The displacement (in meters) of an object moving in a straight line is given by , where

Explain what each of the following means and illustrate with a sketch.

Describe several ways in which a limit can fail to exist. Illustrate with sketches.

State the following Limit Laws. (a) Sum Law (b) Difference Law (c) Constant Multiple Law (d) Product Law (e) Quotient Law (f ) Power Law (g) Root Law

What does the Squeeze Theorem say?

(a) What does it mean to say that the line is a vertical asymptote of the curve ? Draw curves to illustrate the various possibilities. (b) What does it mean to say that the line is a horizontal asymptote of the curve ? Draw curves to illustrate the various possibilities.

Which of the following curves have vertical asymptotes? Which have horizontal asymptotes? (a) (b) (c) (d) (e) (f ) (g) (h) y 1x y sx y e y ln x x y tan1 y tan x x y x y sin x 4 y f

(a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval ? What can you say about the graph of such a function?

What does the Intermediate Value Theorem say?

Write an expression for the slope of the tangent line to the curve at the point .

Suppose an object moves along a straight line with position at time t. Write an expression for the instantaneous velocity of the object at time . How can you interpret this velocity in terms of the graph of f ?

If and x changes from to , write expressions for the following. (a) The average rate of change of y with respect to x over the interval . (b) The instantaneous rate of change of y with respect to x at .

Define the derivative . Discuss two ways of interpreting this number.

(a) What does it mean for to be differentiable at a? (b) What is the relation between the differentiability and continuity of a function? (c) Sketch the graph of a function that is continuous but not differentiable at a = 2.

Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

Find the limit.

Find the limit.

Find the limit.

Find the limit.

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.

Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.

If 2x 1  fx  x 0  for 0  x  3 , find imxl1 fx 2 .

Prove that imxl0 x 2 cos1x 2 0 2x

Prove the statement using the precise definition of a limit.

Prove the statement using the precise definition of a limit.

Prove the statement using the precise definition of a limit.

Prove the statement using the precise definition of a limit.

et (a) Evaluate each limit, if it exists. (i) (ii) (iii) (iv) (v) (vi) (b) Where is discontinuous? (c) Sketch the graph of .

Let (a) For each of the numbers 2, 3, and 4, discover whether is continuous from the left, continuous from the right, or continuous at the number. (b) Sketch the graph of

Show that the function is continuous on its domain. State the domain.

Show that the function is continuous on its domain. State the domain.

Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval.

Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval.

(a) Find the slope of the tangent line to the curve at the point . (b) Find an equation of this tangent line.

Find equations of the tangent lines to the curve at the points with -coordinates and

The displacement (in meters) of an object moving in a straight line is given by , where is measured in seconds. (a) Find the average velocity over each time period. (i) (ii) (iii) (iv) (b) Find the instantaneous velocity when .

According to Boyles Law, if the temperature of a confined gas is held fixed, then the product of the pressure and the volume is a constant. Suppose that, for a certain gas, , where is measured in pounds per square inch and is measured in cubic inches. (a) Find the average rate of change of as increases from 200 in to 250 in . (b) Express as a function of and show that the instantaneous rate of change of with respect to is inversely proportional to the square of .

(a) Use the definition of a derivative to find , where . (b) Find an equation of the tangent line to the curve at the point (2, 4). ; (c) Illustrate part (b) by graphing the curve and the tangent line on the same screen.

Find a function and a number a such that

The total cost of repaying a student loan at an interest rate of r% per year is . (a) What is the meaning of the derivative ? What are its units? (b) What does the statement mean? (c) Is always positive or does it change sign?

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.

(a) If , use the definition of a derivative to find . (b) Find the domains of and . ; (c) Graph and on a common screen. Compare the graphs to see whether your answer to part (a) is reasonable.

(a) Find the asymptotes of the graph of and use them to sketch the graph. (b) Use your graph from part (a) to sketch the graph of . (c) Use the definition of a derivative to find . ; (d) Use a graphing device to graph and compare with your sketch in part (b).

The graph of is shown. State, with reasons, the numbers at which is not differentiable.

The total fertility rate at time t, denoted by , is an estimate of the average number of children born to each woman (assuming that current birth rates remain constant). The graph of the total fertility rate in the United States shows the fluctuations from 1940 to 1990. (a) Estimate the values of , , and . (b) What are the meanings of these derivatives? (c) Can you suggest reasons for the values of these derivatives?

Let be the total value of U.S. banknotes in circulation at time . The table gives values of this function from 1980 to 1998, at year end, in billions of dollars. Interpret and estimate the value of .

Graph the curve and the tangent lines to this curve at the points and

Suppose that  fx   tx x li for all , where imx l a tx 0 2,. Find limx l a fx 

Let . (a) For what values of does exist? (b) At what numbers is discontinuous?