Calculus Practice Test 2

Show all work and justify each step.

(1) Find the difference quotient for

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(2) Find where

    (a)
    
    (b) 2
    
    (c)
    
    (d) 0
    
    (e) does not exist


(3) The derivative of at a point is given by

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


(4) Find if

    (a) 4
    
    (b) 8
    
    (c) 7
    
    (d) does not exist
    
    (e) 3


(5) Evaluate the limit

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


(6) The function is discontinuous

    (a) at only
    
    (b) at only
    
    (c) nowhere
    
    (d) at and only
    
    (e) at only


(7) If then is discontinuous at

    (a) 1 only
    
    (b) only
    
    (c) and only
    
    (d) no point of the domain
    
    (e) every point of the domain


(8) Constants and are chosen such that the function is continuous for all The product is

    (a) 9
    
    (b) 2
    
    (c) 6
    
    (d) 3
    
    (e) undefined


(9) How many distinct real solutions are there to the equation

    (a) 0
    
    (b) 1
    
    (c) 2
    
    (d) 3
    
    (e) 4


(10) If and are differentiable functions such that and find

    (a) 16
    
    (b)
    
    (c)
    
    (d) 8
    
    (e) 12


(11) If and are differentiable functions such that such that and find

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


(12) If then, at is

    (a) differentiable but not continuous
    
    (b) continuous but not differentiable
    
    (c) not continuous and not differentiable
    
    (d) continuous and differentiable
    
    (e) not defined


(13) Show that the function satisfies the equation


(14) Find a value for so that You should solve algebraically; no credit will be given for the use of L'Hospitals's Rule.


(15) Sketch clear, well-labeled graphs that depicit functions and that satisfy the indicated conditions.
    
    (a) The function has domain and is continuous on but is not continuous on
    
    (b) The function is not continuous at but if is changed to then becomes continuous at
    
    (c) The function is continuous on except at the function is continuous on except at and for all in (Sketch and on separate coordinate systems).


(16) Suppose the world's population grows according to the formula where is the number of years since the population is and is the growth rate. If the population is now 6.1 billion people and if it continue to grow according tot he given formula with a growth rate of how long (to the nearest year) will it take before there is only 1 square yard of land per person? (The Earth contains approximately square of land and


(17) Find an equation for the tangent line to the graph of the function that is parallel to the line
        

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