The following list of objectives is designed to give you some guidance at what topics are important from the various sections we will cover from the book.  These objectives are some of the key concepts that will be discussed for each section.  When you take exams in this course you must show me your work to earn full credit for a problem.  Correct answers with no work shown will be penalized. Partial credit will be given on problems therefore showing your work is vital to your grade and it also allows me to better determine where your weakness may be if you need assistance with any of the material.

MATH221 CHAPTER 0 OBJECTIVES

SECTION 0.1 Functions and Their Graphs
1)  Understand interval notation
2) Understand function notation
3) Determine the domain of functions
4) Determine whether a graph is the graph of a function

SECTION 0.2 Some Important Functions
1) Graph various functions including: linear, quadratic, cubics, 1/x, greatest
    integer, and piecewise functions.

              Your graphs should adhere to the following guidelines:

• The x and y axis is to be labeled (or whatever variables are used)
• A scale is to be included even if you mean for each block to count as one unit
•  Linear equations should be determined by at least three points (the third being a double check of your work).
•  Quadratics should include a vertex, axis of symmetry, y intercept, x intercepts, and have at least five points on the graph
•  Otherwise plot a sufficient number of key points to determine the appearance of the graph

SECTION 0.4 Zeros of Functions – The Quadratic Formula and Factoring
1) Be able to factor polynomials including knowing how to factor sums or
      differences of squares and cubes, and to factor by grouping
2) Be able to determine the point(s) of intersection of pairs of curves (Note: a point
      of intersection must be an ordered pair)

SECTION 0.5 Exponents and Power Functions
1) Use the laws of exponents to simplify expressions (Note: there is often more than
      one way the rules can be used to simplify expressions.  You should make clear
      which rules you are using to go from step to step and I should be able to follow
      your steps)

SECTION 0.6 Functions and Graphs in Applications
1) Set up equations for simple word problems (DO NOT SOLVE).  These include
      knowing the area and circumference of a circle, the area and perimeter for
      rectangles and triangles,  the Pythagorean theorem, and the volume and surface
      area of a rectangular box.
 
 

MATH221 CHAPTER 1 OBJECTIVES

SECTION 1.1 The Slope of a Straight Line
1) Be able to find equations of lines given two points on the line, given a point and the slope of the line, given a slope and y intercept, or given the line is parallel or perpendicular to another line. (Note we use the terms undefined slope for a vertical line and zero slope for a horizontal line)
2) Understand the various forms of a line including: standard form, point slope form, and slope intercept form.

SECTION 1.2 The Slope of a Curve at a Point
1) Use the fact that the derivative is the slope of the tangent line to the curve at a given point to help determine the derivatives of simple linear functions.
2) Use the derivative of x² is 2x and of x³ is 3x² to find the equation of the tangent line to these curves at some point (x, y)
Note: Because the derivative gives the slope and you will already know a particular x value to the curve( and thus the ordered pair for the point), I suggest you leave your equations for lines in point slope form for this course.

SECTION 1.3 The Derivative
1) Use the power rule to find the derivative of x raised to some power.
2) Rewrite radicals as expressions with fractional exponents and denominators as negative exponents to apply the “power rule”.
3) Use difference quotients to calculate derivatives the “long way”.

SECTION 1.4 Limits and the Derivative
1) Evaluate limits of polynomials
2) Evaluate limits (if possible) of indeterminate form (0/0) by factoring and reducing expressions
3) Evaluate limits involving infinity (This procedure is related to the precalculus topic of finding the horizontal asymptote of a rational expression )

SECTION 1.5 Differentiability and Continuity
1) Determine whether the function whose graph is shown is differentiable or continuous at a particular value of x
2) Determine whether the equation of a function given is differentiable or continuous at a particular value of x

SECTION 1.6 Some Rules for Differentiation
1) Use the constant multiple rule for differentiation
2) Use the sum rule for differentiation
3) Use the general power rule for differentiation
4) Find the equation of the tangent line to a curve at a particular point

SECTION 1.7 More About Derivatives
1) Determine the second derivatives of functions
2) Evaluate derivatives at a particular x value.
3) Understand the prime and double prime notation
4) Understand derivatives using variables other than x and y
 

MATH221 CHAPTER 2  OBJECTIVES

SECTION 2.1 Describing Graphs of Functions
1) Given a graph you should be able to determine all types of extrema and any inflection points
2) You should be able to determine the intervals where a graph is increasing, decreasing, concave upward, and concave downward
3) You should be able to locate the x and y intercepts, any undefined points, and any asymptotes.

SECTION 2.2 The First and Second Derivative Rules
1) When the first derivative is positive a graph is increasing, when it is negative it is decreasing, and when it is zero it is a possible extrema.  You should be able to use a critical point table to determine the intervals where a given graph is increasing or decreasing and to locate extrema.
2) When the second derivative is positive the graph is concave upward, when the second derivative is negative the graph is concave downward, and when there is a switch in sign there is an inflection point. You should be able to determine this information from a graph.
3) Give a rough sketch of a graph given some properties of the function
4) Interpret the signs of the derivatives from a given graph.

SECTION 2.3 Curve Sketching (Introduction)
1) You should be able to make use of the information from the previous section to sketch a second or third degree polynomial.
2) You should be able to determine intercepts for a graph from the equation.
 Do not forget the properties of polynomials that you learned in Precalculus.  Polynomials are smooth unbroken curves with at most n-1 turning points and goes off to ± infinity at the extremes

SECTION 2.4 Curve Sketching (Conclusion)
1) You should be able to apply the techniques from the previous section to graph a fourth degree polynomial or higher
2) You should determine if there is any symmetry to aid in the graphing process.
3) You should be able to determine asymptotes for rational expressions (we will not go into these graphs in much detail)

SECTION 2.5 Optimization Problems
1) Optimization problems should be checked to verify you are at the minimum or maximum by making sure the first derivative switches sign as it should.
2) Draw a picture if possible to help you determine the objective function (What you are asked to minimize or maximize) and assign variables
3) You should determine the constraint, which will allow you to find a relationship between variables so the objective can be written in terms of a single variable.

SECTION 2.6 Further Optimization Problems
1) Use the fact that revenue is xp (demand times price) to help solve some of these problems
2) Use the fact that marginal means derivative for our purposes to find the marginal revenue or marginal cost.

SECTION 2.7 Applications of Derivatives to Business and Economics
1) Use the fact that Profit = Revenue – Cost to aid in solving these problems. I will combine several concepts into one larger problem that you will have to solve.
2) You will be given a problem of the form:  Suppose the consumer demand for a certain item as a function of its price p is given x = D(p)  = ______. Determine the production level and price that maximizes the profit if the cost output function is given by C(x) = ______.

I suggest will do this by performing the following calculations
                           Find the restrictions on x and p
                           Solve for p in terms of x
                           Use revenue is price times demand to find R(x)
                           Use profit Pr(x) = R(x) – C(x)
                            Set the profit derivative equal to zero
                            Evaluate profit at the endpoints(restrictions on x determined earlier) and at any
                                valid x value where the derivative was found to be zero. Use the best option.
                            Finally state: Produce __ units at $___ each for a maximum profit of $___
 
 

MATH221 CHAPTER 3 OBJECTIVES

SECTION 3.1  The Product and Quotient Rules
1) Be able to use the product or quotient rule to differentiate
2)  Power rules still apply and combination problems such as powers that involve a quotient, a quotient that involves a product will be asked of you to differentiate.
3) Find the equation of the tangent line to these more complex functions at a particular point.

You will be asked to leave the derivatives after the first step, meaning you will not be asked to simplify the expressions.  The algebra involved in simplifying can take some time and is of secondary importance here.  My concern is that you understand how to find the derivatives.

SECTION 3.2  The Chain Rule and General Power Rule
1) Be able to work with composite functions
2) Use the chain rule for composite functions
3) Use the chain rule dy/dx = dy/du * du/dx

SECTION 3.3   Implicit Differentiation and Related Rates
1) Find dy/dx implicitly (y is some unknown function of x)
2) Solve a simple related rate problem

To find dy/dx implicitly you will need to differentiate each term with respect to x treating y as a function of x.  Then you will need to isolate all terms involving dy/dx and factor out the dy/dx and divide by the coefficient to solve for dy/dx.
 

MATH221 CHAPTER 4 OBJECTIVES

SECTION 4.1  Exponential Functions
1) Be able to work with an exponential function
2) Be able to work with the laws of exponents
3) Rewrite functions in the form 2^kx or 3^kx as appropriate
4) Use a^x  = b^x implies a = b and a^x  = a^y implies x = y to work with exponential equations.

SECTION 4.2  The Exponential Function e^x
1) Be able to work with the exponential function e^x
2) Be able to take derivative of e^x
3) Solve exponential equations.

SECTION 4.3  Differentiation of Exponential Functions
1) Find derivatives of e^g(x)
2) Work with a simple differential equation y’ = ky

SECTION 4.4   The Natural Logarithm Function
1) Work with properties of logarithms
2) Work with natural logarithms

SECTION 4.5  The Derivative of lnx
1) Take derivatives of lnx
2) Take derivatives of ln(g(x))

SECTION 4.6  Properties of the Natural Logarithm Function
1) Work with the properties of natural logarithms
                         ln(xy) = lnx + lny
                         ln(x/y) = lnx – lny
                         ln(x^r) = rlnx
2) Use properties of logarithms to simplify sums or differences of logarithms

You should also be able to solve logarithmic equations involving other bases. We will cover this in class but it is not in your book.  You should remember the domains of logarithms are positive real numbers therefore watch out for extraneous solutions when dealing with logarithmic equations.
 

MATH221 CHAPTER 5 OBJECTIVES

SECTION 5.2  Compound Interest
1) Work with simple interest
2)  Work with compounded interest (n times a year)
3) Work with continuous compounding
4) Determine the total amount or the interest accumulated with any of the types of equations dealing with simple interest or compounded interest.

SECTION 5.3  Applications of the Natural Logarithm Function to Economics
1) Be able to determine the percentage rate of change
 

MATH221 CHAPTER 6 OBJECTIVES

SECTION 6.1  Antidifferentiation
1) Be able to integrate x^r
2) Be able to integrate e^kx
3) Be able to integrate 1/x
4) Remember that indefinite integration requires +C

SECTION 6.2  Areas and Riemann Sums
1) Determine the left endpoints, right endpoints or midpoints of subintervals.
2) Use Riemann sums with left endpoints, right endpoints or midpoints to approximate the area under the graph of f(x) on the given interval with selected points as specified.

SECTION 6.3  Definite Integrals and the Fundamental Theorem
1) Calculate definite integrals that may involve logarithms, exponentials, polynomials, and powers by using the Fundamental Theorem of Calculus.

SECTION 6.4  Areas in the xy-Plane
1) Find areas in the xy-plane by using a graph to help you find the x values of the points of intersection and to determine which curve is on top in a given interval.
2) Algebraically find the x values of the points of intersection [by setting f(x) = g(x)] and to determine which curve is on top in a given interval.

When integrating always use the top curve  minus the bottom curve within the interval to guarantee a positive value upon integrating.  This serves as a partial check of your work  Otherwise if you get a negative and the work is correct you would use the absolute value of the result as the solution to the problem.

SECTION 6.5  Application of the Definite Integral
1) Find the average value of a continuous function over an interval.
2) Find the consumer or producer surplus for a commodity having a given demand curve.
 

MATH221 CHAPTER 7 OBJECTIVES

SECTION 7.1  Examples of Functions of Several Variables
1)  Evaluate a function of several variables at a specific point.
2)  Be able to draw level curves.

SECTION 7.2  Partial Derivatives
1) Find  first partial derivatives of multivariable functions
2) Find first partial derivatives at a specific point
3) Find the second partial derivatives of multivariable functions.

SECTION 7.3  Maxima and Minima of Functions of Several Variables
1) Use the second derivative test for a function of two variables to find any local extrema of the bivariate function.

Note: This means you will set the first partial derivatives to equal zero.  Therefore you will need to be comfortable with solving systems of equations which may not involve linear functions.
 

MATH221 CHAPTER 9 OBJECTIVES

SECTION 9.1  Integration by Substitution
1) Be able to integrate using u substitution.  If u substitution is appropriate the integration becomes one of our three basic types, e^x, 1/x, or x^r.  This means try using u as the exponent if e is involved, the denominator if a fraction is involved , or as what is being raised to a power if powers are involved (higher power)

SECTION 9.2  Integration by Parts
1) When u substitution is not appropriate, you will need to know how to integrate by parts.  If e^x is involved it is g prime, if lnx is involved it is f(x) otherwise choose f(x) so that it has a “nicer” derivative.

SECTION 9.3  Evaluation of Definite Integrals
1) Evaluate definite integrals involving u substitution or parts.

SECTION 9.6  Improper Integrals
1) Evaluate improper integrals using limits.