Calculus 2 Exam Tips

Are you prepared for your Calculus 2 exam?

It is that time of year again. Students in Calculus 2 (integral calculus) are preparing for their final exams in April. Studying for this exam can be challenging, given she sheer volume of problem types and techniques that you are required to understand and apply. There many avenues that a student can pursue to get support for preparing for this exam, but for now, here are some tips and suggestions for things to focus on to help you with preparing for you Calculus 2 final exam:

• Reimann Sums / Integration – For this section you are going to want to remember the left, right and midpoint formulas to find areas under a curve. You will also want to remember the variations of the Reimann Sum (e.g. Trapezoidal Rule, etc), and how to apply them effectively. If these are not given on a formula sheet (which often they are), you are going to want to simply memorize them.
• Integration Techniques – Be sure to practice the more complicated integration techniques as much as you can. In particular, focus on u-substitutions, integration by parts, and trig substitutions / integrals, as they are some of the more challenging techniques. In particular, focus on how to identify whether a question requires a u-sub or by parts or a trig substitution, as they can look very similar at first glance. In general:
  1. U-Substitutions – use this technique anytime you see a function, AND something that looks similar to that same function’s derivative, being multiplied or divided in the question. Make the original function the “u” and the derivative the “du”, and the integral should become much more straight forward. For example, if you were asked to solve:
     ∫ sin(x)*cos(x) dx
     Make    u =sin(x)             
                    du = cos(x) dx
     And this integral very nicely becomes:
     ∫ u du
  2. By Parts – use this technique anytime you have the product of two functions, one of which would disappear if differentiated enough times (typically polynomial functions) and one of which would not disappear over time if you continued to differentiate it. Examples of such functions are trig functions (e.g. sin(x), cos(x)), and exponential functions (e.g. e^x). In these cases, make “u” the function that would disappear over time (the polynomial) and make dv the function that would not disappear over time, and the integral that results should be much simpler to solve. For example if you were asked to solve:
     ∫ xe^x dx
     Make    u = x                      dv = e^x dx
                    du = dx                  v = e^x
     And this integral very nicely becomes:
     xe^x –  ∫ e^x dx
     *Note: You will need to remember the by-parts equation, if it is not given on your formula sheet:

     ∫ u dv = u*v – ∫ v du

  3. Trig Substitutions – use a trig substitution anytime you have something that looks like a variation of Pythagoras’ theorem in the equation. The two possible variations are:
     a² + b² = c² ->c = sqrt(a² + b²)
     b² = c² – a² -> b = sqrt(c² – a²)
• Differential Equations – The differential equations presented in this course are by no means the most challenging part of the course but you are almost certainly going to be given at least one question which tests the techniques presented for solving them. This is because these techniques are important fundamentals for a new set of calculus-type courses which follow Calculus 2, namely Elementary and Partial Differential Equations. Make sure you are comfortable with these techniques, as the techniques are fairly straight forward, and will be important in future courses.
• Sequences and Series – The key portions of this unit are two fold:
  1. Convergence / Divergence – make sure you are comfortable with the techniques used to determine whether infinite series converge or diverge. In particular, be familiar with the various tests (ratio test, comparison test, etc) and how to apply them.
  2. Power / Taylor Series – I find that the best way to deal with power series and Taylor series is to simply memorize the equations, and apply them to the function presented. In particular, if it is not given on your formula sheet, be sure to remember the formula used to find the Taylor series expansion for a given function, as this will likely be tested.