Problem Set 1—Limits 1

Purpose

This lesson reinforces your understanding of the basic concept of a limit and the formal definition of the idea.

Background

This lesson is based on material in sections 2.1 (“Rates of Change and Tangents to Curves”) through 2.3 (“The Precise Definition of a Limit”) of our textbook. We covered (or will cover) this material in classes between approximately August 28 and September 9.

One question in this exercise asks you to use R as a calculator. You should already have installed R on your computer, but if you still need to do it you can download R from http://cran.r-project.org/.

Doing calculations with R is fairly straightforward. You write your calculation (technically called an “expression” in R and programming documentation) in more or less normal notation (e.g., numbers with operators between them), then press the Enter or Return key and R shows you the resulting value. The characters used to represent arithmetic operations in R however, may not always be the ones you’re used to, since some of the common operator symbols don’t appear on a standard keyboard. Here are the most common operator symbols in R:

These operations have the usual precedence (i.e., exponentiation is done first, then multiplication and division from left to right, and finally addition and subtraction from left to right), which can be over-ridden by enclosing parts of a calculation in parentheses. Thus, the following are examples of arithmetic expressions in R:

4 + 8 (adds 4 plus 8)
6 - 8 * 2 (subtracts 8 times 2 from 6)
3 ^ (8 / 4) (raises 3 to the 2nd (from 8 divided by 4) power)

Activity

Solve each of the following problems:

Average Rate of Change

Calculate the average rate of change of the function f(x) = 3.24 x² + 1.9 x - 2.379 over the interval [6.2, 13.73]. Show how to use R to do the necessary calculations.

Section 2.2, Exercise 24

(lim_x→-3(x+3)/(x²+4x+3).)

Section 2.2, Exercise 54d

(lim_x→4g(x)/(f(x)-1) given that lim_x→4f(x) = 0 and lim_x→4g(x) = -3.)

Section 2.3, Exercise 38

(Prove that lim_x→3(3x-7) = 2.)

Section 2.2, Exercise 78b

(lim_x→-2(f(x)/x) given that lim_x→-2(f(x)/x²) = 1.)

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.