MAT 125: Calculus 2

Fall 2018

Gateway exam information

1. Logistics

Exam format. There will be 6 problems on the Gateway. Notes, calculators, and laptops are not allowed. To pass, you must get at least 5 problems completely correct. (Do not forget "$+C$" for indefinite integrals; the integral will be considered incorrect without it.) There is no partial credit.

The first opportunity to take the Gateway is in class on Friday, 2018-10-05. To motivate you to study for the Gateway, if you pass it then, you will receive 10 extra credit homework points.

Retaking the exam. If you do not pass it in class, you will have 4 more opportunities to take similar (but not identical) exams. It will be offered during scheduled make-up times or instructor office hours. You may retake the exam at most once per day. You should review before retaking the exams.

You must pass the Gateway by Friday, 2018-10-19. If not, in accordance to the syllabus, your final grade will be reduced by one full letter (e.g. a B+ would change to a C+).

2. How to study

Research has shown that the best way to study is to practice. (See tips for Exam 1.) Try problems on pages 408 to 410 in the textbook. Ansewrs to odd-numbered questions are in the back of the book. For even-numbered questions, you may stop by during office hours to check or use Mathematica.

A review session will be planned and announced.

3. Exam content

  1. Basic integrals of $e^x$, $b^x$, $x^n$, $\frac1x$, $\sin x$, $\cos x$, $\sec^2 x$, $\frac1{1+x^2}$, $\frac1{\sqrt{1-x^2}}$.
  2. Integration by substitution: used by itself and as part of all other techniques.
  3. Integration by parts: useful for products like $x\cos(2x)$ and for $\ln x$, $\arctan x$, $\arcsin x$. The more complicated questions where we use by parts multiple times to get the original integral to show up and solve for the answer will not appear on the Gateway.
  4. Partial fractions: long division will not be needed.
  5. $\int\sin^mx\,\cos^nx\,dx$ where at least one of $m$ and $n$ is odd.

4. Sample questions

  1. $\int_0^{\pi/16}\sec^2(4x)dx$
  2. $\int\left(e^{3x}+4+2\sqrt x+\sin2x+\frac5{4x}\right)dx$
  3. $\int\frac4{\sqrt{1-4x^2}}dx$
  4. $\int\frac6{3+6x^2}dx$
  5. $\int\sin^3x\,\cos^4x\,dx$
  6. $\int\frac{2x}{x^2+8x-9}\,dx$
  7. $\int\ln3x\,dx$
  8. $\int2x\cos4x\,dx$
  9. $\int\frac{5-4x}{2x^2+x-1}dx$
  10. $\int\frac5{2x^2+x-1}dx$