Chapter 4 Review: Practice Test & Important Concepts

Hello everyone! I apologize for the late posting of this review sheet. I had a family emergency arise, which prevented me from completing it sooner. To compensate, I will be allowing the use of notes on tomorrow's test. Please use this practice test to guide your studying, and don't hesitate to ask questions.

This chapter focuses on applications of derivatives, including finding extreme values (maxima and minima), analyzing intervals of increasing and decreasing behavior, and understanding concavity.

Key Concepts to Review:

  • Extreme Values: Understanding local (relative) and absolute (global) maximum and minimum values of a function.
  • Critical Numbers: Points where the derivative is zero or undefined; these are key candidates for extreme values. Remember to use the first derivative test!
  • Increasing and Decreasing Intervals: Determine where a function is increasing ($f'(x) > 0$) or decreasing ($f'(x) < 0$).
  • Concavity: Determine where a function is concave up ($f''(x) > 0$) or concave down ($f''(x) < 0$).
  • Inflection Points: Points where the concavity changes ($f''(x) = 0$ or is undefined).
  • Optimization Problems: Using calculus to find the maximum or minimum value of a function in a real-world context.

Practice Problems:

Here are some practice problems similar to what you might see on the test:

  1. Finding Extreme Values: Find the local and absolute extreme values of the function $f(x) = x^3 - 6x^2 + 9x + 2$ on the interval $[2, 4]$.

    Hint: Find the critical points within the interval and evaluate the function at the critical points and endpoints.

  2. Critical Numbers: Find all the critical numbers of the function $g(\theta) = 2\cos(\theta) + \sin^2(\theta)$.

    Hint: Take the derivative and set it equal to zero. Remember your trigonometric identities!

  3. Maximum and Minimum Points: Find the maximum and minimum points of the function $F(x) = (1 + x^2)^3 + 6x^4$.
  4. More Maximum and Minimum Points: Find the maximum and minimum points of the function $F(x) = \frac{2x}{1 + 4x^2}$.
  5. Analyzing the First Derivative: The graph of the first derivative $f'(x)$ of a function $f$ is provided. At what values of $x$ does $f$ have a local maximum or minimum? Remember that a local max/min occurs when f'(x) changes sign.
  6. Estimating Extreme Values: Estimate the extreme values of the function $y(x) = \frac{1}{3}x^3 - 7x^2 + 40x + 15$. Round the answers to the nearest hundredth.
  7. Increasing/Decreasing and Max/Min: Given $f(x) = \frac{10x}{x^2 + 36}$, find the intervals on which $f$ is increasing or decreasing, and find the relative maxima and relative minima of $f$.
  8. Concavity and Inflection Points: Determine where the graph of the function $f(x) = \sin(9x)$ is concave upward and where it is concave downward on the interval $0 \le x \le \frac{2\pi}{9}$. Also, find all inflection points of the function.
  9. Sketching Graphs: Sketch the graph of the function $y = \cos^2(x)$ on $0 \le x \le \frac{\pi}{4}$.
  10. Optimization Problem: The owner of a ranch has 1,800 yd of fencing with which to enclose a rectangular piece of grazing land situated along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area he can enclose? What is the area?
  11. Revenue Maximization: A baseball team plays in a stadium that holds 56,000 spectators. With ticket prices at $9, the average attendance had been 32,000. When ticket prices were lowered to $8, the average attendance rose to 36,000. How should ticket prices be set to maximize revenue? Assume the demand function is linear.

Remember to show your work and explain your reasoning clearly. Good luck with your test! I am confident you will do well.