MAT 281: Exams


Exam outlines contain items that have a high probability of showing up on the exam. However, items not listed in the outline can still show up on the exam and items listed in the outline are not necessarily guaranteed to show up on the exam.

  • Final Exam
  • There will be at least one problem of finding the area between two curves, volume (resp., surface area) of a solid (resp., surface) of revolution, or arc length.
  • Find the antiderivative and/or definite integral of \(f(x)\). This will involve new integration techniques (Trigonometric substitution, Integration-by-Parts, Partial Fractions or Trigonometric Integrals)
  • Find the interval of convergence and radius of convergence.
  • Determine if the series is absolutely convergent, conditionally convergent, or divergent.
  • Find the Maclaurin series (resp., polynomial of \(n\)th degree) or Taylor series (resp., polynomial of \(n\)th degree) centered at \(x = c\).
  • Solve a differential equation.
  • Sketch the graph of the curve with given parametric equations
  • Sketch the graph of a polar equation
  • Find the arc length of the graph of a curve with given parametric equations
  • Find the first, second, and third derivatives, given parametric equations
  • Find the area bounded by two polar curves.
  • Exam 3 Solutions
  • State the Divergence test.
  • State the Integral Test.
  • There will be a graph and you’ll be asked questions related to the graph (i.e. find the limit of a sequence, determine whether the series converges or diverges,…)
  • Find the exact sum of a series. This will be either geometric or telescoping.
  • You will be asked to determine if a series converges or diverges. You will not be told what test to use. It will be up to you to determine which test to apply, to check that all the conditions of the test hold, and finally state the conclusion of the test.
  • You will be asked to find the interval of convergence of a given power series. In this problem, when you are testing the endpoints, you’re only required to state whether the series converges or diverges at these endpoints, and by what test(s). Your final answer must be given in interval notation.
  • Find a Maclaurin Polynomial
  • Find a Taylor Polynomial
  • Find the Taylor series
  • Find the Maclaurin series
  • Use geometric series to rewrite a repeating decimal as a fraction of integers (i.e. \(0.\bar{64} = 64/99\)).
  • Use series to find derivatives.
  • Use series to evaluate limits.
  • Exam 2 Solutions
  • State L’Hospital’s Rule
  • Identify different indeterminate forms
  • There will be a graph and you’ll be asked questions related to the graph (i.e. area of a bounded region, volume of a solid of revolution, arc length, surface area,…)
  • Integration:
    • Integration by Parts
    • Trigonometric Integrals (at least 1 Problem)
    • Partial Fractions (1 Problem)
    • Trigonometric Substitution (1 Problem)
  • Applications of the Definite Integral using new integration techniques.
    • Find the volume of a solid of revolution.
    • Find the area between curves.
    • Find the surface area of a surface of revolution.
  • Evaluate an improper integral.
  • Evaluate a limit with indeterminate form either (\(\infty-\infty\), \(0 \cdot \infty\),\(1^\infty\), or \(0^0\)).
  • Exam 1 Solutions
  • State the Trapezoidal Rule and its assumptions as it appears below.
    Trapezoidal Rule: Let \(f\) be continuous on \([a,b]\). Then
    $$
    \int_a^b f(x) \; dx \approx \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2 f(x_{n-1}) + f(x_n)],
    $$
    where \(\Delta x = (b-a)/n\) and \(x_i = a + i \Delta x\).
  • State the Integration by Parts formula and its assumptions as it appears in the online notes.
  • There will be a graph and you’ll be asked questions related to the graph (i.e. area of a bounded region, volume of a solid of revolution, arc length, surface area,…)
  • Find the antiderivative using new integration techniques: Integration-by-Parts, Trigonometric Integrals, and Trigonometric Substitution. Exactly one of these problems should be done using Trigonometric Substitution.
  • Find the area between curves.
  • Find the volume of a solid of revolution. You are free to use any method you like, but the easiest (possibly only) approach to one of these will be to use cylindrical shells. One of these will also involve a line of revolution different than the \(x\) or \(y\)-axis.
  • Find the arc length.
  • Find the surface area of a surface of revolution.