Department of Mathematical Sciences,
Northern Illinois University

MATH 211 Course Prerequisites

CATALOG DESCRIPTION: Calculus for Business and Social Science (3). An elementary treatment of topics from differential and integral calculus, with applications in social science and business. Except with departmental approval students may not receive credit for both MATH 211 and MATH 229.

Prerequisite: MATH 110 or satisfactory performance on the Mathematics Placement Test.

Below is some detailed information about the algebra you should know before signing up for the course.

OVERVIEW OF CALCULUS: Many of the techniques of calculus were originally designed to help study the physics of moving objects, though now these basic principles can be applied in many different areas. The derivative acts like a speedometer, giving an instantaneous rate of change. It is helpful in graphing, and it can be used to find the maximum and minimum values of a function. The integral uses an averaging process to compute the total output (when only the rate of output is known).

Functions: The techniques of calculus are designed to apply to functions. For example, before it is possible to decide on a strategy to maximize profit, it is necessary to express the profit as a function of the production level. You need to have a good knowledge of functions and their graphs. Here are some sample problems.

1. If f(x) = x2 - 2x - 3, then f(-2) =
(a) 14
(b) 5
(c) 3
(d) -3
(e) -6

2. If f(x) = x2 - 2x, then f(x+h) =
(a) x2 + h2 -2x
(b) x2 + h2 -2x -2h
(c) x2 + h2 -2x +2h
(d) x2 + 2xh + h2 -2x -2h
(e) x2 + 2xh + h2 -2x +2h

3. If f(x) = 3x + 1 and g(x) = x2, then f(g(x)) =
(a) 9x2 + 1
(b) 9x2 + 6x + 1
(c) 3x2 + 1
(d) 3x2 + 3x + 1
(e) None of the above

Graphing: In Math 211 you will learn how to apply techniques of calculus to help you understand the shape of a graph, but you should have had some previous experience with graphing. You should already be familiar with the graphs of functions of the form
f(x) = mx + b (representing a straight line), and
f(x) = ax2 + bx + c (representing a parabola).

Calculators: You may have learned algebra with the help of a graphing calculator. If you depended too much on the calculator, you may not have learned important basic principles.

If you answered no to any of these questions, you will probably need to do more review than most other students.

Exponents: You need to be confident about using the basic rules
am+n = aman and (am)n = amn .
You also need to be able to work with negative and fractional exponents. Try these problems to check your skill level.

4. 100-2 =
(a) 1000
(b) 1
(c) .01
(d) .0001
(e) None of the above

5. 84/3 =
(a) 16
(b) 32
(c) 32/3
(d) 4
(e) None of the above

6. (1/8)-2/3 =
(a) 1/12
(b) 1/4
(c) 1/2
(d) 2
(e) None of the above

7. x4(y2/x3)2 =
(a) x2y4
(b) y4 / x2
(c) y2 / x2
(d) x2 / y2
(e) None of the above

Exponential and logarithmic functions: Chapter 4 and Chapter 5 of the text deal with exponential and logarithmic functions. These functions are important in describing the growth of many natural processes.
The course spends about 3 weeks on this material, and most of the questions on the third hour test come from these sections.
If you have never studied exponential and logarithmic functions, you will find it almost impossible to learn the algebraic facts at the same time you are learning how calculus uses these functions.

8. Solve for x: e5x = e20 .
(a) x = ln 4
(b) x = 4
(c) x = 15
(d) x = 100
(e) None of the above

9. Simplify: ln ( x2 / (3-x)3 ) =
(a) 2x - 3(3-x)
(b) 2x + 3(3-x)
(c) 2 ln x - 3 ln (3-x)
(d) 2 ln x + 3 ln (3-x)
(e) None of the above

Areas and volumes: Do you remember these formulas?

10. A room to be carpeted measures 12 feet wide by 15 feet long. The carpeting costs $5 per square yard, and the installation fee is $50. What is the total cost?
(a) $150
(b) $350
(c) $650
(d) $950
(e) None of the above

Simplifying complicated expressions: Some of the calculus problems require a lot of algebra. Here are some typical problems from the textbook.

       1        1
11.  _____  -  ___  =

     x + h      x

12. (1/2)(1 - x2)-1/2(-2x) =

13. (x2 - 1)45(x2 + 1)4(2x) + (x2 + 1)54(x2 - 1)3(2x) =

Factoring: You should know how to factor polynomials of degree 2. This is important not just in solving equations but in computing limits of functions.
An important fact is that the number c is a root of the polynomial equation f(x)=0 precisely when x-c is a factor of f(x). For example, in computing the limit


         x2  + x - 6
  lim   _____________
  x->2    
         x2  - 6x + 8
the first step is to substitute x=2 in both the numerator and denominator, to get
        
 2 + 22 - 6      4 + 2 - 6    0
 _____________ = __________ = _  .
   
 22 - 6(2) + 8   4 - 12 + 8   0
This shows that both the numerator and denominator have x-2 as a factor, and then it is easy to find the other factors.
        
      x2 + x - 6         (x - 2)(x + 3)        (x + 3)
lim  ____________ = lim  ______________ = lim  _______
x->2                x->2                  x->2
      x2 - 6x + 8        (x - 2)(x - 4)        (x - 4)
Here are some factoring problems for you to try.

14. Factor x2 - 9.

15. Factor 2x2 + 7x - 15 .

16. Factor x5 + 2x4 + x3 .

Answers are given below.





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Answers:

1 (b); 2 (d); 3 (c); 4 (d); 5 (a); 6 (e); 7 (b); 8 (b); 9 (c); 10 (a)
11 -h/(x2+hx); 12 -x/(1-x2)1/2;
13 2x(x2-1)3(x2+1)4(9x2-1) (see p 251); 14 (x-3)(x+3); 15 (2x-3)(x+5); 16 x3(x+1)2


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