MAT 125: Calculus 2

Fall 2021

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 exam will be taken in the Bethel Testing Center. You will get an email from the Testing Center to set up a slot for the exam. Choose a time that works for your schedule.

The first opportunity to take the Gateway is on Monday, 2021-10-11. If you are too busy on that day, you may take it a day later. To motivate you to study for the Gateway, if you pass it on this first attempt, you will receive a small amount of extra credit, roughly equivalent to 2 homework points.

Retaking the exam. If you do not pass it the first time, you will have 4 more opportunities to take similar (but not identical) exams. It will be offered roughly twice a week. You should review before retaking the exams.

You must pass the Gateway by Friday, 2021-10-29. According to the syllabus, the Gateway exam is worth 10% of the course grade. A pass on the first or any subsequent retake attempts will result in full credit. Otherwise, no credit is given at all. By passing the Gateway exam, your final grade will roughly increase by one full letter (e.g. a C+ would change to a B+).

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. Answers to odd-numbered questions are in the back of the book. For even-numbered questions, you may check by using Mathematica or stopping by Math Lab or during office hours.

Here is a video review of some questions.

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. $\int\frac5{2x^2+x-1}dx$