Chapter 2

Forgotten but Not Gone: Review of Algebra and Pre-Calculus

IN THIS CHAPTER

Calculating with fractions and factorials

Working with exponents and simplifying rational expressions

Remembering radian measure

Proving trig identities

Understanding important parent functions and their transformations

Converting an infinite series from sigma notation to expanded notation

Most students have been studying math for at least 10 years before they enter their first calculus classroom. This fact leaves many students overwhelmed by all the math they should know, and perhaps did know at one time, but can’t quite recall.

Fortunately, you don’t need another 10 years of review to be ready for Calculus II. In this chapter, I get you back up to speed on the key topics from your Pre-Algebra, Algebra, and Pre-Calculus classes that will help you the most this semester.

To begin, you go all the way back to middle school for a quick review of fractions. I also give you some practice calculating factorials.

After that, I remind you how to work with exponents, and especially how to use negative and fractional exponents to express rational and radical functions. Then I cover a few important ideas from trigonometry that you’re sure to need, such as radian measure and trig identities.

Next, I give you an overview of how to sketch the most important parent functions on the xy-graph: polynomials, exponentials, radicals, logarithmic functions, and the sine and cosine functions. You use these to work with a variety of function transformations, such as vertical and horizontal transformations, as well as stretch, compress, and reflect transformations.

Fractions

When finding derivatives in Calculus I and integrals in Calculus II, you’ll often need to add 1 to (or subtract 1 from) a fraction. Here’s a trick for doing both of those operations quickly in your head without getting a common denominator:

Q. What is ?

A. . To do this calculation in your head, add the numerator and denominator, and then keep the denominator of 5:

Q. What is ?

A. . To calculate this value in your head, subtract the numerator minus the denominator, and then keep the denominator of 6.

1 Add 1 to the following fractions and express each answer as a proper or improper fraction (no mixed numbers).

2 Subtract 1 from the following fractions and express each answer as a proper or improper fraction (no mixed numbers).

Factorials

In Calculus II, when working with infinite series, you also may need to make use of factorials. Recall that the symbol for factorial is an exclamation point (!). The factorial of any positive integer is that number multiplied by every positive integer less than it. Thus:

Also, by definition, .

When you know how to expand factorials in this way, simplifying rational expressions that include them is relatively straightforward. Always look for opportunities to cancel factors in both the numerator and denominator.

Q. Simplify .

A. 10. Begin by expanding the factorials:

   Now, cancel factors in both the numerator and denominator, and simplify the result:

Q. Simplify .

A. . Expand the factorials as follows:

   Cancel factors in both the numerator and denominator, and simplify the result:

3 Simplify each of the following factorial expressions.

4 Simplify each expression in terms of n.

Negative and Fractional Exponents

When an expression has a negative exponent, you can rewrite it with a positive exponent and place it in the denominator of a fraction. For example:

When an expression has a fractional exponent, you can rewrite it as a radical. For example:

More complicated fractional exponents can be written in two separate and equally valid ways as a combination of a radical and an exponent. For example:

When a fractional exponent is negative, you can rewrite it as a radical in the denominator of a fraction:

Keep track of these types of conversions! In Calculus II, you’ll need to change expressions back and forth between these two forms just about every day.

Q. Rewrite the expression without using a negative fractional exponent.

A. . First, rewrite the expression with a positive exponent in the denominator of a fraction, then change this to radical form:

Q. Express using a negative fractional exponent.

A. . Begin by changing the radical to a fractional exponent, then change this value from negative to positive by bringing the expression out of the denominator:

5 Express each of the following without using negative or fractional exponents.

6 Express each of the following as a negative or fractional exponent of x.

7 Express each of the following using negative or fractional exponents of x.

Simplifying Rational Functions

Another key algebra skill you’ll need to recall is simplifying rational expressions by factoring and then canceling factors.

Q. Simplify the rational expression .

A. Always begin by factoring out the greatest common factor (GCF) in both the numerator and denominator, if possible, then simplify:

   Now, factor the numerator as a difference of squares and the denominator as a quadratic expression:

   Depending on the problem, you may also need to remove all parentheses:

8 Fully simplify each of the following rational functions by factoring, cancelling factors, and then removing all parentheses:

Trigonometry

In Calculus II, you’ll be working with trigonometric functions constantly. Most students are at least a little shy about their trig skills. Fortunately, you don’t have to remember every last trig fact you were ever tested on. Here’s a quick review of the skills you’ll need most to do well in your current class.

When you draw a right triangle and identify one of the two small angles as x, you can distinguish the three sides of the right triangle as the opposite (O), the adjacent (A), and the hypotenuse (H).

The first three trig functions (sine, cosine, and tangent) are all defined in terms of this angle x (recall the mnemonic SOH-CAH-TOA):

The three reciprocal trig functions (cosecant, secant, and cotangent) all follow from these definitions:

Given any trig function of x, you can deduce the other five functions by drawing a triangle, labeling it properly, and finding the remaining side with the Pythagorean Theorem.

Q. If , what is the value of tan x?

A. . Recall that , so draw a right triangle with angle x, an opposite side of 3, and a hypotenuse of 5. Now, use the Pythagorean Theorem to find the length of the adjacent side:

   Thus, the adjacent side has a length of 4, so .

In Calculus II, you’ll almost always use radian measure for angles instead of degrees. Here are the most common conversions from degrees to radians:...