Welcome back to Professor Baker's Math Class! As we approach the Fall 2025 Chapter 4 Test, it is time to consolidate everything we have learned about the Applications of Derivatives. This chapter is a pivotal moment in Calculus where we take the mechanical skills of differentiation and apply them to analyze functions and solve complex real-world problems.

Below is a breakdown of the key concepts covered in the review sheet. I have categorized the practice problems to help you focus your study sessions.

1. Curve Sketching and Function Analysis

A major portion of this test focuses on understanding the behavior of functions. You need to be comfortable finding critical numbers, intervals of increase/decrease, and concavity. Remember the relationship between the function and its derivatives:

  • First Derivative Test: Set $f'(x) = 0$ to find critical numbers and determine where the function is increasing or decreasing (min/max).
  • Concavity: Set $f''(x) = 0$ to find possible points of inflection and intervals where the graph is concave up or down.

Practice Problems to Review:

  • Problem 1: Find the extrema for $y = 2x^4 - 8x$.
  • Problem 3: Analyze the domain and concavity for $y = x(\ln(x))^2$. (Hint: Watch out for the domain of natural log!)
  • Problem 4: Determine increasing/decreasing intervals for $f(x) = x^{\frac{2}{3}}(x-6)$.
  • Problem 5: Locate the point of inflection for the polynomial $f(x) = x^3 - 3x^2 - 6x - 3$.

2. Limits and L'Hôpital's Rule

We are also revisiting limits, specifically those that result in indeterminate forms (like $\frac{0}{0}$). Don't forget to check if L'Hôpital's Rule applies before taking the derivative of the numerator and denominator.

Key Limits:

  • Problem 6: $$\lim_{x \to \frac{\pi}{4}} \frac{\cos(x)-\sin(x)}{\tan(x)-1}$$
  • Problem 7: $$\lim_{x \to 1} \frac{\sin(x-1)}{x^3+2x-3}$$

3. Optimization: Real-World Applications

Perhaps the most challenging (but rewarding) part of Chapter 4 is Optimization. These word problems require you to translate a physical scenario into a mathematical function and then find the absolute maximum or minimum.

Strategies for Success:

  1. Draw a picture: Label your variables (x, y, etc.).
  2. Constraint Equation: What is limited? (e.g., Cost = $7000, Perimeter = 8 ft).
  3. Objective Function: What are you trying to maximize or minimize? (e.g., Area, Light, Length).

Must-Do Practice Problems:

  • Problem 8 (The Farmer): Maximizing the area of a rectangular plot with specific cost constraints. Note the detail about the "west side" cost being shared!
  • Problem 9 (The Norman Window): A classic calculus problem. A rectangle surmounted by a semicircle with a fixed perimeter of 8 feet. You need to maximize the light admitted (Area).
  • Problem 10 (The Ladder): Calculating the shortest ladder to clear an 8ft fence located 4ft from a building. This involves minimizing the length function based on the angle or similar triangles.

Take your time with these problems. If you can handle the algebra in Problem 2 ($f(x) = \sqrt{x^2+2x} - x$) and the geometry in the optimization problems, you are in excellent shape for the exam. Good luck studying—you've got this!