Final Exam Review Packet

Hello everyone! Here is the review packet to help you prepare for the final exam. This packet covers many of the key concepts we've learned throughout the semester. Work through these problems carefully, and don't hesitate to seek help from your classmates or during office hours if you get stuck.

Download the Review Packet: Final Review

Important Notes Regarding Your Note Sheet

  • You are allowed one sheet of handwritten notes (8.5x11 inch paper) during the final exam.
  • No tape, staples, or attachments are permitted on your note sheet. It must be a single sheet of paper.
  • Note cards are not allowed; your notes must be consolidated onto one sheet.

Answer Key

To encourage collaboration and active learning, the answer key to this review packet will not be posted. If you missed the in-class review session where we worked through the problems together, please connect with your classmates to obtain the answer sheet.

Review Problems

Here are some sample problems similar to what you can expect to find on the final exam. These problems cover a range of topics, from applications of integration to series convergence.

  1. Hydrostatic Force: A dam has the shape of a trapezoid. The height is 20 m, the width at the top is 50 m, and the width at the bottom is 30 m. Find the force on the dam due to hydrostatic pressure if the water level is 4 m from the top of the dam. This problem requires you to set up an integral to find the force exerted by the water, $F = \int P dA$, where $P$ is the pressure and $dA$ is the area of a horizontal strip. Remember that $P = \rho g d$, where $\rho$ is the density of water, $g$ is the acceleration due to gravity, and $d$ is the depth.
  2. Consumer Surplus: The demand for a product is given by $p = 1200 - 0.2x - 0.0001x^2$, where $p$ is the price in dollars. Find the consumer surplus when the sales level is 500. Consumer surplus is given by the integral $\int_0^{500} (D(x) - p) dx$, where $D(x)$ is the demand function and $p$ is the market price.
  3. Indefinite Integrals: Evaluate the following integrals:
    • $\int_1^2 \frac{x}{(x+1)^2} dx$ (Use u-substitution or partial fraction decomposition)
    • $\int \frac{dx}{\sqrt{e^x - 1}}$ (Trigonometric Substitution)
    • $\int \frac{\sec^6(\theta)}{\tan^2(\theta)} d\theta$ (Trigonometric Identities and u-substitution)
    • $\int \tan^3(\theta) \sec^3(\theta) d\theta$ (Trigonometric Identities and u-substitution)
    • $\int \frac{e^{\sin x}}{\sec x} dx$ (u-substitution)
  4. Parametric Equations: Given $x = t^2 + 1$ and $y = t^3 + t$, find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$. For what values of $t$ is the curve concave upward? Remember that $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ and $\frac{d^2y}{dx^2} = \frac{d/dt (dy/dx)}{dx/dt}$.
  5. Area in Polar Coordinates: Find the area of the shaded region for the curve $r^2 = \sin(2\theta)$. The area enclosed by a polar curve is given by the formula $A = \frac{1}{2} \int r^2 d\theta$.
  6. Area Enclosed by Polar Curve: Sketch the curve and find the area that it encloses $r = \sqrt{1 + \cos^2(5\theta)}$. The area enclosed by a polar curve is given by the formula $A = \frac{1}{2} \int r^2 d\theta$.
  7. Convergence and Divergence of Series: Determine whether the following series are convergent or divergent:
    • $\sum_{n=1}^{\infty} \frac{n^3}{5^n}$ (Ratio Test or Root Test)
    • $\sum_{n=2}^{\infty} \frac{1}{n \sqrt{\ln n}}$ (Integral Test)
    • $\sum_{n=1}^{\infty} (-1)^{n-1} n^{-3}$ (Alternating Series Test)

Good luck with your studying, and I look forward to seeing you at the final exam!