David Harmsworth

2013

Part I

Functions

Since calculus is primarily concerned with the study of functions, we begin this course with a review of some of the basic concepts. Since most of these ideas should already be familiar to you, we’ll move quite quickly, with a focus on addressing some common misconceptions.

However, we will also introduce some concepts which are usually not discussed until later on in the calculus sequence, so there should be something new for everyone each week.

1

Review of the Basics

A function is simply a rule which assigns a single output value to each input value. You are probably familiar with the “vertical line test”; for a true function there can be only a single output for each input, and this corresponds to the fact that its graph cannot pass through any vertical line more than once. Since we customarily use the name “x” for the independent

(input) variable and the name “y” for the dependent (output) variable, we can describe the action of a function f by writing y = f (x). You may occasionally also see the notation f : x → f (x), which doesn’t require us to assign a name to the output.

Note that there doesn’t have to be a formula for a function. For instance, the temperature at a given location can be regarded as a function of time, but there’s no explicit formula for it! Of course, to use calculus, we might try to invent a formula which approximates the real function. 1

The domain of a function is the set of allowable values for the independent variable, while the range is the set of possible values for the dependent variable. For example, for the function

√

f (x) = x − 1, the domain (unless otherwise speciﬁed) is the set of values of x such that x ≥ 1, and the range is the set of values of x such that x ≥ 0.

Comment:

We’ll often describe such sets in interval notation: an interval such as

{x | 1 < x < 2} can be expressed simply as the interval (1, 2). If we wish to include the endpoints we use square brackets: the interval {x | 1 ≤ x ≤ 2} can be written as [1, 2]. These two types of intervals are referred to as “open” and “closed”, respectively. The two types of parentheses can be combined as needed, so for example the interval [1, 2) is closed on the left and open on the right (that is, the number 1 is included, but the number 2 is not). We use the symbol ∞ for unbounded intervals; it will always be accompanied by a round bracket (because

∞ is not a real number, so it can’t be included in an interval). With this notation we could

√

write the domain and range for x − 1 as [1, ∞) and [0, ∞), respectively.

Comment #2: Of course, we could also impose a restriction on the domain for a given

√

function. For example, we could deﬁne a function as g(x) = x − 1 for x ∈ [1, 5), and then the range would be just [0, 2).

Comment #3: You’ll probably notice that textbooks alternate between two diﬀerent notations for introducing the functions they want you to work on in their exercises. Is there a diﬀerence in meaning between writing, for example, f (x) = x2 , and writing y = x2 ? Well, obviously there isn’t much diﬀerence in the amount of information given; the diﬀerence is really one of emphasis. In the prior notation the emphasis is on the rule; we are given a name for the function, and told that it is the one which squares the input. In the latter notation we are given a name for the output, and the emphasis is on the relationship between variables.

In a sense it is a contraction of two statements: y = f (x) , where f (x) = x2 .

Comment #4: Sometimes other relationships between variables may also be of interest.

For example, the equation x2 + y 2 = a2 should be familiar as the equation of a circle of radius a, but its graph clearly fails the vertical line test! So, why do we make such a fuss about which relationships are functions and which aren’t? Well, it does make a diﬀerence for the theory of

2…