Review Problems

Answer the following on your own blank sheets of paper.
Note: This sheet, while intended to help you review, is NOT intended to be a comprehensive, representative review for your Fina

  1. What is the domain of g(q) = (6-q)-1/2?


  2. "f(x) is an odd function if for any x in Domain(f), ______________."


  3. Let f(x) = x+1, g(x)=x-1, h(x) = x2, k(x)=ex. Find:

    1. f(f(x))
    2. f(g(x))
    3. g(f(x))
    4. g(g(x))
    5. g(k(x))
    6. f(k(x))
    7. k(f(x))
    8. k(h(f(x)))
    9. k(h(g(f(x))))


  4. Give a graph of the following parametric curves (without using your graphing calculator), where t ranges through all values for which the expressions x(t) and y(t) make sense. State what this range is.


    A. x(t)=t and y(t)=t2
    B. x(t)=t and y(t)=sin(t)
    C. x(t)=et and y(t)=et
    D. x(t)=cos2(t) and y(t)=sin2(t)


  5. (a) How can you test whether f(x) is a function from its graph?
    (b) How can you test whether a function f(x) has an inverse, from its graph?


  6. (a)Explain why ln(e)=1.
    (b) If ln(A)=a and ln(B)=b, what is ln(AB)? ln(A/B)? ln(AB)? ln(BB)?


  7. Evaluate (a) the limit, as x approaches 4, of (4-x)(2-x1/2)-1
    (b) The limit, as x approaches infinity, of (3 x2-5x+7) (7 x3 - 5 x2- 3x -1)-1.
    Do not just give an answer; prove that the limit is as you state.


  8. True or false? (If true, why? ; if false, why not? )

    (a) f(x) is continuous at a if (i) the limit of f(x) as x approaches a from the left exists and (ii) the limit of f(x) as x approaches a from the right exists and (iii) these two limits are equal.


    (b) f(x) is differentiable at a if f(x) is continuous at a.


    (c) f(x) is continuous at a if f(x) is differentiable at a.


    (d) If f(x) is differentiable at a, then the limit as x approaches a of f(x) must equal f(a).


    (e) If f(x) is not differentiable at a but is continuous at a, then the limit as x approaches a of f(x) might still equal f(a).


    (f) If f'(x) is negative, then f''(x) is decreasing.


    (g) If f'(x) is increasing, then f(x) is concave up.


    (h) If f'(x) is decreasing then f''(x) is negative.


  9. For each function, list all of its vertical and horizontal asymptotes, if any.

    (a) 1/x
    (b) ln(ln(x))
    (c) (3x-5)/(x-2)

End of review problems.
If you have extra time, choose from among the following:

  • Without using your graphing calculator, graph the following functions. Be sure to label key points on the x-axis and y-axis

    1. x2
    2. (x-4)2 - 4
    3. 3 x2
    4. (3x)2
    5. ((1/3)x)2
    6. sin(x)
    7. cos(-x)
    8. |sin(x)|
    9. ln(x). What is the domain? What is the range?
    10. log2(x) What is the domain? What is the range?

  • Sit in a circle of 3-5 people. Each member of the group sketches the graph of a function f(x) -- labeling the axes, any asymptotes, etc, carefully -- and passes the graph to the person sitting on their left, who must sketch (a) f'(x) (b)The graph of some function F(x) whose derivative is the original function f(x). Repeat this exercise as often as you like, with increasingly complex functions.

  • Play a few friendly rounds of the " I challenge you!" activity, in which the challenger specifies the number of local/absolute maxima/minima and critical numbers, and possibly the points in the domain at which these take place.

  • Play the derivative version of "I challenge you" with the challenger coming up with a function f(x), and the challenge-ee needing to find f'(x). The only rule is: the challenging person must first calculate f'(x) themselves on a separate sheet of paper. At the end, compare your answers, and make sure they equal one another, and maybe have a third person calculate f'(x) to make sure you all have the right answer. Work up towards more and more complex/esoteric functions.

  • Play the implicit-differentiation version of "I challenge you".

  • Play the bloon related-rates version of "I challenge you". This one will require careful planning, so you will not be able to do many rounds of this game.

    A spherical balloon is being inflated, and one has V(t) for the volume at time t, r(t) for the radius at time t, and S(t) for the surface area of the balloon at time t, and d(t) the diameter of the balloon at time it. The challenger must make up a related rates problem in which they specify enough of the values of these functions and/or their derivatives, and asks for the rate of change of ______ (one of the variables) either "when time is _______" or "when [other variable] is _______".