Mt 217 Accelerated Calculus - Fall 2003

Department of Mathematics & Computer Science
CARDINAL STRITCH UNIVERSITY
Sr. Barbara E. Reynolds, Ph.D.

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Study Guide for Final Exam

This test will focus on your conceptual understanding of several major themes which have been part of our work this semester. Although you can expect to do some computation on this test, the questions will focus more on your conceptual understanding of these major themes than on your computation skill.

The major themes of this course are

mathematical tools to study problem situations,
functions,
limits,
difference quotients and derivatives,
Riemann sums, antiderivatives, and definite integrals,
sequences and series,
some useful algebraic and computational strategies,
reading and interpreting material as presented in the textbook.

Using mathematics to study problem situations

You can expect the exam to have some word problems for which you will have to use the ideas of calculus to solve the problem.

Given a situation expressed as a word problem, you should be able to:

make a sketch which illustrates the situation,
identify variables in the situation, and give an appropriate domain,
describe (using English words or mathematical notation) a process for solving the problem,
sketch a graph showing the relationship between two of the variables in the situation,
set up a function using appropriate mathematical notation to represent the situation,
set up a difference quotient to represent a rate of change,
set up a Riemann sum to represent a total of some kind (e.g., a total profit if you are given a profit function),
represent the problem situation as a sequence or as a sequence of partial sums (i.e., as a series), then determine whether the sequence or series converges or diverges to solve the problem,
use the tools of calculus -- that is, the function, the graph of the function, a difference quotient, the derivative, a Riemann sum, the definite integral -- to answer questions about the problem situation.

Functions

Calculus is a study of functions. You should be able to demonstrate some understanding of the idea of function -- a process which accepts inputs and returns outputs. For example, a function, its derivative, and its antiderivative are three different and related functions. The derivative itself is a process which takes in a function, calculates a difference quotient, and returns a function. The Fundamental Theorem of Calculus says that the definite integral is a function which takes in a function, and returns an "area" associated with each value of x in the domain of the function.

You should be able to:

construct functions from problem situations,
In setting up a solution for a word problem, it is often helpful to think about the problem in terms of a function which will take some inputs from the problem situation, and -- through the process defined in the function -- convert those inputs into outputs required for the solution.
represent functions in various ways -- graphs, expressions, tables, verbally, and sequences,
work with functions, which are presented in any of these forms,
work with a function, which is presented in parametric form,
sketch a complete graph of a given function.
In constructing a complete graph, you have to think about properties of functions -- x- and y-intercepts, any local maxima and minima, asymptotes, regions where the graph is increasing or decreasing, the concavity of the curve, the general shape of the graph. You should be able to sketch a graph of a function which clearly shows:
- the general shape of the graph,
- the x and y intercepts,
- the zeros of the function,
- the regions where the function is positive or negative,
- the regions where the function is increasing or decreasing,
- the regions where the slope of the function is positive or negative,
- the behavior of the function as x -> a,
  Does the graph of the function have a "hole" or a vertical asymptote where x = a? Is the function continuous at this point? Does the function have a jump discontinuity at this point?
- the behavior of the function as its variable "goes to infinity",
  What is the ultimate value of the function f(x) as x gets very large? Does the function have any horizontal or oblique asymptotes? Does the function have an asymptotic curve? As the variable x goes to infinity, does the graph of the function come "down" to the asymptote from above, or go "up" to the asymptote from below?
- an appropriate domain and range for the function.

Understanding and Calculating Limits

You should be able to:

calculate limits for a given function,
calculate the limit of a given sequence,
calculate the limit of a function at a finite value, x = a,
Are the left- and right-limits as x -> a the same? (If not, the two-sided limit is not defined.)

You should be able to distinguish between the value of the function at x = a, that is, f(a), and the limit of the function as x -> a.

You should be able to find the location of any vertical asymptotes of a given function, and you should be able to determine whether the function is going to + or - infinity as x --> a.

If x is not defined at x = a, you should be able to determine whether the function has a removable discontinuity, a jump discontinuity, or an infinity discontinuity as x --> a.
calculate the limit of a function or a sequence as x -> infinity,
You should be able to determine whether the graph of the function approaches a horizontal asymptote, an oblique asymptote, or an asymptotic curve as x --> +infinity or -infinity.
evaluate limits using both graphical and algebraic strategies,
use limits in problem situations.

Difference Quotients and Derivatives

You should be able to:

set up and simplify a difference quotient to prove the basic rules of differentiation,
construct a difference quotient to represent a rate of change in a problem situation,
sketch a graph of the derivative of a function, D(f), given a sketch graph of the function f,
use the difference quotient as a problem-solving strategy.

Riemann sums, Antiderivatives, and Definite Integrals

You should be able to perform and discuss the two-step process which is used to set up a Riemann sum; that is, you should be able to construct a partition on a given interval, and then set up a Riemann sum (using either left or right sums).

You should be able to:

set up a simple Riemann sum (e.g., using left sums for a simple function over a specified interval for a small number of subdivisions),
use the process of constructing a Riemann sum to find an approximate total or sum in a given problem situation,
sketch a graph to show (geometrically) what is being calculated when a Riemann sum (using left or right sums) is applied to f,
use the idea which underlies the construction of a Riemann sum to evaluate a definite integral using geometry (instead of finding an antiderivative),
sketch a graph of a function, f, given a sketch graph of its derivative, D(f).

Sequences and Series

You should be able to:

determine whether a given sequence converges or diverges -- and if it converges, you should be able to calculate its limit;
discuss the long-term behavior of sequences --that is, the limit of the sequence as n goes to infinity;
Does the sequence converge to zero? ... converge to a finite non-zero limit? ... oscillate between two different limits? ... become unbounded? ...
recognize and identify those series which have special names: geometric, harmonic, telescoping, p-series, alternating series;
use the n^th-term test to determine whether a given series definitely diverges -- or might converge;
determine whether (or not) a given series converges;
If a given series is a geometric series, how do you determine whether or not it converges? ... How do you find its sum if it does converge?
If a given series is a p-series, how do you determine whether it converges or diverges?
If a given series is a telescoping series, how do you determine whether it converges? ... and how do you find its sum?
What does the n^th-term test tell you if the series is an alternating series? How does this differ from the situation for a series of all positive terms?
construct a Taylor polynomial for a given function around a given point.

Algebraic and Computational Strategies

You should be able to:

use partial fraction decomposition to express a rational function as a sum of two simpler rational functions,
set up a trigonometric substitution to "simplify" an expression involving radicals, such as sqrt(a^2 + b^2),
do an integration by parts.

Reading your calculus textbook -- Special theorems of calculus

One of the objectives of this course is that you become better at reading and interpreting mathematics and technical information. We have taken time is several class periods to read some sections of the text, and to try to interpret what the text is saying. There are a number of special named theorems in calculus, and you should be able to read these theorems and apply them in a problem situation. In particular, you should be able to read and apply the following special theorems:

the Intermediate Value Theorem,
the Mean Value Theorem, and
the Fundamental Theorem of Calculus.

Functions represented in sketch graphs:

Given a graphical representation showing three different (but related) functions, you should be able to determine which is the original function, f(x), which is its derivative, f'(x), and which is a representative of the family of antiderivatives, int(f(x), x), for the original function.

You should be able to sketch a graph showing the important features of any of the curves in the list below (where a, b, and c denote constants). You should be able to recognize or identify the basic shapes of these curves.

`y = a * x + b`	`y = a * sin(x) + b`	`y = a * \|x\| + b`
`y = a * x^2 + b`	`y = a * cos(x) + b`	`y = \|x + c\|`
`y = a * x^3 + b`	`y = a * tan(x) + b`	`y = \|sin(x)\| + b`
`y = a / x`	`y = a / (x + c)`	`y = (a/x) + b`
`y = a / x^2`	`y = a / (x^2 + c)`	`y = (a/x^2) + b`
`y = sqrt(x)`	`y = sqrt(x + c)`	`y = sqrt(x) + b`
`y = 2^x`	`y = 2^(-x)`	`y = e^x`
`x^2 + y^2 = r^2`	`y = (x + a) (x + b) (x + c)`	`y = ln(x)`

Notes

This test will have six to eight problems. These problems will focus on your conceptual understanding of the material we have been studying this semester. Although you will be expected to do some computations, the questions will focus on your understanding of the underlying principles of calculus.

You may use your calculator and/or Maple during the exam. However, the focus on the exam questions will be on your conceptual understanding. If you find yourself needing to do a lot of computations, you may have missed the point of the question.

You should bring your textbook to this exam. One of the expectations I have at this point in Calculus is that you are able to read and work with material that is presented in your textbook. There will be at least one question where you will be asked to work with some material that we have not explicitly covered in class.

Consequently, this will be an open textbook exam.

Because of the schedule during final exam week, we must adhere to the two hour time limit for the exam. The exam is scheduled from 8 - 10 a.m., and you must turn in your paper no later than 10:10.

There is no Phase 2 on the Final Exam.

This test will contribute 20% to your grade for the course. If you do well on this exam, it could help to improve your overall course grade.

Return to Sr. Barbara E. Reynolds Home Page.
Return to course list for 2003 -- 2004.
Go to Mt 217 Accelerated Calculus Syllabus.
Return to the Mt 217 Assignments page.

The easiest way to contact me is to send an email message to Sr. Barbara E. Reynolds.

This page was updated on December 3, 2003.