find the change in f, denoted f, we can simply take f (x + x), the value of the function at the new input x + x, and subtract f (x), the value the function takes at the old input x i.e., f = f (x + x) f (x). The rate of change is defined by f x or written out f x f (x + x) f (x) =. x Geometrically, this rate is the slope of a line that starts at the point (x, f (x)) and ends at the point (x + x, f (x + x)). Van Essen (U of A) Basic Math Concepts 15 / 66

16 Derivative of a Single Variable Function Van Essen (U of A) Figure: BasicRate Math Concepts of Change 16 / 66

17 Derivative of a Single Variable Function For example, suppose a firm was producing 10 units of output with 5 workers and that after they added 3 more workers to the production process output jumped to 12. If we let f (l) be the production function that tells us how much output we can produce when we use l workers, the change in output was f = f (8) f (5) = 2 and the rate of change f l = 2 3. The derivative is the instantaneous rate of change (or the marginal change) i.e., it is the rate of change of a function when we make a small change in the input. Specifically, the derivative is defined as d f (x) = lim dx x 0 f (x + x) f (x). x Van Essen (U of A) Basic Math Concepts 17 / 66

18 Derivative of a Single Variable Function Graphically, the derivative of the function f at the point x = x 0 is the slope of a line that is just tangent to f (x 0 ). This is illustrate below. Van Essen (U of A) Basic Math Concepts 18 / 66

19 Derivative of a Single Variable Function In summary, a derivative tells us how a function is changing at a particular point (i.e., the slope). Graphically, this is just the slope of a particular tangent line. We care about derivatives because they tell us how the function is changing. In particular, the sign of the derivative tells us whether the function increasing ( df dx > 0), decreasing ( df dx < 0), or constant ( df dx = 0). The actual derivative evaluated at a point is just a number which tells us the magnitude of the change i.e., how fast is the function changing. In other words, a function with a derivative equal to 10 at x = 1 is increasing faster than a function with a derivative of 2 at x = 1. Van Essen (U of A) Basic Math Concepts 19 / 66

20 Derivative of a Single Variable Function Now we come to the problem of how to find the derivative of a function. Of course, we have a definition of the derivative and we could always apply this definition to a particular function. The point you should take away from the following example is that this way of calculating a derivative is rather annoying and tedious. Van Essen (U of A) Basic Math Concepts 20 / 66

21 Derivative of a Single Variable Function Suppose we want to compute the derivative of the function f (x) = 25x 1 2 x 2. Using the definition of the derivative we know we have to compute f x and then take a limit of this expression as x 0. First, we compute the pieces of f x. The value of the function f evaluated at the new point x + x is f (x + x) = 25 (x + ) 1 (x + )2 2 = 1 2 x 2 x ( x) + 25x 1 2 ( x) ( x). Van Essen (U of A) Basic Math Concepts 21 / 66

22 Derivative of a Single Variable Function The value of the function f at the original point x is f (x) = 25x 1 2 x 2. Therefore the change in f is f = f (x + x) f (x) = 1 2 ( x)2 ( x) x + 25 ( x) The change in x is just x. Therefore the rate of change of f is f (x + x) f (x) x (25 (x + x) 1 2 (x + x)2) ( 25x 1 2 x 2) = x = 1 2 ( x)2 ( x) x + 25 ( x) x = 25 x 1 2 x Van Essen (U of A) Basic Math Concepts 22 / 66

23 Derivative of a Single Variable Function The derivative of f is the instantaneous rate of change so d f (x) = lim dx 25 x 1 x = 25 x. x 0 2 Finally, it is useful to evaluate this derivative at several points in order to see how the function is changing at different points. At the point x = 5, the derivative of f is 20 so the function is increasing. At the point 25 the derivative is zero so the function is constant. Last, at the point x = 30, the derivative is negative so the function is decreasing. Van Essen (U of A) Basic Math Concepts 23 / 66

24 Rules for Taking Derivatives There are simple rules for calculating most derivatives. Van Essen (U of A) Basic Math Concepts 24 / 66

25 Constant Rule Theorem Suppose f (x) = a, where a is a constant real number, then the derivative of f with respect to x is df (x) = 0. dx Van Essen (U of A) Basic Math Concepts 25 / 66

26 Adding Rule Theorem Suppose f (x) = g(x) + h(x), then the derivative of f with respect to x is df dg dh (x) = (x) + dx dx dx (x). Van Essen (U of A) Basic Math Concepts 26 / 66

27 Power Rule The power rule for taking a derivative applies to polynomial functions. Theorem Suppose f (x) = ax b, where a and b are constant real numbers, then the derivative of f with respect to x is df dx (x) = abx b 1. Van Essen (U of A) Basic Math Concepts 27 / 66

28 More Examples Example Suppose f (x) = 2, then df dx (x) = 0. Example Suppose f (x) = 3x, then df dx (x) = 3. Recall that x 0 = 1. Example Suppose f (x) = 3x 2, then df dx (x) = 6x. Example Suppose f (x) = 3x 1 2, then df dx (x) = 3 2 x 3 2. Van Essen (U of A) Basic Math Concepts 28 / 66

29 Example: Power Rule, Constant Rule, and Adding Rule Example Suppose we have a polynomial function f (x) = 1 + x 1 2 x 2 + 5x 3, then df (x) dx = x + 15x = 1 x + 15x 2. Van Essen (U of A) Basic Math Concepts 29 / 66

30 Product Rule Theorem Suppose f (x) = f 1 (x)f 2 (x), then df dx (x) = df 1 dx (x)f 2(x) + f 1 (x) df 2 dx (x). Van Essen (U of A) Basic Math Concepts 30 / 66

31 Product Rule Example Suppose f (x) = 3 ln x, then Example df dx (x) = 3 x. Suppose f (x) = (3x 2 )(x 4 x + 1), then df dx (x) = (6x) (x 4 x + 1) + (3x 2 )(4x 3 1) = 18x 5 9x 2 + 6x. Van Essen (U of A) Basic Math Concepts 31 / 66

32 Quotient Rule The quotient rule applies when the function we are interested in taking the derivative can be thought of as the ratio of two functions. Theorem Suppose f (x) = f 1(x ) f 2 (x ), then Example f (x) = f 1(x ) f 2 (x ) = x 2 x, then df dx (x) = df1 dx (x)f 2(x) f 1 (x) df 2 dx (x) (f 2 (x)) 2. df dx (x) = (2x)(x) (x 2 )(1) (x) 2 = 2x 2 x 2 x 2 = 1 Van Essen (U of A) Basic Math Concepts 32 / 66

33 Derivative of the Natural Logarithm Function and the Exponential Function The derivative of the natural logarithm function and its inverse, the exponential function have their own rules. Theorem Suppose f (x) = ln x, where x > 0, then df dx (x) = 1 x. Theorem Suppose f (x) = e x, then df dx (x) = ex. Van Essen (U of A) Basic Math Concepts 33 / 66

34 Chain Rule The chain rule applies when the function we are interested in taking the derivative can be thought of as the composition of two functions. For example, the function h(x) = ln(2x 2 ) can be thought of as the composition of the functions f (x) = ln(x) and g(x) = 2x 2 i.e., h(x) = f (g(x)). Theorem Suppose i(x) = f (g(x)), then Example di df (x) = dx dx (g(x))dg dx (x). Suppose f (x) = ln ( 2x 2), then df dx (x) = ( 1 2x 2 ) (4x) = 2 x. Van Essen (U of A) Basic Math Concepts 34 / 66

35 Example: Derivative of an Inverse Example (Derivative of an Inverse Function): Suppose f 1 is the inverse function of f, then f (f 1 (x)) = x. If we take the derivative of both sides with respect to x we have df dx (f 1 df 1 (x)) (x) = 1. dx Thus, the derivative of the inverse function f 1 is df 1 dx (x) = 1 df dx (f 1 (x)). Van Essen (U of A) Basic Math Concepts 35 / 66

36 Example Example Suppose f (x) = e x and f 1 (x) = ln(x). Since the exponential and logarithm functions are inverses of one another we have that df 1 dx (x) = d dx ln(x) = 1 df dx (f 1 (x)) = 1 e ln(x ) = 1 x. Similarly, if we let f (x) = ln(x) and f 1 (x) = e x. Then df 1 dx (x) = d dx ex = 1 1 e x = e x. Van Essen (U of A) Basic Math Concepts 36 / 66

37 Higher Order Derivatives The derivative of a function is also a function which, in some circumstances, can also be differentiated. In particular, the derivative of df dx (x), which is denoted d 2 f (x) or dx 2 sometimes f (x), is called a second order derivative. This is the only higher order derivative we will need in this course. The second derivative of a function tells us how the first order derivative function is changing with x. This information is very useful, for example, when trying to determine the maximum of a function. Van Essen (U of A) Basic Math Concepts 37 / 66

38 Derivative of a Single Variable Function Example Suppose f (x) = x , then df dx (x) = 2x. The second order derivative is d 2 f (x) = 2. This tells us that the function df dx 2 dx (x) is decreasing at a constant rate. Van Essen (U of A) Basic Math Concepts 38 / 66

39 Example: Higher Order Derivatives Suppose f (x) = ln x, where x > 0. Then the first derivative of f is df dx (x) = 1 x > 0 for x > 0. Thus, the logarithm function is an increasing function. The second order derivative is d 2 f dx 2 (x) = 1 x 2 < 0. So the logarithm function is increasing, but at a decreasing rate. Van Essen (U of A) Basic Math Concepts 39 / 66

40 Example y x Van Essen (U of A) Basic Math Concepts 40 / 66

41 Multi-variable Real Valued Functions Almost every function you work with in economics is a multi-variable real valued function f : R N R. Utility functions and production functions are examples of real valued multi-variable functions that will use frequently in this course. We will briefly discuss how to graph 2 variable functions using three popular functions: linear function, Cobb-Douglas function, and the min function. Van Essen (U of A) Basic Math Concepts 41 / 66

42 Multi-variable Real Valued Functions Consider the function f : R 2 + R that is defined by for all (x, y) R 2 +. f (x, y) = x + y This is a relatively simple linear function. It takes, as input, a point (x, y) and adds the x component and the y component together. Van Essen (U of A) Basic Math Concepts 42 / 66

43 Multi-variable Real Valued Functions Its graph, in three dimensions, is the plane illustrated in the figure below. 10 z x y 4 Figure: 3-D Plot of f (x, y) = x + y Van Essen (U of A) Basic Math Concepts 43 / 66

44 Multi-variable Real Valued Functions This 3-D depiction of the function demands too much artistic ability to be useful. We will use level curves to graph a multi-variable function. A level curve is a 2-D way of plotting a 3-D object. For example, a map of a mountain uses contour lines to indicate all of the spots on the mountain that have the same elevation. A level curve is the same thing. In particular, it is a line that connects all of the (x, y) such that function f obtains the same value. Van Essen (U of A) Basic Math Concepts 44 / 66

45 Multi-variable Real Valued Functions In the graph below, we plot three level curves for the function f (x, y) = x + y. Figure: Level Curves for f (x, y) = x + y Van Essen (U of A) Basic Math Concepts 45 / 66

46 Multi-variable Real Valued Functions The level curves f (x, y) = 1, f (x, y) = 3, and f (x, y) = 5. We can solve for the equation that defines the level curve. For instance, the level curve f (x, y) = 1 is defined by the equation if we solve for y we get y = 1 x. f (x, y) = x + y = 1 Van Essen (U of A) Basic Math Concepts 46 / 66

47 Multi-variable Real Valued Functions Cobb-Douglas functions are the family of real valued functions f : R 2 R of the form f (x, y) = Ax a y b with parameters A, a, b R +. For example, the function f (x, y) = x 2 y is a member of the Cobb-Douglas family with A = 1, a = 2, and b = 1. A 3-D plot of the Cobb-Douglas function f (x, y) = x 2 y is given below: z y x 4 Van Essen (U of A) Basic Math Concepts 2 47 / 66

48 Multi-variable Real Valued Functions Again, to solve for a level curve f (x, y) = k we set the function equal to k. For the example, the level curves will be defined by the equation x 2 y = k or y = k x 2. A few of these level curves from this same function are: y x Figure: Level Curves for the Cobb-Douglas Function f (x, y) = x 2 y Van Essen (U of A) Basic Math Concepts 48 / 66

49 Multi-variable Real Valued Functions Finally, the minimum function is a real valued function of the form f (x, y) = c min{ax, by}, where parameters a, b, c R +. The 3-D plot of this function looks like a side of a pyramid. 10 z y x Figure: 3-D Plot of f (x, y) = min{2x, y} Van Essen (U of A) Basic Math Concepts 49 / 66

50 Multi-variable Real Valued Functions The level curves of the minimum function look like the letter L. The corners of these level curves are along the line y = a b x. This equation is found by setting the ax component in the min function equal to the by component of the min function. Figure: Level Curves of the Min Function f (x, y) = min{2x, y} Van Essen (U of A) Basic Math Concepts 50 / 66

51 Partial Derivatives If you know how to take derivatives of single variable functions then you know how to take partial derivatives of multi-variable functions. The trick is that we just pretend that all variables, other than one we are differentiating with respect to, are constants and then proceed as before. Example Suppose f (x, y) = x 2 y, then the partial derivatives with respect to x and y are f x f y = 2xy = x 2 respectively. Van Essen (U of A) Basic Math Concepts 51 / 66

52 Higher Order Partial Derivatives Suppose f : R N R, then the partial derivative n f x i is a function of N variables. Moreover, if this function is differentiable, then we can take partial derivatives of it as well. These are higher order derivatives. They tell us how the partial derivative function is changing with respect to different variables. Van Essen (U of A) Basic Math Concepts 52 / 66

53 Second Order Partial Derivatives We will mostly be concerned with first and second order derivatives of a function for a reason which we will learn about tomorrow-ish. A function of N variables has N first order derivatives and N 2 second order derivatives. Example Suppose f (l, k) = 4l 1 4 k 3 4, then the first order partial derivatives are f l = l 3 4 k 3 4 and f k = 3l 1 4 k 4 1 the second order partial derivatives of f are 2 f l 2 = 3 4 l k 4 2 f = 2 f l k = 3 4 l 3 4 k 1 4 k l 2 f k 2 = 3 4 l 1 4 k 4 5 Van Essen (U of A) Basic Math Concepts 53 / 66

54 Second Order Partial Derivatives In the example we saw that 2 f k l = 2 f l k, this was not a coincidence. Theorem (Young s Theorem) Suppose that y = f (x 1,..., x N ) is C 2 on an open region J in R N. Then for all x J and for each i,j we have that 2 f x i x j = 2 f x j x i Van Essen (U of A) Basic Math Concepts 54 / 66

55 Total Differential The partial derivative tells us how a function changes when we vary one variable and hold the others fixed. Sometimes we wish to know how a function changes when more than one variable is changed. In particular, suppose z = f (x, y) and x goes from x 1 to x 2 and y goes from y 1 to y 2, then z = f (x 2, y 2 ) f (x 1, y 1 ). We can approximate z by considering a plane that is tangent to f at (x 1, y 1 ). The changes on this plane are denoted the by total differential formula dz = f f dx + x y dy The total differential is an approximation for z when x changes by x and y changes by y i.e., z f f x + x y y Van Essen (U of A) Basic Math Concepts 55 / 66

56 Total Differential: Example Approximate via differentials. Well let then (4.1) 3 (2.95) 3 (1.02) 3 f (x, y, z) = x 3 y 3 z 3 df = f f f dx + dy + x y z dz Van Essen (U of A) Basic Math Concepts 56 / 66

57 Total Differential: Example df = f f f dx + dy + x y z dz Choose (x 1, y 1, z 1 ) = (4, 3, 1), then f x (x 1, y 1, z 1 ) = 4 f y (x 1, y 1, z 1 ) = 9 4 f z (x 1, y 1, z 1 ) = 1 4 The differential is df = (4)(0.1) + ( 9 ( 4 )( 0.05) + 1 ) (0.02) 4 = Van Essen (U of A) Basic Math Concepts 57 / 66

58 Total Differential: Example The differential is df = (4)(0.1) + ( 9 ( 4 )( 0.05) + 1 ) (0.02) 4 Thus, = f = f (x 2, y 2, z 2 ) f (x 1, y 1, z 1 ) Since f (x 1, y 1, z 1 ) = 6, we have f (4.1, 2.95, 1.02) The actual value is Van Essen (U of A) Basic Math Concepts 58 / 66

59 Another Differential Example What is d ln y d ln x? Well, Thus, d ln y = 1 y dy d ln x = 1 x dx d ln y d ln x = dy x dx y which is the price elasticity formula. Van Essen (U of A) Basic Math Concepts 59 / 66

60 Total Derivative/ Chain Rule How do you find the change of the function f (x, y) with respect to y when x and y are related. Use the total derivative. Suppose z = f (x, y) and x = g(y) First, find the total differential Next, divide both sides by dy dz = f f dx + x y dy dz dy = f dx x dy + f y This decomposes the change in y into a direct effect f y indirect effect f dx x dy. and an Van Essen (U of A) Basic Math Concepts 60 / 66

61 Total Derivative/ Chain Rule Example Suppose f (x, y) = x 2 y, where y = g(x) = 3x 2 Then the total derivative of f with respect to x is df dx = f x + f dy y dx = 2xy + x 2 (6x) = 6x 3 + 6x 3 = 12x 3 Note: that if we replaced y with g(x) and tool the derivative we would get the same answer i.e., d ( 3x 4 ) = 12x 3. dx Van Essen (U of A) Basic Math Concepts 61 / 66

62 Antiderivatives Definition An antiderivative of a function f (x) is a function F (x) whose derivative is the original i.e., d dx F = f. Definition The function F is also called the indefinite integral of f and written F (x) = f (x)dx Van Essen (U of A) Basic Math Concepts 62 / 66

63 Common Indefinite Integrals af (x)dx = a f (x)dx (f (x) + g(x)) dx = f (x)dx + g(x)dx x n dx = x n+1 n+1 + C 1 x dx = ln x + C e f (x ) f (x)dx = e f (x ) + C f (x ) dx = ln f (x) + C f (x ) Van Essen (U of A) Basic Math Concepts 63 / 66

64 Example ( x 2 + x ) dx = x 3 x 3 + x ln(x) + C Van Essen (U of A) Basic Math Concepts 64 / 66

65 Definite Integral For fixed numbers a and b, the definite integral of f (x) from a to b is F (b) F (a) where F is the anti-derivative of f. b a f (x)dx = F (b) F (a) Graphical Interpretation of a Definite Integral: Area under a curve. Useful when we talk about Consumer and Producer Surplus. Van Essen (U of A) Basic Math Concepts 65 / 66

66 Example 1 0 ( x 2 + x 3) dx = = [ x x 4 4 [ ] ] x =1 x =0 [ ] = 7 12 Van Essen (U of A) Basic Math Concepts 66 / 66

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

In economics, the amount of a good x demanded is a function of a person s wealth and the price of that good. In other words,

In economics, the amount of a good x demanded is a function of a person s wealth and the price of that good. In other words, LABOR NOTES, PART TWO: REVIEW OF MATH 2.1 Univariate calculus Given two sets X andy, a function is a rule that associates each member of X with exactly one member ofy. Intuitively, y is a function of x

More information

Items related to expected use of graphing technology appear in bold italics.

Items related to expected use of graphing technology appear in bold italics. - 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

More information

Main page. Given f ( x, y) = c we differentiate with respect to x so that

Main page. Given f ( x, y) = c we differentiate with respect to x so that Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

6.4 Logarithmic Equations and Inequalities

6.4 Logarithmic Equations and Inequalities 6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.

More information

2.7. The straight line. Introduction. Prerequisites. Learning Outcomes. Learning Style

2.7. The straight line. Introduction. Prerequisites. Learning Outcomes. Learning Style The straight line 2.7 Introduction Probably the most important function and graph that you will use are those associated with the straight line. A large number of relationships between engineering variables

More information

The data is more curved than linear, but not so much that linear regression is not an option. The line of best fit for linear regression is

The data is more curved than linear, but not so much that linear regression is not an option. The line of best fit for linear regression is CHAPTER 5 Nonlinear Models 1. Quadratic Function Models Consider the following table of the number of Americans that are over 100 in various years. Year Number (in 1000 s) 1994 50 1996 56 1998 65 2000

More information

Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.

Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson. Lecture 17 MATH10070 - Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx

More information

WEEK #16: Level Curves, Partial Derivatives

WEEK #16: Level Curves, Partial Derivatives WEEK #16: Level Curves, Partial Derivatives Goals: To learn how to use and interpret contour diagrams as a way of visualizing functions of two variables. To study linear functions of two variables. To

More information

Learning Objectives for Math 165

Learning Objectives for Math 165 Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

More information

1 Cobb-Douglas Functions

1 Cobb-Douglas Functions 1 Cobb-Douglas Functions Cobb-Douglas functions are used for both production functions Q = K β L (1 β) where Q is output, and K is capital and L is labor. The same functional form is also used for the

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

Mathematics for Business Economics. Herbert Hamers, Bob Kaper, John Kleppe

Mathematics for Business Economics. Herbert Hamers, Bob Kaper, John Kleppe Mathematics for Business Economics Herbert Hamers, Bob Kaper, John Kleppe Mathematics for Business Economics Herbert Hamers Bob Kaper John Kleppe You can obtain further information about this and other

More information

2.4 Motion and Integrals

2.4 Motion and Integrals 2 KINEMATICS 2.4 Motion and Integrals Name: 2.4 Motion and Integrals In the previous activity, you have seen that you can find instantaneous velocity by taking the time derivative of the position, and

More information

Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then

Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,

More information

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were: Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

More information

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4. MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

More information

f(x) = lim 2) = 2 2 = 0 (c) Provide a rough sketch of f(x). Be sure to include your scale, intercepts and label your axis.

f(x) = lim 2) = 2 2 = 0 (c) Provide a rough sketch of f(x). Be sure to include your scale, intercepts and label your axis. Math 16 - Final Exam Solutions - Fall 211 - Jaimos F Skriletz 1 Answer each of the following questions to the best of your ability. To receive full credit, answers must be supported by a sufficient amount

More information

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 110 Review for Final Examination 2012 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the correct graph. 1) y = -

More information

Linear and quadratic Taylor polynomials for functions of several variables.

Linear and quadratic Taylor polynomials for functions of several variables. ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

More information

Chapter 1 Quadratic Equations in One Unknown (I)

Chapter 1 Quadratic Equations in One Unknown (I) Tin Ka Ping Secondary School 015-016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real

More information

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

More information

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

More information

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) = Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

MATH301 Real Analysis Tutorial Note #3

MATH301 Real Analysis Tutorial Note #3 MATH301 Real Analysis Tutorial Note #3 More Differentiation in Vector-valued function: Last time, we learn how to check the differentiability of a given vector-valued function. Recall a function F: is

More information

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y) Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function

More information

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm. PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

More information

Extra Problems for Midterm 2

Extra Problems for Midterm 2 Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

More information

1 Calculus of Several Variables

1 Calculus of Several Variables 1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 300-31. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined

More information

Calculus Card Matching

Calculus Card Matching Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information

More information

Objectives. Materials

Objectives. Materials Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways

More information

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x) SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

More information

Limits and Continuity

Limits and Continuity Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

More information

Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

More information

x 2 + y 2 = 25 and try to solve for y in terms of x, we get 2 new equations y = 25 x 2 and y = 25 x 2.

x 2 + y 2 = 25 and try to solve for y in terms of x, we get 2 new equations y = 25 x 2 and y = 25 x 2. Lecture : Implicit differentiation For more on the graphs of functions vs. the graphs of general equations see Graphs of Functions under Algebra/Precalculus Review on the class webpage. For more on graphing

More information

The Derivative. Philippe B. Laval Kennesaw State University

The Derivative. Philippe B. Laval Kennesaw State University The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition

More information

Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

More information

Techniques of Differentiation Selected Problems. Matthew Staley

Techniques of Differentiation Selected Problems. Matthew Staley Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4

More information

The Simple Linear Regression Model: Specification and Estimation

The Simple Linear Regression Model: Specification and Estimation Chapter 3 The Simple Linear Regression Model: Specification and Estimation 3.1 An Economic Model Suppose that we are interested in studying the relationship between household income and expenditure on

More information

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

More information

Math 181 Spring 2007 HW 1 Corrected

Math 181 Spring 2007 HW 1 Corrected Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)

More information

Calculus 1: Sample Questions, Final Exam, Solutions

Calculus 1: Sample Questions, Final Exam, Solutions Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

Constrained optimization.

Constrained optimization. ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

More information

Inverse Functions and Logarithms

Inverse Functions and Logarithms Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that

More information

MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

More information

Differential Equations

Differential Equations Differential Equations A differential equation is an equation that contains an unknown function and one or more of its derivatives. Here are some examples: y = 1, y = x, y = xy y + 2y + y = 0 d 3 y dx

More information

Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if

Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if f(x)) f(x 0 ) x x 0 exists. If the it exists for all x 0 (a, b) then f is said

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

More information

MATHS WORKSHOPS. Functions. Business School

MATHS WORKSHOPS. Functions. Business School MATHS WORKSHOPS Functions Business School Outline Overview of Functions Quadratic Functions Exponential and Logarithmic Functions Summary and Conclusion Outline Overview of Functions Quadratic Functions

More information

2 Integrating Both Sides

2 Integrating Both Sides 2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

More information

Change of Continuous Random Variable

Change of Continuous Random Variable Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

Curriculum Map. Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein

Curriculum Map. Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein Curriculum Map Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein August/September: State: 8.B.5, 8.C.5, 8.D.5 What is a limit? What is a derivative? What role do derivatives and limits

More information

Multi-variable Calculus and Optimization

Multi-variable Calculus and Optimization Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus

More information

Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

Average rate of change of y = f(x) with respect to x as x changes from a to a + h: L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,

More information

An Introduction to Calculus. Jackie Nicholas

An Introduction to Calculus. Jackie Nicholas Mathematics Learning Centre An Introduction to Calculus Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 Some rules of differentiation and how to use them

More information

Official Math 112 Catalog Description

Official Math 112 Catalog Description Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A

More information

Blue Pelican Calculus First Semester

Blue Pelican Calculus First Semester Blue Pelican Calculus First Semester Teacher Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

5 Indefinite integral

5 Indefinite integral 5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7

Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7 About the Author v Preface to the Instructor xiii WileyPLUS xviii Acknowledgments xix Preface to the Student xxi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number

More information

1 Lecture 19: Implicit differentiation

1 Lecture 19: Implicit differentiation Lecture 9: Implicit differentiation. Outline The technique of implicit differentiation Tangent lines to a circle Examples.2 Implicit differentiation Suppose we have two quantities or variables x and y

More information

Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

More information

Numerical methods for finding the roots of a function

Numerical methods for finding the roots of a function Numerical methods for finding the roots of a function The roots of a function f (x) are defined as the values for which the value of the function becomes equal to zero. So, finding the roots of f (x) means

More information

Covariance and Correlation. Consider the joint probability distribution f XY (x, y).

Covariance and Correlation. Consider the joint probability distribution f XY (x, y). Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Section 5-2 Covariance and Correlation Consider the joint probability distribution f XY (x, y). Is there a relationship between X and Y? If so, what kind?

More information

2.2 Derivative as a Function

2.2 Derivative as a Function 2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x

More information

Math 103: Secants, Tangents and Derivatives

Math 103: Secants, Tangents and Derivatives Math 103: Secants, Tangents and Derivatives Ryan Blair University of Pennsylvania Thursday September 27, 2011 Ryan Blair (U Penn) Math 103: Secants, Tangents and Derivatives Thursday September 27, 2011

More information

100. In general, we can define this as if b x = a then x = log b

100. In general, we can define this as if b x = a then x = log b Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,

More information

Mathematical Economics: Lecture 15

Mathematical Economics: Lecture 15 Mathematical Economics: Lecture 15 Yu Ren WISE, Xiamen University November 19, 2012 Outline 1 Chapter 20: Homogeneous and Homothetic Functions New Section Chapter 20: Homogeneous and Homothetic Functions

More information

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate

More information

Clear & Understandable Math

Clear & Understandable Math Chapter 1: Basic Algebra (Review) This chapter reviews many of the fundamental algebra skills that students should have mastered in Algebra 1. Students are encouraged to take the time to go over these

More information

DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

More information

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

Midterm I Review:

Midterm I Review: Midterm I Review: 1.1 -.1 Monday, October 17, 016 1 1.1-1. Functions Definition 1.0.1. Functions A function is like a machine. There is an input (x) and an output (f(x)), where the output is designated

More information

Solving Logarithmic Equations

Solving Logarithmic Equations Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

More information

MATH 2300 review problems for Exam 3 ANSWERS

MATH 2300 review problems for Exam 3 ANSWERS MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

More information

Introduction to Calculus for Business and Economics. by Stephen J. Silver Department of Business Administration The Citadel

Introduction to Calculus for Business and Economics. by Stephen J. Silver Department of Business Administration The Citadel Introduction to Calculus for Business and Economics by Stephen J. Silver Department of Business Administration The Citadel I. Functions Introduction to Calculus for Business and Economics y = f(x) is a

More information

To give it a definition, an implicit function of x and y is simply any relationship that takes the form:

To give it a definition, an implicit function of x and y is simply any relationship that takes the form: 2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to

More information

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145: MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an

More information

INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

More information

Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

More information

The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result.

The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, do the function f to x, then do g to the result. 30 5.6 The chain rule The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result." Example. g(x) = x 2 and f(x) = (3x+1).

More information

Calculus 1st Semester Final Review

Calculus 1st Semester Final Review Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Exponential and Logarithmic Functions Exponential Functions Overview of Objectives, students should be able to: 1. Evaluate exponential functions. Main Overarching Questions: 1. How do you graph exponential

More information

Handout on Growth Rates

Handout on Growth Rates Economics 504 Chris Georges Handout on Growth Rates Discrete Time Analysis: All macroeconomic data are recorded for discrete periods of time (e.g., quarters, years). Consequently, it is often useful to

More information

Lecture 4: Equality Constrained Optimization. Tianxi Wang

Lecture 4: Equality Constrained Optimization. Tianxi Wang Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x

More information