• You will be asked to sketch a possible antiderivative of a function that is given in a graph.
• You will also be asked to write down a particular definite integral of a function, given a graph of an antiderivative of that function.
• There will be a problem where you need to compute several indefinite integrals and definite integrals.  These involve substitutions and integration by parts.
• There will be a problem dealing with motion subjected to gravitational acceleration.
  • You will be asked to derive the solution to a particular initial value problem.

Click here for page 1 of answers to the first exam, second version
Click here for page 2 of answers to the first exam, second version
Click here for a PDF copy of the first exam, second version

The second exam:

Caution: Although this exam will be open book, on-computer, you will need to know what you are doing to finish during the class time.

• There will be three improper integrals.  You will be asked to state whether each converges, and to find the values of those that converge.  You will need to explain your answers.
• There will be a problem involving pressure or density.  You will need to compute a force or a weight as an integral.  The integral may be an improper integral (for example, if the object or region extends out indefinitely).
• You will be given a density function as a graph, and you will be asked to estimate certain values for the associated population distribution.

The second exam, part 2:

• You will have a word problem involving work.  Pay particular attention to the problems involving pumping water or another liquid out of the top a container of some shape.
• You will have a word problem involving arc length.
• You will need to compute the volume of a solid of revolution.

The third exam

• You will need to write down the Taylor polynomial of specified degree to a specified function.  You may need to use some of the special techniques from §9.3 of the text.
• You will need to write down an error term for the Taylor polynomial. This is best done with the error formula derived in class on Friday, October 29.
• You will need to say whether the Taylor polynomial is an overestimate or an underestimate for the value of the function over a specified interval.
• You will need to use the error term you derived to bound, both below and above, the possible actual errors in the approximation, over the specified interval.
• You will need to use a computer program or calculator to numerically compute the actual approximation error at a particular point, and to compare that to the error in the Taylor polynomial.
• You will need to compute the radius of convergence of a power series.  Based on this radius of convergence, you will need to write down an interval of values on the variable within which the series converges.

The fourth exam
 

You will sketch the slope field of a particular differential equation, and sketch an approximate graph of a particular solution to that differential equation, using your slope field sketch as a guide.

You will solve an initial value problem with separation of variables, showing all your work.

You will use Euler's method to obtain approximate solutions to a particular initial value problem.  You will be able to get an analytical solution in this case, and you will compare the different approximate solutions by computing ratios of errors.

There will be a 15 point extra credit problem (added to any exam grade).  In this problem, you will compute approximate solutions to a differential equation with Euler's method, and compare the approximate solutions to the exact solution by graphing the solutions in Matlab.  (You will hand in printouts of your graphs.)

You will solve a word problem involving either dilution (as explained in class on Friday 11/23/1999) or computation of temperature equilibria.

The final exam
The final exam will have problems that are similar to problems on the first four exams, with the following exceptions and admonishments.

• You will need to use Mathematica, Matlab, or a programmable calculator for one computation, involving numerical approximation of a definite integral with a large number of subintervals. In this problem, you will need to analyze the convergence by computing a ratio of errors.
• Pay close attention to the methods of proving improper integrals convergent or divergent by comparison with simpler integrands.
• Pay close attention to the problems involving work required to pump fluids into and out of tanks.  (Many people missed this type of problem on previous exams.)
• There will be a problem involving the differential equation model of heating and cooling introduced in the text.