Northern Illinois University

**
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.**

- If you took Math 110 at NIU and got at least a solid C, then you should have an adequate background for Math 211.
- Try the sample problems as you are reading (answers are given at the end of this page).
- If you cannot answer the sample questions, you should consider taking Math 110, College Algebra, rather than enrolling in Math 211.
- If you are already registered for Math 211, but think this placement may be incorrect, you can get additional information from the departmental OnLine Placement Test.
- If you have any questions about placement, you can call the NIU Math Office at (815) 753-0566.

**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) = x^{2} - 2x - 3, then f(-2) =

(a) 14

(b) 5

(c) 3

(d) -3

(e) -6

**2.**
If f(x) = x^{2} - 2x, then f(x+h) =

(a) x^{2} + h^{2} -2x

(b) x^{2} + h^{2} -2x -2h

(c) x^{2} + h^{2} -2x +2h

(d) x^{2} + 2xh + h^{2} -2x -2h

(e) x^{2} + 2xh + h^{2} -2x +2h

**3.**
If f(x) = 3x + 1 and g(x) = x^{2}, then f(g(x)) =

(a) 9x^{2} + 1

(b) 9x^{2} + 6x + 1

(c) 3x^{2} + 1

(d) 3x^{2} + 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) = ax^{2} + 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.

- Can you divide 25.6 by 100 without using your calculator?
- Can you tell whether the line y = -3x +2 slopes up or down, without using your calculator?
- Can you graph y = x
^{2}+ 3 without using your calculator?

**Exponents:**
You need to be confident about using the basic rules

a^{m+n} = a^{m}a^{n} and
(a^{m})^{n} = a^{mn} .

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.**
8^{4/3} =

(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.**
x^{4}(y^{2}/x^{3})^{2} =

(a) x^{2}y^{4}

(b) y^{4} / x^{2}

(c) y^{2} / x^{2}

(d) x^{2} / y^{2}

(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: e^{5x} = e^{20} .

(a) x = ln 4

(b) x = 4

(c) x = 15

(d) x = 100

(e) None of the above

**9.**
Simplify: ln ( x^{2} / (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?

- The area of a rectangle is the length times the width.
- The volume of a box is the length times the width times the height.
- The area of a circle is pi times the square of the radius.

**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.

1 111._____ - ___ = x + h x

**12.**
(1/2)(1 - x^{2})^{-1/2}(-2x) =

**13.**
(x^{2} - 1)^{4}5(x^{2} + 1)^{4}(2x) +
(x^{2} + 1)^{5}4(x^{2} - 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

xthe first step is to substitute x=2 in both the numerator and denominator, to get^{2}+ x - 6 lim _____________ x->2 x^{2}- 6x + 8

2 + 2This shows that both the numerator and denominator have x-2 as a factor, and then it is easy to find the other factors.^{2}- 6 4 + 2 - 6 0 _____________ = __________ = _ . 2^{2}- 6(2) + 8 4 - 12 + 8 0

xHere are some factoring problems for you to try.^{2}+ x - 6 (x - 2)(x + 3) (x + 3) lim ____________ = lim ______________ = lim _______ x->2 x->2 x->2 x^{2}- 6x + 8 (x - 2)(x - 4) (x - 4)

**14.**
Factor x^{2} - 9.

**15.**
Factor 2x^{2} + 7x - 15 .

**16.**
Factor x^{5} + 2x^{4} + x^{3} .

Answers are given below.

| | | \ / | | | \ / | | | \ / | | | \ /

**1** (b);
**2** (d);
**3** (c);
**4** (d);
**5** (a);
**6** (e);
**7** (b);
**8** (b);
**9** (c);
**10** (a)

**11**
-h/(x^{2}+hx);
**12**
-x/(1-x^{2})^{1/2};

**13**
2x(x^{2}-1)^{3}(x^{2}+1)^{4}(9x^{2}-1)
(see p 251);
**14** (x-3)(x+3);
**15** (2x-3)(x+5);
**16** x^{3}(x+1)^{2}