Calculus II Final Exam Sample Problems and Hints

  1. Give short (no more than 2 sentences) definitions of each of the following:
    1. lim f(x)
      x -> a

      (Hint: Describe it numerically or graphically!)

    2. +ab f(x) dx
      (Hint: It has something to do with Riemann sums)
    3. A solution to a differential equation
      (Hint: It's a function that...)
    4. A convergent improper integral
      (Hint: Improper means what? Convergent means what?)
    5. An interval of convergence of a power series
      (Hint: It helps to start by thinking that you have a function that's hard to evaluate, that you come up with a power series for the function that's easy to plug into that 'looks like' the function, or that is 'close' to the function. Now what?)

  2. A function y = f(x) is increasing and concave up; below are four numerical approximations of the area under the graph. Which could be trapezoidal, which midpoint, which Simpson's rule, which right or left? Remember to explain your answer.

    (Hint: Which would be furthest? Closest? Above? Below? In a pinch, you could use the fact that SIMP(N) = (2*MID(N) + TRAP(N))/3...)

  3. Draw a function where the trapezoidal rule would overestimate the area, the midpoint rule would underestimate the area, and where the left rule would give a more accurate answer than the right rule.
    (Hint: Try increasing and concave up; does it work? If not, how could you fix it?)
  4. Draw a histogram or a probability distribution function where the mode, the mean, and the median are all in substantially different places.
    (Hint: Start with a bell curve centered at x = 50; it has the mean, median, and mode all in the same place. Now leave the left half of the graph alone, and take some of the area just to the right of the x = 50 line, and move it way to the right, and make a very narrow, very high bump at like x = 100. What happened to the mean, median, and mode when we did that?)
  5. If MID(7) = 17 and MID(49) = 20, approximate the error in TRAP(7).
    (Hint: This is the number line thing, where you put MID(7) on it, MID(49) on it, then put another mark where you think the real value should go. Then you use the fact that each time you increase the number of divisions (for MID) by N, you divide the error by a factor of N2. How do you finish it?)
  6. Show that the improper integral
    +1 1/(x3+2) dx

    converges, and approximate its value to one decimal place.
    (Hint: Compare it to +1 1/(x3) dx, whose convergence or divergence we already know. Next, you need to locate the 'fencepost' value. Then what?)

  7. This question refer to the integral
    +310 sin(2*Pi*x/5) dx:
    1. Draw TRAP(4).
      (Hint: Divide the interval [3,10] into four equal parts, then connect up what?)
    2. How many different intervals would you have to split [3,10] into in order to use TRAP and MID to approximate correctly to one decimal place? Note that you are not being asked for the number of divisions!
      (Hint: TRAP is an overestimate when? an underestimate when?)

  8. The region bounded by y=0, x=0 and y=4-x2 is revolved about the line x = 5. Sketch the solid generated, and give the approximate volume of one arbitrary chop. (This means you can not find a specific chop and give its volume.)
    (Hint: The answer is the easiest hint to give; if you're stuck, try to work to it: (52 - (5-²(4-y))2)(Delta x))
  9. Set up, but do not evaluate, an integral which is equal to the area bounded by the graphs of y=x1/3 and y=x/7 + 6/7.
    (Hint: To find the points of intersection, you could use your calculator to get a hint (you would still have to show the work ;) )
  10. Set up, but do not evaluate, an integral equal to the amount of work done (against gravity), pumping all of the water out of an inverted cone of height 10 ft, radius 3 ft, to a height 7 feet above the top of the cone.
    (Hint: Take a slice, which should be a disk. Find its volume, multiply by 62.4, multiply that by the distance your disk has to go. Remember the limits of integration are determined by 'h', the height of the chop.)
  11. Show that the function f(x) = xe-x is or is not a solution to the differential equation,
    dy/dx = -y + e-x

    by any method.
    (Hint: Checking to see if a particular function is a solution is much easier than solving the differential equation. Each side of the equation is a set of directions telling you what to do to the function. If you get the same answer on both sides?)

  12. Recall the triangle probability distribution, which is used when there is a lower bound x = a, a most likely value x = b, and an upper bound x = c. They then draw the diagram and then adjust the height so that the area of the triangle is exactly 1.
    1. Draw such a diagram for the following situation:
      `The student works on calculus x hours daily (at non-exam times)' Remember to explain how you chose a, b, and c.
    2. Use your diagram to approximate the mean and median amounts of time students work on calculus daily (at non-exam times).

    (Hints: Guess at a, b, and c, and draw the triangle. Use the fact that the area under a pdf is 1 to figure out c. The median you can find by guessing at which x-value the area on the left is equal to the area on the right. The mean is harder: Start at your median point, and try to figure out whether the population is further distributed to the left or to the right. Then move a little in whatever direction you figured out! Alternatively, it's the center of gravity of the triangle. Alternatively, it's the integral of xf(x) from a to c, and you could just do it!)

  13. Is the function graphed below a pdf or a cdf? If a pdf, sketch the graph the corresponding cdf. If a cdf, sketch the graph of the corresponding pdf. Use the graphs to approximate the mean, median and the mode.
    (Hint: cdf's always start at height 0 and end at height 1. What's the relationship between a pdf and a cdf? Finding the median and mode are easy; finding the mean you get either by doing an integral, or guessing at where the center of gravity is...)
  14. A detective discovers a corpse in an abandoned building, and finds its temperature to be 27° C. An hour later, its temperature is 21°. Assume that the air temperature is 8°, that normal body temperature is 37°, and that Newton's Law of Cooling applies to the corpse. Write a differential equation satisfied by the temperature, H of the corpse at time t, so that any constants that appear in the equation are positive.
    (Hint: Don't forget! Newton's law of cooling is: the rate of change of the body temperature is proportional to the difference between the corpse temperature at time t and the room temperature. To figure out whether you should put a 'k' or a '-k', figure out whether each side is positive or negative.)
  15. The missing differential equation problem! This should be a problem like the decaying leaf problem...
  16. Find the length of the sine wave y = sin(t), for 0 d t d Pi, correct to one decimal place.
    (Hint: Remember the formula for arclength, and how to approximate integrals!)
  17. Give an integral, but do not evaluate, for the force that water is exerting at the vertical end of a tank, if the end is an isosceles triangle, 6 ft across the bottom, 8 feet from top to bottom, but the tank is only full up to the 5 foot mark.
    (Hint: This is the Venetian blind thing: Make a horizontal slice, making a small rectangle at height h. Use similar triangles to figure out the area of the rectangle. If you flip this rectangle to make it horizontal, you will have the bottom of a box of water, whose weight is the force of the water on our strip. Find the weight of this water, then add up all these weights, then take the limit, etc...)
  18. If f(x) = x1/4,
    1. Find the first four terms of the Taylor series for the function f(x) about a =16, and use it to approximate (16.1)1/4.
      (Hint: Take three derivatives, then use the formula for Taylor series)
    2. f(4)(x) = -231/(256x15/4), find the maximum the error could be if we use the third degree Taylor polynomial about a = 16 to approximate (16.1)1/4.
      (Hint: Use the formula that says that if f(4)(x) d M between 16 and 16.1, then the difference between our polynomial and the function value is at most M(x-0)4/4!. In order to find M, graph the fourth derivative, and use the largest value of f that you see between 16 and 16.1)


  19. Which of the following integrals are improper? Of the improper ones, which ones converge?

    (Hint: To see if they're improper, graph them! (Two of them are, one is not.) To see if the improper integrals converge, remember the definitions! (These are harder than the usual ones to use STFIF and BTIII))

  20. Bob likes to run marathons. In fact, he's made a bet with a friend that he will run his latest marathon in under four hours. If Bob's speed after running s miles is given by
    v(s) = 10 - s/4 miles/hour

    and the marathon is exactly 26 miles long, does Bob win his bet? (Hint: You want to take a small slice of distance Delta s, and use Bob's rate in the form 1/(10-s/4) hours/mile)

  21. Find at least the first six nonzero terms of a Taylor series for the function
    f(x) = +0x (t*et2-t)/(1 + 2t) dt

    (Hint: Start with the series for ex, plug in t2. Multiply that series by t, then subtract t from the whole thing. To deal with the dividing by (1+2t), remember that dividing by (1+2t) is the same as multiplying by 1/(1+2t) = 1 + (-2t) + (-2t)2 + (-2t)3 + .... After all that, you need to integrate from 0 to x (WHEW!))

  22. For the Taylor series
    T(x) = x + 2x2/2 + 22x3/3 + 23x4/4 + ...
    1. Determine which of T(0), T(1/3), T(1) make sense.
      (Hint: For each, plug in the value of x, then go through the list of convergence tests to see if the series converges or diverges.)
    2. Determine the approximate interval of convergence from the graphs of the successive Taylor polynomials.
      (Hint: Start graphing the polynomials you get by adding more and more terms on top of each other. (Start by graphing the polynomial you get with just 2 terms. Next, on top of that, graph the polynomial you get with 3 terms. Then 4 terms. The interval of convergence should be that interval that the graphs of these polynomials stay with each other. For what it's worth, the correct interval is -1/2 d x < 1/2)


  23. Find the interval of convergence for each of the following series, whose nth terms are given, including checking the endpoints:
    1. an = (n+1)/(2n+1) * (x-2)n/3n
    2. an = xn/²n

    (Hint: Do the ratio test to find the basic interval. Then plug each of the endpoints into the series, and go through the list to see which of them applies!)

  24. One Taylor series for ln((1+x)/(1-x)) is 2(x + x3/3 + x5/5 + ...
    1. Find a formula for the nth term.
    2. Graphically, give a rough estimate of the interval of convergence.

    (Hint: What do the terms have in common? What is changing (this should tell you where the 'n' goes?) Graphically, see the hint above.)

  25. For the differential equation
    dy/dx = x + xy:
    1. Use Euler's method to approximate y(3) if y(2) = 1 using n=4.
    2. Solve the equation algebraically.
    3. On a sketch of the slope field, put work pertaining to your answer from (a), and work from your answer from (b).

    (Hint: Don't forget Euler's method! To separate x and y, factor out the x first...)

    1. Find a Taylor series about a = 0 for the function f(x) = x cos(3x).
    2. The graph of the 4th derivative of f(x) on [0, Pi] is graphed below: Use your answer to (a) and information from this graph to both find an approximation for 3cos 9, and to give an upper bound for the error of your approximation.

    (Hint: For (a), remember the Taylor series formula for cos(x), and manipulate it. If you are given the graph of the 4th derivative, find its largest  value, and call it M. Then we know that |f(x) - P(x)| < M(x-0)4/4!)

    1. Find a Taylor series for the function f(x) = cos(x2) about a = 0. (or x/(1+x2), or x2 sin(x2), etc.
    2. By graphing more and more of the Taylor polynomials, develop a guess for the interval of convergence.
    3. Use your Taylor series to approximate f(1).
    4. Find a bound for the third derivative of your function by any means.
    5. Use your bound to explain roughly how good your approximation is.

    (Hint: See all the above hints!)


The End!