Professor Baker's Math Class - Chapter 8 Test Answer Sheet

Welcome to the Chapter 8 Test Answer Sheet! Use this resource to check your work and deepen your understanding of the material. Remember, the goal is not just to find the right answer, but to understand why it's the right answer. Good luck!

Key Concepts Covered:

  • Arc Length: Calculating the length of a curve using integration. Remember the formula: $L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} dx$
  • Surface Area of Revolution: Finding the surface area when a curve is rotated about an axis. The formula for rotation about the x-axis is: $S = 2\pi \int_a^b y \sqrt{1 + (\frac{dy}{dx})^2} dx$
  • Probability Density Functions: Understanding the properties of PDFs and calculating probabilities. A PDF, $f(x)$, must satisfy $f(x) \ge 0$ and $\int_{-\infty}^{\infty} f(x) dx = 1$
  • Exponential Density Functions: Used for modeling waiting times. An exponential density function is given by $f(t) = \frac{1}{\mu}e^{-t/\mu}$, where $\mu$ is the mean.
  • Normal Distribution: Calculating probabilities using the normal distribution with mean $\mu$ and standard deviation $\sigma$.
  • Differential Equations: Solving differential equations, which are equations involving derivatives.

Answers and Explanations:

Below are the answers to the Chapter 8 test questions, along with some brief explanations. Be sure to review your notes and examples from class if you're struggling with any particular concept. Remember that math builds on itself, so keep practicing!

  1. Arc Length: For $y = 4 + 8x^{3/2}$, $0 \le x \le 1$, the arc length is $\frac{1}{216}(145\sqrt{145} - 1)$. This involves finding the derivative, squaring it, adding 1, and then integrating. Practice similar problems to master this skill.
  2. Arc Length: For $y = \frac{x^2}{4} - \frac{1}{2}ln(x)$, $1 \le x \le 4$, the arc length is $\frac{15}{4} + \frac{1}{2}ln(4)$. Be careful with the algebra when simplifying the expression under the square root!
  3. Surface Area of Revolution: For $y = \sqrt{1 + e^x}$, $0 \le x \le 9$ rotated about the x-axis, the surface area is $\pi(e^9 + 17)$. Don't forget the $2\pi y$ factor in the surface area integral.
  4. Surface Area of Revolution: Rotating $y = \frac{x^2}{4} - \frac{1}{2}ln(x)$, $2 \le x \le 3$ about the y-axis yields a surface area of $\frac{22\pi}{3}$. Remember to integrate with respect to $x$ since we are given the function in terms of $x$.
  5. Probability Density Function: To make $f(x) = \frac{c}{1 + x^2}$ a PDF, $c = \frac{1}{\pi}$. The probability $P(-6 < X < 6) \approx 0.895$. Make sure to use the limits of integration properly!
  6. Exponential Density Function: (a) $0.487$, (b) $0.135$, (c) $4.494$ sec. Remember the formula for the exponential density function and how to integrate it.
  7. Normal Distribution: (a) $2.28\%$ (b) $2.28\%$. Practice using z-tables or calculators to find probabilities associated with the normal distribution.
  8. Differential Equation: The solution is $y = (\frac{5x^2}{4} + C)^2$ for $y \neq 0$. Also, $y=0$ is a solution. Separating variables is a common technique for solving differential equations.
  9. Differential Equation: The solution is implicitly defined and cannot be expressed explicitly for $u$.
  10. Differential Equation: $u = -\sqrt{t^2 + tan(t) + 100}$

Keep up the great work, and don't hesitate to ask questions during office hours or in the discussion forum if you need further clarification!