Calculus Review Sheet 1

    This topic is a collection of problems and concepts that might help someone understand their working knowledge of Calculus 1.

(1) Estimate the limits, if they exist, by using a table of values to two decimal places:

    (a)         (b)     (c)

(2) Find so that the function satisfies

(3) Consider the function Estimate by evaluating at -values near 0. Sketch the graph of

(4) Explain why does not exist.

(5) The tabular approach is a convenient device for discussing limits informally, but if it is not used carefully, it can be misleading. For example, for let



    (a) Construct a table showing the value of and for and Based on this table what would you say about
    
    (b) Construct a table showing the value of and for Based on this table what would you say about
    
    (c) Based on your results in (a) and (b) what do you conclude about

(6) Determine so that

(7) Determine so that

(8) Determine so that

(9) Evaluate and Conjecture a formula for and explain.  

(10) Find the one-sided limit algebraically,

(11) Find the one-sided limit algebraically,

(12) Find the two-sided limit,

(13) Find the two-sided limit,

(14) Find the limit of the function as and .

(15) Find the limit of the function as and .

(16) Find the limit of the function as and .

(17) Find the limit of the function as and

(18) Find the limit of the function as and

(19) Find the limit of the function as and

(20) Graph the rational function and include the graphs and equations of the asymptotes.

(21) Sketch the graph of a function that satisfies the conditions: and

(22) Find constants and so that the function will be continuous for all in the domain.

(22) Find constants and so that the function will be continuous for all in the domain.

(23) Find constants and so that the function will be continuous for all in the domain.

(24) Let and . Show that

(25) Prove that the function must have at least one real root.

(26) Prove that the function must have at least one real root.

(27) Find an equation for the tangent to the curve at the point Then sketch the curve and tangent together.

(28) Find an equation for the tangent to the curve at the point Then sketch the curve and tangent together.

(29) At what points does the graph of the function have horizontal tangents?

(30) What is the rate of change of the area of a circle with respect to the radius when the radius is

(31) Does the graph of have a tangent at the origin? Give reasons for your answer.

(32) Using the definition of the derivative, calculate the derivative of the function and then evaluate the derivative function at and

(33) Using the definition of the derivative, calculate the derivative if   

(34) Find the value of the derivative when

(35) Using the definition of the derivative of a function find the derivative function given

(36) Using the definition of the derivative of a function find the derivative function given

(37) Determine so that given

(38) Determine so that given

(39) Find the first, second, and third derivatives of the following functions.

    (a)

    (b)
    

(40) Show that the curve has no tangent lines with slope

(41) At what points on the curve is the tangent line parallel to the line

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