Calculus Practice Test 4

Show all work and justify each step.

(1) Find where and when

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


(2) Find the differential

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


(3) Let be a function continuous on for and and for Then

    (a) has neither a relative maximum nor a relative minimum point.
    
    (b) has a relative maximum but not a relative minimum point.
    
    (c) has a relative maximum and a relative minimum point.
    
    (d) has a relative minimum but not a relative maximum point.
    
    (e) It is not possible to determine whether has a relative maximum or a relative minimum point.
    
    
(4) Given , find the value of such that is continuous on

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


(5) The functions and and their first and second derivative and are defined on At they take on the values given in the following table



If compute

    (a)
    
    (b)
    
    (c)
    
    (d) 2
    
    (e) 10


(6) Referring to the table in Question 5, if compute

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


(7) Referring to the table in Question 5, find an equation of the tangent line to the graph of at the point with -coordinate

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


(8) Find all real numbers that satisfy

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


(9) Let . Find if it exists

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


(10) Find all critical number(s) of

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


(11) Find all critical number(s) of

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


(12) Find the smallest and largest values of on

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


(13) Find the slope of the tangent line to the graph of at

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(14) Which of the following functions satisfy

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


(15) A basketball is dropped from the height of 6 feet. Use the formula for the position function, seconds. How long (in seconds) does it take for the ball to be at height of feet?

    (a) 2
    
    (b) 4
    
    (c)
    
    (d)
    
    (e)


(16) A ladder 13 ft long rests against a vertical wall and is sliding down the wall at the rate of 3 ft/sec at the instant the foot of the ladder is 5 ft from the base of the wall. At this instant, how fast is the foot of the ladder moving away from the wall?


(17) Let be a function on and for and and and for Sketch the graph of labeling any relative extrema and inflection points.


(18) Find an equation for the tangent line to the graph of the function at the point with -coordinate


(19) For the function if find


(20) A box with an open top and a square base is to be built so that the height of the box plus the length of one of the sides is to be 8 meters. Find the exact dimensions for such a box that yields the maximal volume, justify your reasoning by showing all steps in your solution.     

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