Calculus Review Sheet 2

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

(1) Given determine the function such that  
    
    

(2) Find an equation of the tangent line to the given curve at the given point

     

(3) Determine all possible real numbers so that the graph of the function



has a horizontal tangent for at least one

(4) Differentiate the function given by

(5) Differentiate the function given by

(6) Differentiate the function given by

(7) If where is a differentiable function, find

(8) Suppose is a differentiable function on Let and Find expressions for and

(9) Let be a differentiable function of Use to prove that



when Use the formula to find given

(10) Show that What do you think is the importance of the exercise?

(11) Use implicit differentiation to find   at for

(12) Use implicit differentiation to find   at for

(13) Use implicit differentiation to find   at for

(14) Use implicit differentiation to find   at for

(15) Show that the sum of the - and -intercepts of any tangent line to the curve is equal to

(16) Use logarithmic differentiation to find   for

(17) Use logarithmic differentiation to find   for

(18) All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each side is (a) 1 centimeter and (b) 10 centimeters?

(19) A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep?

(20) A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. What is the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground?

(21) Find the differential of the function

(22) Find the differential of the function

(23) Use differentials to find an approximate value for the real number

(24) Use differentials to find an approximate value for the real number

(25) Use differentials to find an approximate value for the real number

(26) Find the linearization of the function at

(27) Find the linearization of the function at

(28) Find the critical numbers of the given functions.

(29) Find the absolute maximum and absolute minimum values of the given function on the given interval.

(30) Show that is a critical number of the function but does not have a relative extremum at

(31) Consider the cubic function where Show that can have zero, one, or two critical numbers and give examples of each.

(32) Explain why the function must attain a minimum in the open interval

(33) Determine the values of the constants and such that the following functions satisfies the hypotheses of the Mean Value Theorem on the given interval.

    (a)   on   

    (b)   on   
    

(34) Assume Let Prove that for any interval the value of guaranteed by the Mean Value Theorem is the midpoint of the interval.

(35) Show that the following equations have exactly one real root.
    
    (a)         (b)

(36) Use the Mean Value Theorem to show the following

    (a)         (b) when
    

(37) Let and Show that for all in their domains. Can we conclude that is constant?

(38) What are the critical points of given . On what intervals is increasing or decreasing? At what points, if any, does assume local maximum and minimum values?

(39) Given find the intervals on which the function is increasing and decreasing. Then identify the function's local extreme values, if any, saying where they are taken on. Which, if any, of the extreme values are absolute? Support your finding with a graph of the function.

(40) Given find the intervals on which the function is increasing and decreasing. Then identify the function's local extreme values, if any, saying where they are taken on. Which, if any, of the extreme values are absolute? Support your finding with a graph of the function.

(41) Identify the function's local extreme values for in the given domain of and say where they are assumed. Which of the extreme values, if any, are absolute? Support your finding with a graph of the function.  

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