What is the average velocity between times and ?

(b) The table gives over a three second interval.

The average velocity over the interval is

(c) Select all reasonable approximations for the instantaneous velocity at time second:

Review questions for MIDTERM 2.

(a) Suppose is the position (in feet) of an object moving along a line at time
seconds.

What is the average velocity between times and ?

(b) The table gives over a three second interval.

The average velocity over the interval is

(c) Select all reasonable approximations for the instantaneous velocity at time second:

The graph of a function is given below

(1) (a) Select the figure which has a secant line drawn through the points and .

(b) Select the figure which has the line tangent to the curve at

(2) Choose the correct statement:

(i)

(ii)

(iii) Select the best approximation of

Let be a function such that and .

(a) Find the limit or say it does not exist (DNE).

(b) An equation of the tangent line to the curve at the point where is given by

(i) Given
use implicit differentiation to find
(b)

(ii) Given y, select :

(a)

(c)

(d)

(ii) Find the following values or state the value is undefined (DNE):

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

A table of values for and , along with a graph of a function is shown below.

(i) =

12 4 DNE None of the above

(ii) =

-1 -4 4 DNE None of the above

(iii) =

-1 -5 0 DNE None of the above

(iv)

4 1 2 DNE None of the above

(v) =

2 4 DNE None of the above

The function is defined by .

Answer the following question about the function .

(a) The domain of is

(b) The function is

odd even neither

(c)

(d)

(e) Select the correct statement for each interval:

(f) Select the correct graph for :

(g) Is the function one-to-one:

Yes No

The graph of (the derivative of ) on the interval is shown in the figure.

Use the given graph of to answer the following questions about f:

(a) On what interval(s) is decreasing?

ANSWER: and

(b) List the x- coordinates of all critical points of (in ascending order).

ANSWER:

(c) List the x-coordinates of all critical points of that correspond to local
maxima?

ANSWER:

(d) List the x-coordinates of all critical points of that correspond to neither local
maxima nor local minima?

ANSWER:

(e) On what intervals is concave up?

ANSWER: and

(f) List the x-coordinates of all inflection points of (in ascending order).

ANSWER

Given that , , and , find the following values or state ‘cannot be determined’
(CBD):

(a)

(b)

(c)

(d)

(e)

(f)

The position, , of an object moving along a horizontal line is given by , , where is
measured in feet and in seconds.

Complete the statements below.

(a) The position of the particle at time is ft.

(b) The average velocity, , of the object over the interval is

.

(c) Using the expression found in part (b), evaluate

(e) What does the limit in part (c) represent?

The position at The acceleration
at The instantaneous velocity at

(f) Find the velocity of the particle:

(g) Find the acceleartion of the particle:

The (entire) graph of a function is shown in the figure below.

Give answers for the following, or write ‘DNE’.

(a) Find the x-coordiantes of all points in the interval where has a local minimum.

ANSWER:

(b) Find the x-coordinate of all points in the interval where has a local
maximum.

ANSWER:

(c) Find all critical points in the interval (in ascending order).

ANSWER:

Assume that a function is continuous on its domain, . The graph of , the derivative
of , is shown in the figure below.

(a) Write the x-coordinates of all critical points of (or write NONE), in ascending
order.

ANSWER:

(b) Write the x-coordinates of all local maxima of (or write NONE).

ANSWER:

(c) Write the x-coordinates of all local minima of (or write NONE).

ANSWER:

(d)Find the interval(s) on which is increasing.

ANSWER:

(e) Find the interval(s) on which is concave down.

ANSWER:

(f) Write the x-coordinates of all inflection points (or write NONE), in ascending
order.

ANSWER:

A curve is given by the equation
.

(i) (a) Use implicit differentiation to find the derivative.

(b) Check (algebraically), that the point lies on the curve.

(c) Compute at the point .

ANSWER:

(ii) Part of the curve that contains the point is shown in the figure below:

(a) Find an explicit expression for , , represented by the graph above.

(b) Using the explicit expression in part (a), find .

(c) Evaluate the derivative.

(d) Write an equation of the tangent line to the graph of at the point where .

ANSWER:

A function (the derivative of ) is given by . Answer the following, or write ”DNE”
(does not exist).

(a) List all interval(s) on which is increasing.

ANSWER:

(b) List x-coordinates of all points where has a local maximum.

ANSWER:

(c) List x-coordinates of all points where has a local minimum.

ANSWER:

(d) Find .

ANSWER:

(e) List all interval(s) on which is concave up.

ANSWER:

(f) List x-coordinates of all inflection points of .

ANSWER:

The function , measured in meters, gives the position of an object moving along a
line at time , measured in seconds.

(a) Is the function one-to-one?

Yes No

(b) Find the expression for inverse of , .

ANSWER:

(c) Find .

ANSWER:

(d) Interpret your answer to part (c), by selecting the correct answer.

gives the
speed of the object at time 5 gives the position of the object at time 5 gives the
time when the object was at position 5 gives the time when the object had speed 5

(e) Find the average velocity between times and .

ANSWER:

(f) Find an expression for the instantaneous velocity at time .

ANSWER:

(g) Find an expression for the instantaneous acceleration at time .

ANSWER:

(h) Is the velocity increasing or decreasing for ? Select the correct answer.

Increasing Decreasing

(i) Is the speed increasing or decreasing for ? Select the correct answer.

Increasing Decreasing

A ladder ft long rests against a vertical wall. If the bottom of the ladder
slides away from the wal at a speed of ft/s, how fast is the angle between
the top of the ladder and the wall changing when the angle is radians?

The position function (in feet) for an object at time (in minutes), , is given by
.

(a) Find the velocity at any time .

ANSWER:

(b) Find the acceleration at any time .

ANSWER:

(c) At what time is the object furthest from the origin in the positive direction?

ANSWER: At the time min.

(d) What are the velocity and acceleration at that time?

ANSWER: ,

(e) At what (positive) time is the object at the origin?

ANSWER: At the time min.

(f) What are the velocity and acceleration at that time?

ANSWER: ,

(g) Find the time interval(s) when the velocity is decreasing.

ANSWER:

The (entire) graph of a function is given in the figure below:

(a) Find the value.

(b) List the x-coordinates of all critical points of .

ANSWER:

(c) List the x-coordinates of all local minima of , or say ‘none’.

ANSWER:

(d) List the x-coordinates of all local maxima in of , or say ‘none’.

(g) Select the correct graph of .

(i) Let .

List the x-coordinates of all critical points of in ascending order.

ANSWER:

(ii) Let

(a) Find the domain of the function .

ANSWER:

(b) List the x-coordinates of all critical points of in ascending order.

ANSWER:

(iii) Let

(a) Find the domain of the function .

ANSWER:

(b) State the interval(s) of continuity of .

ANSWER:

(c) List the x-coordinates of all critical points of in ascending order.

ANSWER:

A water tank is to be drained for cleaning. There are liters of water left in the tank
minutes after the draining began, where

(a) Find the average rate at which water drains during the first minutes.

ANSWER:

(b) Find the rate at which the volume of water is changing minutes after the
draining began.

ANSWER:

(c) Compute the derivative.

(d) What are the units of the answer to part (c)?

(e) Find the rate of the rate at which the volume of water is changing 10 minutes after draining begins.

ANSWER:

(f) Is the rate at which the volume of the water is changing increasing or decreasing (during the draining)?

Increasing Decreasing

(g) Assume that the tank has the shape of a rectangular box m long, m wide, and m
high. What is the rate of change of the water depth when the water depth is m?
(HINT: 1 liter = .001 )

ANSWER: