Math 132, Calculus 2, Exam 2 Practice Problems,
 Tom Linton, Fall 2003
The following questions contain many of the important concepts for our second exam that covers sections 6.3 to 7.6 in the Hughes-Hallett 3rd edition calculus textbook. This practice sheet is by no means a complete list of the topics you should know, it is simply a collection of problems to help prepare you for the exam. Most likely, none of these questions will appear on the exam, but if you can do all of these problems without much trouble, you should be well prepared for the test.
  1. A car starts with initial velocity of 5 meters / sec and accelerates at -0.6t + 4 meters / sec2 for t = 0 to 12 seconds.
    1. Find formulas for the car's velocity after t seconds, v(t), and distance traveled after t seconds, x(t).
    2. How far does the car go in 12 seconds?
    3. How long does it take the car to travel 100 meters? How fast is it going at this time? You'll probably need some sort of technology, like your calculators root or zero finder, to solve for this time.
  2. Consider F[x]
    1. What is F'[x]? How about F'[3]?
    2. Make a plot of the integrand, f[x] from t = 2 to 5.
    3. Use MID[8] to estimate F[3]. Is this an over-estimate, or an under-estimate? How do you know?
    4. Use MID[16] to give another estimate of F[3].
    5. Assume that the error in MID[16] is exactly one fourth the error in MID[8] to give an even better estimate of F[3].
    6. Now calculate F[3] exactly as follows:
      1. Make the substitution t = 5sin(theta), given by a right triangle with lower-left angle theta, base the square root of 25 - t2, height t, and hypotenuse 5. You can leave the bounds in terms of t.
      2. Use the trigonometric identity sin2(theta) = 1 - cos2(theta) on the numerator of the resulting integrand, and split the integral into two pieces.
      3. Use the tables to integrate 1 / cos(theta), the second integral is elementary.
      4. Draw 2 new triangles, one with t = 2 inserted, the other with t = 3 inserted, to get an exact value for F(3). You should be able to evaluate all of the standard trig functions (sine, cosine, etc.) using these 2 new triangles.
  3. Use the substitution u = 1 + x3 to completely transform (bounds and all) over to terms of u. Then evaluate the integral.
  4. Evaluate with as little use of the tables as possible (I think you should be able to do these without using the tables at all) the integrals on page 339 and 340 numbers 12,13,14,60,61,62,115,116,117,118. You should show enough work to convince me that you did these integrations "by hand", and not with technology.
  5. Find the most general antiderivative F(x) for each of the following functions f(x), using the tables of integrals (and perhaps a substitution or integration by parts).
    1. f(x) = xe2x^2cos(4x2), hint, start by letting u = 2x2.
    2. f(x) = x4 ln(x).
    3. f(x) = sin3(3x)cos2(3x).
  6. Three graphs of f(x) are shown below, with vertical lines drawn at x = 1 and x = 4. Approximating using LEFT(3), MID(3), and TRAP(3) gave the values (listed from smallest to largest, NOT necessarily in the order stated above), I got 11.8198, 12.0904, and 13.8701.


    1. On the first graph above, draw in the 3 rectangles for LEFT(3).
    2. On the 2nd graph above, draw in the 3 rectangles for MID(3).
    3. On the 3rd graph above, draw in the 3 trapezoids for TRAP(3).
    4. Match each of the rules (LEFT, MID, and TRAP) with the approximate value it produced. State how you decided which rule produced which value, and then decide where the true value of the integral lies (between which two approximations).
    5. What is RIGHT(3)? Hint: you know LEFT(3) and TRAP(3) and hopefully you know a formula that relates TRAP(3) to LEFT(3) and RIGHT(3), which you can solve for RIGHT(3).
    6. What is SIMP(3)?
    7. The y tickmark in the graphs is at an integer value (the same one for all 3 plots). It is also at the average value of f(x) from x = 1 to 4. What is the y-value of that tickmark?