The first exam

Numerical grades will be retained for use in computing the final letter grade in the course. Students with grades of D or F on this exam should be very concerned about their likely success in this course. Indeed, students with low grades should evaluate the amount of time and effort they can devote to this course as part of their entire plan of work for the balance of the semester. Such students may wish to drop this course or drop another course in order to increase their chances of success in what remains. I strongly recommend considering such actions.
Class attendance and watching the instructors do problems is not adequate practice. Individual time and effort must be given.
We will continue to use improper integrals, estimation techniques, integration by parts, and familiarity with functions and with algebraic manipulation a great deal in this course, and much of what we will do is important to your later technical education and practice. Please don't waste your time and effort with false expectations. I want students to be successful and will work diligently to help this occur, but I don't want to deceive them.
Here are approximate letter grade assignments for this exam:

Discussion of the grading

Minor errors (missing factor in a final answer, sign error, etc.) will be penalized minimally. Students whose errors materially simplify the problem will not be eligible for most of the problem's credit.

Problem 1 (12 points)
2 points for setting up the pieces in the partial fraction expansion. 2 points for getting an equation relating the top to the constants in the pieces. 3 points for correct solution of this equation. 3 points for antidifferentiation. 2 points for the answer (1 of these for using ln properties to verify the displayed result).

Problem 2 (14 points)
a) (10 points) 3 points for the formula: Pi₀²(sin(x))⁵dx. 5 points for the antidifferentiation and 2 points for the answer.
b) (4 points) 2 points for the correct integrand and 2 points for the correct bounds on the definite integral.

Problem 3 (12 points)
This is a force·distance problem. Modeling the force (the weight of the bucket) is worth 5 points. This should be a linear function: A+B(Variable). Also this function should be at least 1 in the variable's interval (the student is advised that the weight of bucket+water is at least 1 pound). The limits of integration should be the length that the bucket is raised: 2 points. The distance part is dVariable and earns 2 points if the limits of integration are correct.
3 points remain for the integration and answer (which is worth 1 of the 3 points). A student's answer is eligible for these three points if the function being integrated is linear, not constant, and positive everywhere in the interval of integration whose length must be the length the bucket is lifted, otherwise the student is not doing the requested problem. An integration of any random function (a number of people had quadratic integrands) on any random interval will not earn credit.

Problem 4 (10 points)
Stating one useful substitution gets 4 points (2 points if the dx and x are not consistent!). Correctly carrying out the substitution to get an antiderivative (including substituting back to x's!) is 6 points.
Beginning a good integration by parts strategy also will earn 4 points, and then carrying it out can earn the remaining 6 points.

Problem 5 (14 points)
a) (3 points) The sketch should have a straight line segment and a curve which intersect at (0,0) and (1,Pi/2) for arcsin and which intersect at (0,0) and (1,Pi/4) for arctan. The curve should be increasing and concave up for arcsin and increasing and concave down for arctan. 1 point will be deducted for each indicated item which is not correct up to a possible 3 points.
b) (9 points) A correct setup (definite integral and integrand) earns 2 points. 5 points for correct antidifferentiation, and 2 points for evaluation.

Problem 6 (12 points)
6 points for correct integration by parts and getting the antiderivative (4 points for the first u,v step and then 2 points for the final antiderivative). 3 points for setting up an integral from 1 to a large number and substituting (1 of these points was earned for writing the improper integral as a limit of a proper integral with a parameter). 1 point for finding the limit correctly, and 2 points for showing where L'H is used and using it correctly. Just "plugging in" infinity in the antiderivative is not an acceptable strategy!
Note No points were earned if the student ignored the integral and applied L'H to the integrand and presented some related result as the "solution". This is a totally different and much easier problem. It is not the problem students were asked to solve.

Problem 7 (14 points)
a) (6 points) There should be seven terms, values of square root at the appropriate x's, "weighted" appropriately (1/4/2/4/2/4/1), and these should be multiplied by (6-0)/(6·3). The multiplier is worth 1 point, the weights are worth 2 points, the appropriate x's are worth 2 points, and square root appearing correctly is worth 1 point. No function evaluations need to be done. If the function in the sum is only identified as "f" then the 1 point for square root is not earned.
b) (8 points) Use of the second derivative graph to get some estimate of the size of |f´´(x)| on [1,3] is worth 2 points. Use of the Trapezoidal Rule error term is worth 2 points. The correct N earns 1 point, and getting some correct numerical error estimate is 1 point. A correct (interval!) answer earns the last 2 points and this can only be earned with a correct error estimate.
Comment Some answers confuse the estimate of the integral and the error estimate. These are not the same.

Problem 8 (12 points)
2 points for trying the correct substitution (some multiple of tangent); 1 point for getting the multiple correct; 1 point for getting secant out of the quadratic term; 1 point for the correct dx; 1 point for getting to the integral of secant; 1 point for applying integral of secant correctly; 3 points for getting back to x correctly; 2 points for the correct answer (evaluating and using log properties).
The student also could earn all points by evaluating the integral correctly in the substituted variable (no student did this).

The second exam

Numerical grades will be retained for use in computing the final letter grade in the course.
Here are approximate letter grade assignments for this exam:

Discussion of the grading

Minor errors (missing factor in a final answer, sign error, etc.) will be penalized minimally. Students whose errors materially simplify the problem will not be eligible for most of the problem's credit.

Problem 1 (12 points)
a) (6 points) 2 points for horizontal; 4 points for the others (1 point off for each element which is tilted up where it should be down or vice versa, and 1 point off if the elements do not vary correctly, more or less tilt, as scanned horizontally: max of 2 off for "tilt" errors) and max of 2 off for sign errors.
b) (2 point) one equilibrium solution and specification of the constant with the right equation (so x=CONSTANT loses a point because it does not describe a solution to the differential equation).
c) (8 points) 1 point for correct separation; 3 points for integration (1 of these is for a correct "+C"); 2 points for use of the initial condition; 2 points for solving for y as a function of x.

Problem 2 (12 points)
Use of the arc length formula: 1 point; correct computation of f´(x) earns 2 points; 2 points for the correct integrand; 2 points for the change to sec(x); 2 points for integrating secant; 2 points for evaluating sec and tan at the endpoints; 1 point for the final answer recognizing ln(1).
Note This is essentially problem 10 in section 8.1 of the textbook.

Problem 3 (12 points)
a) (5 points) 2 points for the first and second derivatives; 2 points for correct evaluation of the function value, and the first and second derivatives; 1 point for final assembly of T₂(x). ?????
b) (1 point) Earned for the answer. If the student gives a reasonable polynomial as answer for a) this is earned by the value of the student's polynomial at 5.
c) (6 points) The correct third derivative earns 2 points; estimation of K in the correct way earns 2 points (selecting the correct endpoint earns 1 of the 2 points but just selecting the incorrect endpoint earns 0; the other point is earned for some reason why the correct endpoint was selected); 2 points for putting everything together (factorials, powers, etc.).

Problem 4 (10 points)
2 points for starting with a valid and relevant polynomial related to e^x; 2 points for correct substitution of -x²; 2 points for indicating multiplication of this polynomial by the appropriate linear polynomial; 2 points for carrying out the multiplication; 1 point for further algebraic expansion into standard form (sum of constants times powers of x); 1 point for the answer. The answer point is not earned if the polynomial does not have the correct degree.

Problem 5 (12 points)
a) (6 points) Transforming the sequence to a form where L'Hôpital's Rule could be used is worth 2 points; 1 point for remarking on eligibility; 2 points for using L'Hôpital's Rule (differentiation of the top and bottom); 1 point for the answer.
b) (6 points) 2 points for separating into two geometric series; 2 points each for the correct use of geometric series and the answer.

Problem 6 (12 points)
3 points for realizing or using the "key observation" that dropping the square root increases the size of the fraction (using the other part of the formula doesn't earn credit since it cannot be estimated usefully); 3 points for showing that the infinite tail must be estimated (here either a geometric series or a relevant [improper] definite integral can be given); 3 points for estimating the infinite tail by finding the sum of the relevant geometric series or with a correct antiderivative; 3 points for using the sum and the tabular information (or a correct integral!) to get a correct answer. Merely asserting an answer is not sufficient to get credit here (lots of data was supplied). Asserting that an infinite tail is small because one or a few terms are small also earns no credit.
No credit will be given for a purely arithmetic approach to this problem because this is a calculus course.

Problem 7 (12 points)
4 points for some evidence connecting the sum to the integral. One acceptable item would be a picture similar to that displayed on the answer sheet. An explicit inequality connecting the N^th partial sum with a definite integral would also be acceptable. Also useful would be mention of a relevant function decreasing. But some evidence should be given.
4 points for evaluating a relevant definite integral. 4 points for a correct answer with evidence showing that specific N is valid. Students who use N in place of N+1 in an otherwise correct solution will be penalized 2 points.

Problem 8 (14 points)
a) (8 points) 4 points for computing the ratio and correctly obtaining a simple fraction (1 point is lost if the absolute value is missing); 3 points for obtaining the limit of the ratio; 1 point for the answer.
b) (6 points) 3 points for considering the case x=RIGHT END POINT. 1 of these 3 points is for the answer, and 2 points for correct support of the answer.
3 points for considering the case x=LEFT END POINT. 1 of these 3 points is for the answer, and 2 points for correct supporting of the answer.

Maintained by greenfie@math.rutgers.edu and last modified 4/20/2008.