Calculus Practice Test 6

Show all work and justify each step.

(1) Give an expression in terms of a limit that gives the area under the curve on the interval

    (a)

    (b)

    (c)

    (d)

    (e)


(2) If and what is the value of

    (a)
    
    (b)

    (c) 8
    
    (d) 2
    
    (e) not information is given


(3) If find the value of

    (a)
    
    (b)
    
    (c)

    (d)
    
    (e) not defined


(4) The graph of a certain function has slope at each point and contains the point Find the function

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


(5) Find the values of and so that the function has a local maximum at the point Give the value of

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(6) Let where and are nonzero constants. If how are and related?

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


(7) Find the limit

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


(8) The cost in dollars of producing units of a particular commodity is and the selling price is dollars per unit when units are produced, where How many units should be produced in order to maximize the profit?

    (a) 3
    
    (b) 5
    
    (c) 7
    
    (d) 10
    
    (e) 13
    

(9) Given that and is a function of find

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

    
(10) Let and be functions of Find where and when and

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

    
(11) When a spherical balloon is inflated, the radius of the balloon is increasing at the are of how fast is the volume changing when the radius is 4 cm? The volume of a sphere with radius is

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


(12) Evaluate the limit: where is a positive constant.

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


(13) The limit represents

    (a) where
    
    (b) where

    (c) where

    (d) where

    (e) where


(14) Suppose Given that find the equation of the tangent line to the graph of at the point

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) there is no such equation


(15) Let . Find if it exists.

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

(16) Find the -coordinate of the point on the graph of where the tangent line is horizontal.

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


(17) Suppose is a function with the property that Find where

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) undefined
    
    
(18) Sketch a carefully labeled graph of a function with ALL the following properties:
    (i) for and
    
    (ii) for
    
    (iii) for and
    
    (iv) for
    
    (v) and

    
(19) A ladder 13 ft long rests against a vertical wall and is sliding down the wall at the rate the rate of 2 ft/s at the instant the bottom of the ladder is 5 ft from the base of the wall. At this instant, how fast is the bottom of the ladder moving away from the wall?


(20) Find a constant such that is continuous for all real numbers In particular, using your value of justify why is continuous on


(21) Suppose that is a function that is differentiable for all real and satisfies Find at where


(22) Evaluate , showing all steps as if you do not have a calculator.

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Cite this as:
Calculus Practice Test 6
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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