Review questions for MIDTERM 3.

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?

and

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

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

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

(e) On what intervals is concave up?

and

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

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.

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

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

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

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

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

Draw a possible graph of given that it satisfies all of the following conditions.

(a) Domain of ,

(b) f is continuous on its domain and differentiable all all points in the domain except
at

(c) ,

(d) , , ,

(e) on , and on ,

(f) on ,

(g) on and on ,

(h) on and on

Once you’ve finished, select ‘Done’, and compare your answer with the one
shown

Done

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.

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

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

(d) Find .

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

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

Evaluate the following limits. You may use L‘Hospital’s Rule

(a)

(b)

(c)

(d)

(e)

(f)

The figure shows a right triangle in the first quadrant. One side of the
triangle is on the x-axis; its hypotenuse runs from the origin to a point on
the parabola . Find the coordinates and that maximize the area of the
triangle.

(i) The point on the curve that is closest to the point occurs at

(ii) A cruise line offers a trip for $1000 per passenger. If at least 100 passengers
sign up, the price is reduced for all passengers by $5 for every additional
passenger (beyond 100) who goes on the trip. The boat can accommodate 250
passengers.

The number of passengers which maximizes the cruise line’s total revenue is

What price does each passenger pay if that number of passengers goes on the cruise?

(iii) Find the dimensions of the right circular cylinder of maximum volume that can
be placed inside a sphere of radius R:

(iv) A certain tank consists of a right circular cylinder with hemispherical ends. For a
given surface area S, find the dimensions (radius and length) of the tank with
maximum volume (your answer should include S):

(v) A square piece of tin 24 in on each side is to be made into an open-top box by
cutting a small square from each corner and bending up the flaps to form the sides.
What is the side length of the square that should be cut from each corner to make
the volume of the box as large as possible?

A toy roller-coaster has been designed so that the rail has the shape of the
curve given in the figure below, where (x in inches, gives the altitude)

The average rate of change of the altitude of the roller coaster on the interval is
.

Select the best interpretation of for .

is the altitude at . is the average rate of
change of the altitude on the interval . is the velocity of an object at . is the
instantaneous rate of change of the altitude at . is the slope of the secant line
passing through and .

Because is continuousdifferentiable
on the interval and is continuousdifferentiable
on the interval , satisfies the conditions of the Mean Value Theorem.

By the Mean Value Theorem, there exists in such that . In face this happens twice,
when and when (assume ).

The steepest point on the roller coaster is . (Hint: maximize on .)

The linear approximation, , to at is

Using this linear approximation we estimate that is approximately This estimate is
an overestimateunderestimate
because is concave upconcave down
between 5 and 7.

When changes from to the change in , is We can approximate this change with
the differential which is

Select all correct answers for each question below.

(i) At what point(s) does the conclusion of the Mean Value Theorem hold for on the
interval ?

0 1 2 Such a point does not exit None of the above
(ii) The equation of the line that represents the linear approximation to the function
at is

Such a line does not exit None of the above
(iii) Determine the following indefinite integral

(iv) Evaluate the expression

(i) The population of a culture of cells grows according to the function , where is
measured in weeks.

(a) What is the average rate of change in the population during the time interval ?

(b) Assume that is continuous on and differentiable on . Is there a moment in which
the instantaneous rate of change of is equal to the average rate of change computed
in part (a)?

YesNo

If yes, select the theorem which guarantees the existence of such a point. If no, select
‘No Theorem’.

Intermediate Value TheoremMean Value TheoremL‘Hospital’s
RuleNo Theorem

(ii) Let . The graph of is given in the figure below.

(a) The linear approximation L to the function at is

(b) Select the figure which includes the graph of :

(c) Use the linear apprxoimation to estimate the value of .

(d) The approximation in part (c) is an overestimate underestimate
because is concave upconcave down
.

Given on , and , complete the following steps:

(a) Calculate and the grid points (in ascending order)

(b) Select the figure with the right Riemann sum drawn in

(c) Calculate the right Riemann sum:

The figure illustrates the Riemann sum for on the interval . Use the figure to
answer the following questions.

(i) Given the interval , what is ?

n=3 n=8 n=5 n=4 None of the above

(ii) What is ?

4 0 1 2 None of the above

(iii) Given that , select the expression for the grid points , for ?

None of
the above

(iv) What is for .

None of the above

(v) Evaluate this Riemann sum

None of the above

(vi) This Riemann sum is

midpoint Riemann sun right Riemann sum left
Riemann sum None of the above

Determine the following indefinite integrals. Check your work by differentiation.

(a)

(b)

(c)

(d)

(e)

(f)

The figure shows the graph of a function . At what point(s) does the conclusion of
the Mean Value Theorem hold for on the interval ? (c is an integer)

Consider the following limit of Riemann sums for a function on .

Express the limit as a definite integral.

None of the above

Consider an object moving along a straight line with the velocity on . Find the
distance traveled over the given interval.

None of the above

Suppose that on , on , , and .

(i) Evaluate the following integrals

(a)

(b)

(c)

(ii) Assume that is odd. Evaluate

(ii) Assume that is even. Evaluate

Given that , and , evaluate the following integrals:

(a)

(b)

(c)

The acceleration function (in m/) is for a particle moving along a line is given by , ,
.

(a) Find the velocity at time t:

(b) Find the distance traveled during the given time interval:

(c) Find the position at time t:

The acceleration function (in m/) is for a particle moving along a line is given by , ,
.

(a) Find the velocity at time t:

(b) Find the position at time t:

Find the particular solution of which satisfies the intial condition .

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

(a) Use geometry to evlaluate

(b) Select the graph of a rectangle whose net area is equal to

(ii) The graph of on the interval is given in the figure

(a) Use geometry to evaluate

(b) Compute

An object is moving along a straight line. The graphs of its position function , its
velocity , and its acceleration , on the time interval are given below.

Which curve is which: ; ;

Consider an object moving along a line. The graph of the velocity, , of the object is
shown in the figure below.

Select the graph of given that