Calculus Review 3

    This topic is a collection of problems and concepts that might help someone understand their working knowledge of Calculus 1.

Show all work and justify each step.

(1) Find the limit,


(2) Find the limit,


(3) A rectangular swimming pool is to be built with an area of 1800 square feet. Additionally, the owner wants 5-foot wide decks along either side of the pool and 10-foot wide decks at the two ends. Draw a labelled picture depicting this situation and find the dimensions of the smallest piece of property on which the pool and decks can be built that satisfies these conditions. (Of course, you should justify that the dimensions you give are indeed for the smallest piece of property.)


(4) Find the numbers  whose product is a minimum, given that one of the numbers is nine less than one-fifth the other.


(5) Compute where is any nonzero real number.
    
    (a)
    
    (b)
    
    (c)
    
    (d) 0
    
    (e)
    

(6) If then the value of at is

    (a)
    
    (b)
    
    (c)
    
    (d) does not exist
    
    (e) not enough information is given
    

(7) If then the value of if and

    (a) 0
    
    (b) e
    
    (c)
    
    (d)
    
    (e) does not exist


(8) All the critical numbers of are

    (a) 0
    
    (b) &
    
    (c) 0 & 1
    
    (d)
    
    (e) 1


(9) If a function has derivative then has a local maximum at

    (a) 0
    
    (b) 1
    
    (c)
    
    (d) does not exist
    
    (e) not enough information is given


(10) A function has derivative and second derivative Find the -coordinate(s) of the inflection points(s) of the original function

    (a) 0.8, 2
    
    (b)
    
    (c)
    
    (d)
    
    (e) 0.8


(11) Evaluate the limit

    (a) 0
    
    (b)
    
    (c) 1
    
    (d) does not exist
    
    (e) not enough information given


(12) Evaluate the limit

    (a) 0
    
    (b)
    
    (c) 1
    
    (d) does not exist
    
    (e) not enough information


(13) Given that and there is a function with the property that for all and Find

    (a) 1
    
    (b) 6
    
    (c) 10
    
    (d) 16
    
    (e) 4


(14) One piece of the River Soda is 10 miles wide and flows from the west to the east. Missy Smith is at a point A on the north bank of the river. Directly across the river from Missy, on the south bank, is a point B, and Missy wishes to reach a cabin C located 5 miles down the river from B. Missy can row at 5 mph (including the effect of the current), and run at 13 mph on the ground. Find the route that will take Missy the least amount of time to get from A to C.

    (a)  row to a point on the south bank that is east of B and within  mile of B and then run along the south bank to C
    
    (b) row directly from A to C
    
    (c) row directly from A to B and then run along the south bank to C
    
    (d) row to a point on the south bank that is west of C and within a mile of C and then run along the south bank to C
    
    (e) not enough information is given


(15) A piece of wire 30 cm long is cut into two pieces. One piece is bent into a square and the other piece is shaped into a circle. To maximize the total are enclosed, the length of wire for the circle should be

    (a) 0 cm
    
    (b)
    
    (c) 30 cm
    
    (d) does not exist
    
    (e) not enough information given


(16) A piece of wire 30 cm long is cut into two pieces. One piece is bent into a square and the other piece is shaped into a equilateral triangle. To minimize the total are enclosed, the length of wire for the triangle should be

    (a) 0 cm
    
    (b)
    
    (c) 30 cm
    
    (d) does not exist
    
    (e) not enough information given


(17) A function has derivative Given that find

    (a) 4.3
    
    (b) 4
    
    (c)
    
    (d) not enough information given
    
    (e) none of these  


(18) A stone is flung vertically downwards from the top of a cliff with an initial velocity of 10 m/s. If the stone hits the ground with a speed of 59 m/s, how high is the cliff? You may assume acceleration due to gravity is 9.8 downwards, that the stone lands at the base of the cliff and that air resistance is negligible.

    (a) 3.0 m
    
    (b) 52.8 ft
    
    (c) 52.8 m
    
    (d) 172.5 ft
    
    (e) 172.5 m
    
    
(19) Given an expression in terms of a limit that gives the area between the graph of the -axis and the lines and

     (a)

     (b)
    
     (c)

     (d) no such limit exists

     (e) none of these


(20) If and what is the value of

    (a)
    
    (b)
    
    (c) 8
    
    (d) 2
    
    (e) not enough information is given


(21) Suppose the combined population of coyotes and roadrunners is increasing at a rate of per year (where is measured in years). Find an expression that gives the population's increase between the fifth and ninth years.

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)


(22) The value of at is

    (a)
    
    (b) 1
    
    (c) 2
    
    (d) 3
    
    (e) undefined


(23) Find

    (a)
    
    (b) 1
    
    (c) 2
    
    (d) 3
    
    (e) undefined
    
    
(24) The value of at is

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) undefined


(25) The value of is

    (a)
    
    (b) 2
    
    (c)
    
    (d)
    
    (e) none of these


(26) The value of is

    (a)
    
    (b) 6
    
    (c)

    (d)
    
    (e) none of these    
    
    
(27) If then the value of is

    (a)
    
    (b)
    
    (c)
    
    (d) undefined
    
    (e) none of these
    
    
(28) The value of is

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of these
    
    
(29) The following graph give the velocity for a certain object as a function of time Find the total distance travelled by, and the total displacement of, the object from to in that order.



    (a) 13 units & units
    
    (b) 5 units & 13 units
    
    (c) 13 units & 5 units
    
    (d) 11 units & units
    
    (e) 9 units & 4 units
    

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