2.1 - #1

What is the slope of the line tangent to the graph of f at (a,f(a))?

2.1 - #2

The position of an object moving along a line is given by the function s(t) = -4.92 + 33t + 23. Find the average velocity of the object over the following intervals.

(a) [0, 3](b) [0, 2](c) [0, 1](d) [0, 1 + h], where h > -1 is any real number

2.1 - #2 Example

2.1 - #3

The table gives the position s(t) of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals.

2.1 - #3 Example

2.1 - #4

Consider the position function s(t) = sin(πt) representing the position of an object moving along a line on the end of a spring. Sketch a graph of s with the secant line passing through (0, s(0)) and (0.4, s(0.4)). Determine the slope of the secant line and explain its relationship to the moving object.

2.1 - #5

For the position function s(t)=22/t+1, complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=0.

2.1 - #5 Example

2.1 - #6

a. Graph the function f(x)=x2 – 12x + 35.

b. Identify the point (a, f(a)) at which the function has a tangent line with zero slope.

c. Confirm your answer to part (b) by making a table of slopes of secant lines to approximate the slope of the tangent line at this point.

2.1 - #6 Example

2.1 - #7

A rock is dropped off the edge of a cliff and its distance s (in feet) from the top of the cliff after t seconds is s(t) = 16t2. Assume the distance from the top of the cliff to the water below is 1296 ft. Answer part (a) and (b).

2.1 - #7 Example

2.2 - #1

When (x>>a)lim f(x) exists, it always equals f(a). State whether this statement is true or false.

2.2 - #2

Use the graph of h in the given figure to find the following values, if they exist.

2.2 - #2 Example

2.2 - #3

a. Graph f to estimate (x>>-8)lim f(x).

b. Evaluate f(x) for values of x near -8 to support your conjecture in part (a).

2.2 - #3 Example

2.2 - #4

Use the graph to find the following limits and function value.

2.2 - #4 Example

2.2 - #5

Sketch a possible graph of a function that satisfies the conditions below.

2.2 - #5 Example

2.2 - #6

Estimate the value of the following limit by creating a table of function values for h=0.01, 0.001, and 0.0001, and h=-0.0001, -0.001, and -0.01.

2.2 - #6 Example

2.2 - #7

For any real number x, the floor function (or greatest integer function), [x], is defined to be the greatest integer less than or equal to x (see figure). Answer parts (a) through (e).

2.2 - #7 Example

2.2 - #8

2.2 - #9

Assume that postage for sending a first-class letter in the United States is $0.33 for the first ounce (up to and including 1 oz) plus $0.35 for each additional ounce (up to and including each additional ounce). Complete parts a through d.

2.2 - #9 Example

2.2 - #10

A function is even if f(-x) = f(x) for all x in the domain of f. If f is even, with (x>>7+)lim f(x)=8 and (lim>>x7-) f(x)=-3, find the following limits.

2.2 - #10 Example

2.3 - #1

How are (x>>b+)lim f(x) and (x>>b-) lim f(x) calculated if f is a polynomial function?

2.3 - #2 and Example

Evaluate the following limit.

2.3 - #3 and Example

Suppose p and q are polynomial functions.

2.3 - #4 and Example

Find constants b and c in the polynomial …

2.3 - #5

Give an example of functions f and g such that …

2.3 - #6 and Example

Evaluate the following limit.

2.3 - #7

Evaluate the following limit.

2.3 - #7 Example

2.3 - #8 and Example

Evaluate the following limit.

2.3 - #9

Determine the value of the constraint a for which (x>>-1)lim g(x) exists and state the value of the limit, if possible.

2.3 - #9 Example

2.3 - #10

Use the following function to answer questions (a) and (b) below.

2.3 - #10 Example

2.3 - #11

For the function g(x) shown below, compute the following limits or state that they do not exist.

2.3 - #11 Example

2.3 - #12

Answer parts (a) through (c).

2.3 - #12 Example

2.3 - #13 and Example

The magnitude of the electric field at a point x meters from the midpoint of…

2.4 - #1

Compute the values of … in the table to the right and use them to determine …

2.4 - #1 Example

2.4 - #2

The graph of f shown below has vertical asymptotes at x = -3 and x = 6. Find the following limits. Use … or … when appropriate.

2.4 - #2 Example

2.4 - #3

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.

2.4 - #4

Evaluate the following limits, using … or … when appropriate, or state that they do not exist.

2.4 - #4 Example

2.4 - #5

Determine the following limit or state that it does not exist.

2.4 - #5 Example

2.4 - #6

Find all vertical asymptotes, x = a, of the following function. For each value of a, evaluate …

2.4 - #6 Example

2.4 - #7

Match each function with its graph without using a graphing utility. Answer parts (a)-(f).

2.4 - #8

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following function.

2.4 - #8 Example

2.4 - #9

Let f(x) = … Complete parts (a) through (c) below.

2.4 - #9 Example

2.5 - #1 and Example

Evaluate the following limit.

2.5 - #2 and Example

Evaluate the limit of the polynomial p(x) as x approaches infinity.

2.5 - #3

Evaluate … and … for the following rational function. Use … or … where appropriate. Then give the horizontal asymptote(s) of f (if any).

2.5 - #3 Example

2.5 - #4

Evaluate … and … for the rational function. Then give the horizontal asymptote of f (if any).

2.5 - #4 Example

2.5 - #5 Evaluate … and … for the following rational function. Use … or … where appropriate. Then give the horizontal asymptote(s) of f (if any).

2.5 - #5 Example

2.5 - #6

Complete the following steps for the given function.

a. Use polynomial long division to find the oblique asymptote of f.b. Find the vertical asymptotes of f.c. Graph f and all of its asymptotes with a graphing utility.

2.5 - #6 Example

2.5 - #7

Evaluate … and … for the following function. Then give the horizontal asymptote of f (if any).

2.5 - #7 Example

2.5 - #8

Determine whether the following statements are true and give an explanation or counterexample.

2.5 - #9

Consider the function … Complete parts a and b.

2.5 - #9 Example

2.5 - #10 and Example

Sketch the graph of a function that satisfies the conditions given below.

2.5 - #11

Find the vertical and horizontal asymptotes of …

2.5 - #11 Example

2.5 - #12

If a function f represents a system that varies in time, the existence of … means that the system reaches a steady state (or equilibrium). For the system of the population of a culture of tumor cells given by …, determine if a steady state exists and give the steady-state value.

2.5 - #12 Example

2.5 - #13

A sequence is an infinite, ordered list of numbers that …

2.6 - #1

Give the three conditions that must be satisfied by a function to be continuous at a point.

2.6 - #2

Determine the points at which the function f below has discontinuities. For each point state the conditions in the continuity checklist that are violated.

2.6 - #2 Example

2.6 - #3

Determine whether the following function is continuous at a. Use the continuity checklist to justify your answer.

2.6 - #3 Example

2.6 - #4 and Example

Determine the intervals on which the following function is continuous.

2.6 - #5

Evaluate the limit.

2.6 - #6 and Example

Determine where the function f(x) is continuous.

2.6 - #7

Determine the interval(s) on which the following function is continuous, then evaluate the given limits.

2.6 - #7 Example

2.6 - #8

a. Use the intermediate value theorem to show that the following equation has a solution…

2.6 - #8 Example

2.6 - #9 and Example

Use the continuity of the absolute value function (|x| is continuous for all values of x) to determine the interval(s) on which the following function is continuous.

2.6 - #10

Evaluate the following limit.

2.6 - #10 Example

2.6 - #11

a. Sketch the graph of a function that is not continuous at …, but is defined at …

2.6 - #11 Example

2.6 - #12

a. Determine the value of a for which g is continuous from left at …

b. Determine the value for a for which g is continuous from the right at …

c. Is there a value of a for which g is continuous at …

2.6 - #12 Example

2.6 - #13

Classify the discontinuities in the function … at the points …

2.7 - #1 and Example

Suppose … lies in the interval … What is the smallest value of e such that … for all possible values of f(x)?

2.7 - #2

The function f in the figure satisfies … Determine the maximum value of … satisfying each statement.

2.7 - #2 Example

2.7 - #3

a. For …, find a corresponding value of … satisfying the following statement.

b. Verify that … as follows. For …, find a corresponding value of … satisfying the following statement.

2.7 - #3 Example

2.7 - #4

Use the definition of one-sided infinite limits to prove the infinite limit below.

2.7 - #5

We say that … if for each positive number M, there is a corresponding N > 0 such that f(x) > M whenever x > N. Use this definition to prove …

2.7 - #5 Example

2.7 - #6

Assume f is defined for all values of x near a, except possibly at a. The limit … if for some … there is no value of … satisfying the condition…

2.7 - #6 Example