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Page 3.1: Difference Quotient

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A big part of calculus is being able to understand the notation when it comes to dealing with functions. For this reason, we will spend some time on discussing how to think about our notation.

Suppose we have the function f:RR specified by f(x)=x2. To recap a few things, this notation is telling us that the domain is R meaning that the x values can be any real number. When we plug in these x values into f, the outputs are also real numbers and hence our codomain is R.

Allow us to consider the following exercise.

Exercise Page3.1.1

Suppose f:RR specified by f(x)=x2. Find the following:

1) f(1)

2) f(2)

3) f(3)

4) f(a)

5) f(b)

6) f(a+h)

7) f(a+h)f(a)h

Answer

1) The notation f(1) is telling us that wherever we see x in the function f(x)=x2, we should replace the x by a 1. Doing so yields f(1)=12=1

2) The notation f(2) is telling us that wherever we see x in the function f(x)=x2, we should replace the x by a 2. Doing so yields f(2)=22=4

3) The notation f(3) is telling us that wherever we see x in the function f(x)=x2, we should replace the x by a 3. Doing so yields f(3)=32=9

4) The notation f(a) is telling us that wherever we see x in the function f(x)=x2, we should replace the x by an a. Doing so yields f(a)=a2.

5) The notation f(b) is telling us that wherever we see x in the function f(x)=x2, we should replace the x by a b. Doing so yields f(b)=b2.

6) The notation f(a+h) is telling us that wherever we see x in the function f(x)=x2, we should replace the x by an a+h. Doing so yields f(a+h)=(a+h)2=a2+2ah+h2.

7) Putting Parts 4 and 6 together yields: f(a+h)f(a)h=(a+h)2a2h=a2+2ah+h2a2h=2ah+h2h=2a+h

The quantity we considered in Part 7 has a special name - the difference quotient.

Definition: The Difference Quotient

Definition: Given a function f, we refer to f(a+h)f(a)h as the difference quotient.

More will be said about the difference quotient in the coming days but for now, allow us to compute a few more difference quotients.

Exercise Page3.1.1

Compute the difference quotient of f(x)=x.

Answer

1.

Exercise Page3.1.1

Compute the difference quotient of f(x)=x3x.

Answer

3a2+3ah+h21.

Exercise Page3.1.1

Compute the difference quotient of f(x)=x. Hint: to simplify, rationalize the numerator!

Answer

1a+ha.

Exercise Page3.1.1

Compute the difference quotient of f(x)=1x.

Answer

1(a+h)(a).

Exercise Page3.1.1

Compute the difference quotient of f(x)=1x2+x.

Answer

3(2+a+h)(2+a).


Page 3.1: Difference Quotient is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.

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