Calculus For Dummies, 2nd Edition (2014)

Part II. Warming Up with Calculus Prerequisites

IN THIS PART …

Algebra review: Richard Feynman, the great 20th century physicist, said (tongue-in-cheek) that calculus was the language that God spoke. Well, I don’t know about that, but I do know that algebra is the language of calculus. If you want to learn calculus, you’ve got to know your algebra.

Logarithm review: What’s log 10? And ln 1? Hint for the first one: It’s the loneliest number. Hint for the second: There’s nothing to it.

Function review: Even and odd functions, exponential functions, inverse functions, function transformations, and so on.

Some trig: The all-important unit circle. And the related geometry of the 30°-60°-90° and 45°-45°-90° triangles.

More trig: SohCahToa and the graphs of sine, cosine, and tangent.

Chapter 4. Pre-Algebra and Algebra Review

IN THIS CHAPTER

Winning the fraction battle: Divide and conquer

Boosting your powers and getting to the root of roots

Laying down the laws of logarithms and having fun with factoring

Hanging around the quad solving quadratics

Algebra is the language of calculus. You can’t do calculus without knowing algebra any more than you can write Chinese poetry without knowing Chinese. So, if your pre-algebra and algebra are a bit rusty — you know, all those rules for algebraic expressions, equations, fractions, powers, roots, logs, factoring, quadratics, and so on — make sure you review the following basics.

Fine-Tuning Your Fractions

Open a calculus book to any random page and you’ll very likely see a fraction — you can’t escape them. Dealing with them requires that you know a few rules.

Some quick rules

First is a rule that’s simple but very important because it comes up time and time again in the study of calculus:

You can’t divide by zero! The denominator of a fraction can never equal zero.

equals zero, but is undefined.

It’s easy to see why is undefined when you consider how division works:

This tells you, of course, that 2 goes into 8 four times; in other words, . Well, how many zeros would you need to add up to make 5? You can’t do it, and so you can’t divide 5 (or any other number) by zero.

Here’s another quick rule.

Definition of reciprocal: The reciprocal of a number or expression is its multiplicative inverse — which is a fancy way of saying that the product of something and its reciprocal is 1. To get the reciprocal of a fraction, flip it upside down. Thus, the reciprocal of is , the reciprocal of 6, which equals , is , and the reciprocal of is .

Multiplying fractions

Adding is usually easier than multiplying, but with fractions, the reverse is true — so I want to deal with multiplication first.

Multiplying fractions is a snap — just multiply straight across the top and straight across the bottom:

Dividing fractions

Dividing fractions has one additional step: You flip the second fraction and then multiply — like this:

Note that you could have canceled before multiplying. Because 5 goes into 5 one time, and 5 goes into 10 two times, you can cancel a 5:

Also note that the original problem could have been written as .

Adding fractions

You know that

You can add these up like this because you already have a common denominator. It works the same with variables:

Notice that wherever you have a 2 in the top equation, an a is in the bottom equation; wherever a 3 is in the top equation, a b is in the bottom equation; and ditto for 7 and c. This illustrates a powerful principle:

Variables always behave exactly like numbers.

If you’re wondering what to do with variables in a problem, ask yourself how you would do the problem if there were numbers in it instead of variables. Then do the problem with the variables the same way, like this:

You can’t add these fractions like you did in the previous example because this problem has no common denominator. Now, assuming you’re stumped, do the problem with numbers instead of variables. Remember how to add ? I’m not going to simplify each line of the solution. You’ll see why in a minute.

1. Find the least common denominator (actually, any common denominator will work when adding fractions), and convert the fractions.

The least common denominator is 5 times 8, or 40, so convert each fraction into 40ths:

2. Add the numerators and keep the common denominator unchanged:

Now you’re ready to do the original problem, . In this problem, you have an a instead of a 2, a b instead of a 5, a c instead of a 3, and a d instead of an 8. Just carry out the exact same steps as you do when adding . You can think of each of the numbers in the above solution as stamped on one side of a coin with the corresponding variable stamped on the other side. For instance, there’s a coin with a 2 on one side and an a on the opposite side; another coin has an 8 on one side and a d on the other side, and so on. Now, take each step of the previous solution, flip each coin over, and voilà, you’ve got the solution to the original problem. Here’s the final answer:

Subtracting fractions

Subtracting fractions works like adding fractions except instead of adding, you subtract. Insights like this are the reason they pay me the big bucks.

Canceling in fractions

Finishing calculus problems — after you’ve done all the calculus steps — sometimes requires some pretty messy algebra, including canceling. Make sure you know how to cancel and when you can and can’t do it.

In the fraction, , three xs can be canceled from the numerator and denominator, resulting in the simplified fraction, . If you write out the xs instead of using exponents, you can more clearly see how this works:

Now cancel three xs from the numerator and denominator:

That leaves you with , or .

Express yourself

An algebraic expression or just expression is something like xyz or , basically anything without an equal sign (if it has an equal sign, it’s an equation). Canceling works the same way with expressions as it does for single variables. By the way, that’s a tip not just for canceling, but for all algebra topics.

Expressions always behave exactly like variables.

So, if each x in the preceding problem is replaced with , you’ve got

And three of the expression cancel from the numerator and denominator, just as the three xs canceled. The simplified result is

The multiplication rule for canceling

Now you know how to cancel. You also need to know when you can cancel.

The multiplication rule: You can cancel in a fraction only when it has an unbroken chain of multiplication through the entire numerator and the entire denominator.

Canceling is allowed in a fraction like this:

Think of multiplication as something that conducts electricity. Electrical current can flow from one end of the numerator to the other, from the to the , because all the variables and expressions are connected with multiplication. (Note that an addition or subtraction sign inside parentheses — the “+” in for instance — doesn’t break the current.) Because the denominator also has an unbroken chain of multiplication, canceling is allowed. You can cancel one a, three bs, and three of the expression . Here’s the result:

When you can’t cancel: But adding an innocuous-looking 1 to the numerator (or denominator) of the original fraction changes everything:

The addition sign in front of the 1 breaks the electrical current, and no canceling is allowed anywhere in the fraction.

Absolute Value — Absolutely Easy

Absolute value just turns a negative number into a positive and does nothing to a positive number or zero. For example,

It’s a bit trickier when dealing with variables. If x is zero or positive, then the absolute value bars do nothing, and thus,

But if x is negative, the absolute value of x is positive, and you write

For example, if .

can be a positive number. When x is a negative number, (read as “negative x,” or “the opposite of x”) is a positive.

Empowering Your Powers

You are powerless in calculus if you don’t know the power rules:

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This is the rule regardless of what x equals — a fraction, a negative, anything — except for zero (zero raised to the zero power is undefined). Let’s call it the kitchen sink rule (where the kitchen sink represents zero):

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For example, . This is huge! Don’t forget it! Note that the power is negative, but the answer of is not negative.

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You can use this handy rule backwards to convert a root problem into an easier power problem.

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You add the powers here. (By the way, you can’t do anything to plus . You can’t add to because they’re not like terms. You can only add or subtract terms when the variable part of each term is the same, for instance, . This works for exactly the same reason — I’m not kidding — that 3 chairs plus 4 chairs is 7 chairs; and you can’t add unlike terms, just like you can’t add 5 chairs plus 2 cars.)

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Here you subtract the powers.

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You multiply the powers here.

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Here you distribute the power to each variable.

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Here you also distribute the power to each variable.

·  NOT!

Do not distribute the power in this case. Instead, multiply it out the long way: . Watch what happens if you erroneously use the preceding “law” with numbers: equals , or 64, not , which equals , or 34.

Rooting for Roots

Roots, especially square roots, come up all the time in calculus. So knowing how they work and understanding the fundamental connection between roots and powers is essential. And, of course, that’s what I’m about to tell you.

Roots rule — make that, root rules

Any root can be converted into a power, for example, , , and . So, if you get a problem with roots in it, you can just convert each root into a power and use the power rules instead to solve the problem (this is a very useful technique). Because you have this option, the following root rules are less important than the power rules, but you really should know them anyway:

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But you knew that, right?

No negatives under even roots. You can’t have a negative number under a square root or under any other even number root — at least not in basic calculus.

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You multiply the root indexes.

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If you have an even number root, you need the absolute value bars on the answer, because whether a is positive or negative, the answer is positive. If it’s an odd number root, you don’t need the absolute value bars. Thus,

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·  NOT!

Make this mistake and go directly to jail. Try solving it with numbers: , which does not equal .

Simplifying roots

Here are two last things on roots. First, you need to know the two methods for simplifying roots like or .

The quick method works for because it’s easy to see a large perfect square, 100, that goes into 300. Because 300 equals 100 times 3, the 100 comes out as its square root, 10, leaving the 3 inside the square root. The answer is thus .

For , it’s not as easy to find a large perfect square that goes into 504, so you’ve got to use the longer method:

1. Break 504 down into a product of all of its prime factors.

2. Circle each pair of numbers.

3. For each circled pair, take one number out.

4. Simplify.

The last thing about roots is that, by convention, you don’t leave a root in the denominator of a fraction — it’s a silly, anachronistic convention, but it’s still being taught, so here it is. If your answer is, say, , you multiply it by :

Logarithms — This Is Not an Event at a Lumberjack Competition

A logarithm is just a different way of expressing an exponential relationship between numbers. For instance,

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·  (read as “log base 2 of 8 equals 3”).

These two equations say precisely the same thing. You could think of as the way we write it in English and as the way they write it in Latin. And because it’s easier to think and do math in English, make sure — when you see something like — that you can instantly “translate” it into . The base of a logarithm can be any number greater than zero other than 1, and by convention, if the base is 10, you don’t write it. For example, means . Also, log base e is written ln instead of .

You should know the following logarithm properties:

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With this property, you can compute something like on a calculator that only has log buttons for base 10 (the “log” button) and base e (the “ln” button) by entering , using base 10 for c. On many newer-model calculators, you can compute directly.

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Factoring Schmactoring — When Am I Ever Going to Need It?

When are you ever going to need it? For calculus, that’s when.

Factoring means “unmultiplying,” like rewriting 12 as . You won’t run across problems like that in calculus, however. For calculus, you need to be able to factor algebraic expressions, like factoring as . Algebraic factoring always involves rewriting a sum of terms as a product. What follows is a quick refresher course.

Pulling out the GCF

The first step in factoring any type of expression is to pull out — in other words, factor out — the greatest thing that all of the terms have in common — that’s the greatest common factor or GCF. For example, each of the three terms of contains the factor , so it can be pulled out like this: . Make sure you always look for a GCF to pull out before trying other factoring techniques.

Looking for a pattern

After pulling out the GCF if there is one, the next thing to do is to look for one of the following three patterns. The first pattern is huge; the next two are much less important.

Difference of squares

Knowing how to factor the difference of squares is critical:

If you can rewrite something like so that it looks like , then you can use this factoring pattern. Here’s how:

Now, because , you can factor the problem:

A difference of squares, , can be factored, but a sum of squares, , cannot be factored. In other words, , like the numbers 7 and 13, is prime — you can’t break it up.

Sum and difference of cubes

You might also want to memorize the factor rules for the sum and difference of cubes:

Trying some trinomial factoring

Remember regular old trinomial factoring from your algebra days?

Several definitions: A trinomial is a polynomial with three terms. A polynomial is an expression like where, except for the constant (the 2 in this example), all the terms have a variable raised to a positive integral power. In other words, no fraction powers or negative powers allowed (So, is not a polynomial because it equals ). And no radicals, no logs, no sines or cosines, or anything else — just terms with a coefficient, like the 4 in , multiplied by a variable raised to a power. The degree of a polynomial is the polynomial’s highest power of x. The polynomial at the beginning of this paragraph, for instance, has a degree of 5.

It wouldn’t be a bad idea to get back up to speed with problems like

where you have to factor the trinomial on the left into the product of the two binomials on the right. A few standard techniques for factoring a trinomial like this are floating around the mathematical ether — you probably learned one or more of them in your algebra class. If you remember one of the techniques, great. You won’t have to do a lot of trinomial factoring in calculus, but it does come in handy now and then, so, if your skills are a bit rusty, check out Algebra II For Dummies by Mary Jane Sterling (Wiley).

Solving Quadratic Equations

A quadratic equation is any second degree polynomial equation — that’s when the highest power of x, or whatever other variable is used, is 2.

You can solve quadratic equations by one of three basic methods.

Method 1: Factoring

Solve .

1. Bring all terms to one side of the equation, leaving a zero on the other side.

2. Factor.

You can check that these factors are correct by multiplying them. Does FOIL (First, Outer, Inner, Last) ring a bell?

3. Set each factor equal to zero and solve (using the zero product property).

So, this equation has two solutions: and .

The discriminant tells you whether a quadratic is factorable. Method 1 will work only if the quadratic is factorable. The quick test for that is a snap. A quadratic is factorable if the discriminant, , is a perfect square number like 0, 1, 4, 9, 16, 25, etc. (The discriminant is the stuff under the square root symbol in the quadratic formula — see Method 2 in the next section.) In the quadratic equation from Step 1, , for example, , , and ; equals, therefore, , which equals 121. Since 121 is a perfect square , the quadratic is factorable. Because trinomial factoring is often so quick and easy, you may choose to just dive into the problem and try to factor it without bothering to check the value of the discriminant. But if you get stuck, it’s not a bad idea to check the discriminant so you don’t waste more time trying to factor an unfactorable quadratic trinomial. (But whether or not the quadratic is factorable, you can always solve it with the quadratic formula.)

Method 2: The quadratic formula

The solution or solutions of a quadratic equation, , are given by the quadratic formula:

Now solve the same equation from Method 1 with the quadratic formula:

1. Bring all terms to one side of the equation, leaving a zero on the other side.

2. Plug the coefficients into the formula.

In this example, a equals 2, b is , and c is , so

This agrees with the solutions obtained previously — the solutions better be the same because we’re solving the same equation.

Method 3: Completing the square

The third method of solving quadratic equations is called completing the square because it involves creating a perfect square trinomial that you can solve by taking its square root.

Solve .

1. Put the x² and the x terms on one side and the constant on the other.

2. Divide both sides by the coefficient of x² (unless, of course, it’s 1).

3. Take half of the coefficient of x, square it, then add that to both sides.

Half of is -4 and is 16, so add 16 to both sides:

4. Factor the left side into a binomial squared. Notice that the factor always contains the same number you found in Step 3 (–4 in this example).

5. Take the square root of both sides, remembering to put a sign on the right side.

6. Solve.