Calculus Practice Test 3

Show all work and justify each step.

(1) Let and be functions of Find if and when and

    (a)
    
    (b) 3
    
    (c) 2
    
    (d)
    
    (e)
    
    
(2) The volume of a sphere of radius is When a spherical balloon is inflated, the radius of the balloon is increasing at the rate of cm/min, how fast is the volume changing when the radius is cm?

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(3) Find the differential

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(4) Find all critical number(s) of on the interval

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(5) Let The first step in finding the derivative of is

    (a) and
    
    (b) with and
    
    (c) with and
    
    (d) with   and
    
    (e) with   and
    
    
(6) Assume that the position at time of an object moving along a line is given by on the interval The total distance traveled by the object during the indicated time interval is

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(7) Evaluate the limit: where and are nonzero constants.

    (a)
    
    (b)
    
    (c)
    
    (d) does not exist
    
    (e)
    
    
(8) If you drive from here to Dallas, which theorem implies that your average speed for the whole journey will equal your instantaneous speed attained at some moment(s) during your journey?

    (a) The Mean Value Theorem
    
    (b) The Intermediate Value Theorem
    
    (c) The Squeeze Theorem
    
    (d) The Extreme Value Theorem
    
    (e) The Constant-Difference Theorem
    
    
(9) You are given where and are nonzero constants. Find the value of that gives a relative extremum at

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) no such value
    
    
(10) For every function the definition of an inflection point on the graph of is a point where

    (a) is defined but is not defined
    
    (b) is defined and
    
    (c)
    
    (d) the concavity of the graph changes
    
    (e) both and are not defined
    
    
For problems 11 and 12, refer to the table below.




(11) If compute

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(12) If compute

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(13) A function has domain and derivative Find the critical numbers of and identify which type of extremum (if any) each one yields.

    (a) there is a relative minimum at and a relative maximum at
    
    (b) there is a relative maximum at and a relative minimum at
    
    (c) there is a relative maximum at is a critical number which is not a relative extremum
    
    (d) there is a relative minimum at is a critical number which is not a relative extremum
    
    (e) there is a relative maximum at is a critical number which is not a relative extremum
    
    
(14) Two people start from the same point. One walks east at 3 mile/hour and the other walks north at 2 mile/hour. How fast is the distance between the people changing after 15 minutes?


(15) Find an equation for the tangent line to the graph of the equation at the point


(16) A box with a square base is constructed so that the length of one side of the base plus the height is 10 in. What is the largest possible volume of such a box?


(17) Sketch the graph of a function with ALL the following properties:

    (i)
    
    (ii)
    
    (iii)
    
    (iv) does not exist,
    
    (v) if
    
    (vi) if
    
    (vii) if
    
What can be said about the inflection points of Explain.


(18) A person standing at the edge of a cliff throws a rock directly upward. It is observed that 2 seconds later the rock is at its maximum height (in ft) and that 5 seconds after that, it hits the ground at the base of the cliff. The height of the rock is given by the formula where is the initial velocity and is the initial position.

    (a) Find the height of the cliff.

    (b) Determine the velocity of the rock when it hits the ground.

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