Practice-Final-Exam

Practice Final Exam Overview

  • The final exam may include problems related to the following objectives:

    • Evaluate integrals using different techniques of integration.

    • Evaluate improper integrals.

    • Test the convergence and divergence of a given series using any tests.

    • Find the radius and interval of convergence of a power series.

    • Use the Maclaurin series table to approximate definite integrals, evaluate limits, and find the exact sum of series.

    • Find the length of a curve (in polar or parametric form).

    • Determine the equation of the tangent line at a point on a given curve (in polar or parametric form).

    • Calculate the area of the region bounded by the curve (in polar or parametric form).

Sample Problems

  • Instructions: Show work for each problem. Final answers alone will not be accepted. Solutions to sample problems (1-5) are available in the Practice Midterm exam solutions.

Integral Evaluations

  1. Evaluate the following integrals:

    • a. ( \int_{2}^{1} w^2 \ln(w) , dw )

    • b. ( \int (x^3 + 3x + 1) / (x^2 + 1) , dx )

    • c. ( \int_{1}^{\infty} (x^2 - 1)^{-3/2} , dx )

Convergence Tests

  1. Determine whether the integral converges or diverges. If it converges, find its value:

    • ( \int_{1}^{\infty} \frac{\ln x}{x^2} , dx )

Series Convergence

  1. Determine whether the given series converges or diverges:

    • a. ( \sum_{n=1}^{\infty} \frac{5}{\pi n} )

    • b. ( \sum_{n=1}^{\infty} \frac{1}{n^4} )

    • c. ( \sum_{n=1}^{\infty} \frac{(-1)^n 3n - 1}{2n + 1} )

Power Series Convergence

  1. Find the radius of convergence and the interval of convergence for the given power series:

    • ( \sum_{n=1}^{\infty} \frac{(-1)^n (x - 8)^n}{\sqrt{n}} )

Approximation Using Maclaurin Series

  1. Use the Maclaurin series table to approximate the definite integral:

    • ( \int_{0}^{0.5} x^2 e^{-x^2} , dx ) with an error of (|error| < 0.001)

Evaluating Limits Using Series

  1. Use series to evaluate the following limit correct to three decimal places:

    • ( \lim_{x \to 0} \frac{\sin x - x + 1}{6 x^3 + x^5} )

Exact Summation of Series

  1. Find the exact sum of the series:

    • ( \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n}}{6^{2n} (2n)!} )

Equations of Tangents to Curves

  1. Find the equation of the tangent line to the curve at the point corresponding to the given parameter value:

    • Curve: ( x = t \cos(t), , y = t \sin(t) );

    • Parameter: ( t = \frac{\pi}{2} )

Length of the Curve

  1. Set up an integral that represents the length of the curve given by:

    • ( x = 4 + 9t^2, , y = 2 + 6t^3, , 0 \leq t \leq 3 )

    • Find the length correct to four decimal places using a calculator.

Area Bounded by Curves

  1. Find the area of the region that lies inside the first curve and outside the second:

    • Curves: ( r = 4 \sin(\theta), , r = 2 )