Math 191        Study Guide for Hour Quiz #1 on Chapter 2       C.S.Davis

You are responsible for the subject matter in sections 2.1 - 2.6. The following topics are central in their importance. I have assigned a minimal number of exercises at the end of each topic from Calculus, by James Stewart, 6^th ed. . Work more similar exercises if you need to do so in order to master the topics.

2.1

1.

General.  Give an overview of calculus--differential calculus and integral calculus–with descriptions and diagrams for the two principal problems being solved in each branch. Class notes

2.

Given a table, graph or formula for a function, estimate the slope of the tangent line.   P65#1,7

3.

Find the equation of the tangent line to a curve at a given value of x.   P65#3

4.

Given a table, a graph or formula for a position function, find the average velocity and estimate the instantaneous velocity .  P65#5

2.2

Your task in determining limits can be stated as follows.

For these varieties of limits:

(ordinary) limits and infinite limits,

Taken in these directions as applicable:

one sided limits (i.e., limits from the left and limits from the right) and two sided limits,

Determine

* numerically (from a table I give or from a table you make from a formula),

* graphically (from a graph I give or from a graph you draw from a formula),

* algebraically (using algebra on a formula and applying theorems) and describe

* verbally (saying or writing in English what is happening)

a.

whether the limit exists (meaning, "finite limit") and if so what it is, –or--

b.

whether the limit does not exist and either is infinite (equals plus or minus infinity in the answers in Stewart) or does not exist (labeled "Does not exist" in the answers in Stewart).

Include examples involving "rats, roots and trig" (those functions involving addition, subtraction, multiplication, division, powers, roots, polynomials, rational functions, and trigonometric functions) and piecewise defined functions.   P74#1-31 odd

2.3 and 2.4                 

Your task in verifying limits is as follows.

a.

State the definition of limit (epsilon-delta definition).

b.

Draw two diagrams depicting two functions with epsilons and deltas: one function having a limit at x=a and the other function not having a limit at x=a.

c.

Use the limit theorems both to determine and verify limits. P84#1,2,3,7,11,17,21,25,29,33,41

d.

Using the definition of limit (the epsilon-delta process), 
prove lim f(x) = L for a given f, a and L.    P96#15,17
        x->a

2.5

1.

State the definition of  f continuous at a point.

2.

Given the graph, formula or verbal description for a function, determine whether the function is continuous at a point (especially piecewise defined functions). Identify the kinds of discontinuities: removable discontinuity, infinite discontinuity, jump discontinuity. P105#3,4.15,17,19,21,35,37,43

3.

Apply the theorems about continuity: including sums, differences, products, quotients, polynomials, rational functions, roots and composite functions. P105#11

4.

Determine whether a function is continuous on an interval. P105#13

5.

State, illustrate and apply the Intermediate Value Theorem (IVT) for functions continuous on a closed interval.. P105#47,49