Calculus Review 2

    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) Suppose that a bacteria population starts with 600 cells of bacteria and triples every hour.

    (a) What is the population after one hour?
    
    (b) What is the population after two hours?
    
    (c) What is the population after three hours?
    
    (d) What is the population after hours?
    
    (e) Find the rate of increase of the bacteria population after five hours.
    
    
(2) Functions and and their first and second derivatives, are defined on and at they take on the values in the following table.



    (a) Let Compute if it exists; if it does not exist, explain why not.
    
    (b) Let Compute if it exists; if it does not exist, explain why not.
    
    (c) Let Compute if it exists; if it does not exist then explain why not.
    
    (d) Let Compute   if it exists; if it does not exist then explain why not.     

    (e) Let Compute   if it exists; if it does not exist then explain why not.
    
    (f) Let Compute   if it exists; if it does not exist then explain why not.
    
    
(3) Functions and and their first and second derivatives, are defined on and at they take on the values in the following table.



    (a) Let Compute if it exists; if it does not exist, explain why not.
    
    (b) Let Compute if it exists; if it does not exist, explain why not.
    
    (c) Let Compute if it exists; if it does not exist then explain why not.


(4) Functions and and their first and second derivatives, are defined on and at they take on the values in the following table.



    (a) Let Compute if it exists; if it does not exist, explain why not.
    
    (b) Let Compute if it exists; if it does not exist, explain why not.
    
    (c) Let Compute if it exists; if it does not exist then explain why not.
    
    (d) Let Compute   if it exists; if it does not exist then explain why not.     
    
    
(5) Some function and their corresponding derivatives are shown. Give the steps needed to compute the derivatives (NOT by using a limit) and state which rule you used.

    (a) ;
    
    (b) ;

    (c) ;

    (d) ;


(6) Given that compute at the point


(7) Given that and that compute at the point

    (a)
    
    (b)
    
    (c)

    (d)
    
    (e) none of these


(8) A block of ice, in the shape of a cube, originally having volume is melting in such a way that the length of each of its edge is decreasing at the rate of Assuming that the block of ice maintains its cubical shape, find the rate of change of the surface area of the cube at the time the volume is In so doing, sketch a picture of the situation, labelling relevant items, and lay out your work very clearly.


(9) A balloon is rising vertically at a constant speed of 5 meters per second. A dog is running along a straight line at 15 meters per second, chasing the balloon, and overshoots it. When the dog passes under the balloon, the balloon is 45 meters above the dog. How fast is the distance between the dog and the balloon increasing three seconds after the dog passes under the balloon?

    (a) 12.5 m/s
    
    (b) 13 m/s
    
    (c) 13.5 m/s
    
    (d) m/s

    (e) none of these
    
    
(10) Suppose that we know that a function has derivative for all and that Use a tangent line approximation to estimate the value of

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above
    
    
(11) If and then is

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


(12) Find the absolute maximum of

    (a)

    (b) 2

    (c) 5

    (d) 7

    (e) none of these


(13) Sketch the graph of one function with all of the following properties:

    (i) for
    
    (ii) for
    
    (iii) for
    
    (iv) for
    
    (v) for
    
    
(14) Suppose is a function with the property that Find where

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


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

    (a)
    
    (b)
    
    (c)
    
    (d) undefined
    
    (e) none of these
    
    
(16) Interpret as a derivative.

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


(17) Interpret as a derivative.

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


(18) If then is

    (a) 0

    (b)

    (c) 1

    (d) does not exist

    (e) not enough given information


(19) If then is

    (a) 0

    (b)

    (c) 1

    (d) does not exist

    (e) not enough given information


(20) If a function has deriviative then has a local maxmium at

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


(21) If a function has deriviative then has a local minimum at

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


(22) If then the absolute minimum of over the interval is

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


(23) If , then the absolute minimum of over the interval is

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


(24) All the critical numbers of are
     
     (a)
     
     (b)
     
     (c)
     
     (d)
     
     (e)


(25) A television camera is positioned 4,000 ft from the base of a rocket-launching pad. A rocket rises vertically and its speed is 600 ft/sec whne it has rises 3,000 ft. How fast is the distance from the television camera to the rocket changing at the moment?

     (a)
     
     (b)
     
     (c)
     
     (d)
     
     (e) not enough given information


(26) If then the value of at is
     
     (a)
     
     (b)
     
     (c)
     
     (d) does not exist
     
     (e) not enough given information
     
     
(27) If for then is

     (a) 0
     
     (b)
     
     (c)
     
     (d) does not exist
     
     (e) not enough given information
     
     
(28) If then is

     (a) 0
     
     (b)
     
     (c)

     (d) does not exist
     
     (e) not enough given information


(29) If or some differentiable function then is

     (a)

    (b)
    
    (c)

     (d) does not exist
     
     (e) not enough given information


(30) If or some positive-valued differentiable function then is

     (a)

    (b)
    
    (c)

     (d) does not exist
     
     (e) not enough given information


(31) Find the smallest and largest values of on

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


(32) 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. The total area enclosed by the square and circle is given by the formula where is the length of the peice that is bent to form the square. To maximize the toal area enclosed, the length should be

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


(33) Find two numbers whose product is a minimum, given tht one of the numbers is nine less than one-fifth of the other. Fully justify your answer mathematically.

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

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Cite this as:
Calculus Review 2
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/calculus-review-2.html