#1. Use the graph of given below (with domain [-3,12]) to find the following.
A. Find the values of x where .
B. Find the values of x where is undefined.
C. Find the inflection points of .
D. Find all the local maximums and minimums of .
E. Find all the global maximums and minimums of .
2. Using the graph and your answers to the questions above, do the following:
A. Label all the critical points on a number line. Determine the sign of between each critical point. How does the information on this number line help you determine which critical points correspond to local maximums, minimums, or neither?
B. Label all the points on a number line where is either zero or undefined. Determine the sign of between each marked point. How does the information on this number line help you determine which points correspond to inflection points?
#2.
1. Suppose is a continuous function with domain all real numbers. Answer true or false. If false, explain why or give an example.
A. Every critical point of corresponds to either a local maximum or minimum.
B. Every local maximum or minimum of must be a critical point.
C. A global maximum must also be local maximum.
D. Local maximums and minimums can occur when is undefined.
E. must have both a global maximum and minimum if we restrict x to some closed interval.
F. Inflection points of can only occur when .
G. An inflection point can never be a local maximum or minimum.
2. Use Calculus to determine 1) critical points, 2) local maximums and minimums, 3) inflection points, and 4) intervals where is concave up or down. Include an accurate graph that illustrates these features.
A. B. C.
3. In each case, sketch a graph of a continuous function with the given properties.
-- + --
A. and ||
-13
+--
|
2
+ -- --
B. and is undefined ||
14
--+
|
4
+ -- +--
C. and || |
is undefined -20 2
----
|
0
#4.Suppose is a continuous function whose domain is all real numbers. Answer TRUE or FALSE. If FALSE give a reason or provide a counterexample.
1. Every critical point of corresponds to either a local maximum or local minimum.
2. Every local maximum or local minimum of must be a critical point.
3. All global maximums (and minimums) of must also be local maximums (and minimums).
4. A local maximum or minimum cannot occur if is undefined there.
5. If we restrict the domain of so that we have a closed interval [a,b], then must have
both a global maximum and a global minimum.
6. Inflection points of can only occur where .
7. An inflection point of can never be a local maximum or local minimum.
#5. FINDING EXTREMA FOR FAMILIES OF FUNCTIONS
1. Assume is a positive constant and .
2. Assume and are positive constants.
3. . Assume is a positive constant.
4. . Assume is a positive constant.
#6. Families of Functions
1.
A. For positive values of a, answer the following:
i) How many critical points doeshave? Find the coordinates of all local maximums
and minimums. Your answer will be in terms of a.
ii) How does increasing the value of a affect the position of the critical points? Illustrate by
sketching 3 well-chosen family members, indicating the value of a for each.
B. For negative values of a, answer the following:
i) How many critical points does have? ? Find the coordinates of all local maximums
and minimums. Your answer will be in terms of a.
ii) How does decreasing the value of a affect the shape of the graph? Illustrate by
sketching 3 well-chosen family members, indicating the value of a for each.
C. Find the value of a so that has a local maximum value of 5.
2.
A. For positive values of b, answer the following:
i) How many inflection points does have?
ii) How does increasing the value of b affect the shape of the graph and the position of the
inflection points? Illustrate by sketching 3 well-chosen family members, indicating the
value of b for each.
B. For negative values of b, answer the following:
i) How many inflection points does have?
ii) How does decreasing the value of b affect the shape of the graph and the position of the
inflection points? Illustrate by sketching 3 well-chosen family members, indicating the
value of b for each.
C. Find values of b and c so that (1,10) is an inflection point of .
#7. OPTIMIZATION PROBLEMS
1. A wire of length 12 inches can be bent into a circle, a square, or cut to make both a circle and a
square. How much wire should be used for the circle if the total area enclosed by the figure(s) is
to be a minimum? A maximum?
|
circle square
2. A window consisting of a rectangular topped by a semicircle is to have a perimeter P. Find the
radius of the semicircle if the area of the window is to be a maximum.
3. A rectangular field as shown is to be bounded by a fence. Find the dimensions of the field with
maximum area that can be enclosed with 1000 feet of fencing.
Building
20 ft
4. The operating cost of a truck is cents per mile when the truck travels miles per hour.
If the driver earns $6 per hour, what is the most economical speed to operate the truck on a 400
mile turnpike? Due to construction, the truck can only travel between 35 and 60 miles per hour.
5. A furniture business rents chairs for conferences. A contract is drawn to rent and deliver up to
400 chairs for a particular meeting. The exact number would be determined by the customer later.
The price will be $90 per chair up to 300 chairs. If the order goes above 300 chairs, the price
would be reduced by $0.25 per chair for every additional chair ordered above 300. This reduced
price would be applied to the entire order. Determine the largest and smallest revenues this business
can make under this contract.
6. The speed of traffic through the Lincoln Tunnel depends on the density of the traffic. Let S be the speed in miles per hour and D be the density in vehicles per mile. The relationship between S and D is approximately for . Find the density that will maximize the hourly flow.
7. A commercial cattle company currently allows 20 steer per acre of grazing land. On average a steer weighs 2000 pounds at the market. Estimates by the Department of Agriculture indicate that the average weight per steer will be reduced by 50 pounds for each additional steer added per acre of grazing land. How many steer per acre should be allowed in order to optimize the total market weight of the cattle?
8. The Can-O-Rad Company manufactures cylindrical barrels to store nuclear waste. The top and bottom of the barrels are to be made with material that costs $10 per square foot and the rest is made with material that costs $8 per square foot. If each barrel is to hold 5 cubic feet, find the dimensions of the barrel that will minimize the total cost.
9. A rectangular dance floor of width w and length l feet is to be built inside a semicircular part of a room of radius 30 feet. Find the values of w and l that produce a floor of maximum area. Find the maximum area too.