Welcome back to Professor Baker's Math Class! As we approach the Spring 2025 Chapter 4 Test, it is time to synthesize everything we have learned about the applications of derivatives. This review sheet covers the essential skills you need to master, from analyzing the behavior of complex functions to solving real-world optimization challenges.

1. Curve Sketching and Function Analysis

A major portion of this chapter focuses on using the First and Second Derivatives to understand the shape of a graph. You should be comfortable finding critical numbers, intervals of increase/decrease, and points of inflection.

  • Extrema: Problem #1 asks you to find the minimum/maximum values for $y = 2x^4 - 8x$. Remember to set $y' = 0$.
  • Concavity: In Problem #3, you deal with the function $y = x(\ln(x))^2$. Pay close attention to the domain of $\ln(x)$ and use the Second Derivative Test to find concavity intervals.
  • Inflection Points: Problem #5 requires finding the point of inflection for a cubic polynomial: $f(x) = x^3 - 3x^2 - 6x - 3$.
  • Rational Exponents: Be careful with Problem #4, $f(x) = x^{\frac{2}{3}}(x-6)$, as the rational exponent often indicates a cusp or vertical tangent.

2. Limits and L'Hôpital's Rule

We are also reviewing how to handle indeterminate forms using L'Hôpital's Rule. The review sheet includes two excellent trigonometry-based limits:

$$ \lim_{x \to \frac{\pi}{4}} \frac{\cos(x)-\sin(x)}{\tan(x)-1} $$

and

$$ \lim_{x \to 1} \frac{\sin(x-1)}{x^3+2x-3} $$

Tip: Always verify that the limit results in $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying the rule!

3. Optimization Applications

The most challenging part of Chapter 4 is often Optimization. These word problems require you to translate a physical situation into a mathematical model to find a maximum or minimum value.

  • The Fencing Problem (Q8): This is a classic variation where costs differ by side. Note that the west side cost is split with a neighbor, which changes your cost function constraint.
  • The Norman Window (Q9): You are maximizing light (area) for a window shaped like a rectangle surmounted by a semicircle, given a fixed perimeter.
  • The Ladder Problem (Q10): A minimization problem involving trigonometry or similar triangles to find the shortest ladder reaching over an 8ft fence to a building.

Study Tips

Make sure to review the domain restrictions for square roots and logs (as seen in Problem #2). For the optimization problems, always draw a diagram first and clearly label your variables. Good luck studying, and remember to check the video resources for step-by-step solutions to these specific problems!