Calculus Review 1

    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) If

then

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e)
    
    
(2) Some money, dollars, is invested so that, after years, the total amount of money is given by dollars, where is a constant. The total amount doubles every 6 years. Find the exact value of


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


(4) Let and be constants and define

    (a) How should and be related to each other in order for to be a continuous  function on
    
    (b) How should the values and be restricted in order for to be continuous on


(5) (a) Find the exact limit, if it exists; explain. (b) Find the exact limit, if it exists, where is defined by explain. (c) Find the exact limit, if it exists; explain.


(6) Justify the claim that has no horizontal tangent line.


(7) In this question,

    (a) The derivative, of is given by Show that this formula for is correct by using only the limit of a difference quotient.
    
    (b) As in (a), Find the equation of the tangent line to the graph of at the point


(8) Suppose Given that the derivative, of is given by find the equation of the tangent line to the graph of at the point

    (a)
    
    (b)
    
    (c)
    
    (d)
    
    (e) none of the above


(9) If then is discontinuous at

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


(10) If then is continuous at

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


(11) If then is

    (a)
    
    (b) 0
    
    (c)
    
    (d)
    
    (e)


(12) If then is

    (a)
    
    (b)
    
    (c) 0
    
    (d)
    
    (e)


(13) If then is

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


(14) If then is

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


(15) If then is

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


(16) If then is

    (a)
    
    (b)
    
    (c) 0
    
    (d) 1
    
    (e)


(17) The functions and their derivatives, are defined on and at they take on the values given in the following table.



    (a) Let and find if it exists; of it does not exist explain why not.
    
    (b) Let and find if it exists; if it does not exist, explain why not.
    
    (c) Let and find if it exists; if it does not exist, explain why not.
    
    (d) Let and find if it exists; if it does not exist, explain why not.
    
    (e) Let find if it exists; if it does not exist, explain why not.


(18) Referring to the previous question, find the equation of the tangent line to the graph of at the point with -coordinate 2.


(19) Let

    (a) Sketch the graph of
    
    (b) Justify whether of not is continuous on
    
    (c) Justify whether of not is differentiable on

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Cite this as:
Calculus Review 1
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/calculus-review-1.html