Welcome to this week's focus on Chapter 4, Sections 5 and 6. We are moving from learning the rules of differentiation to applying them in powerful ways. Tonight, we will synthesize everything we know about limits and derivatives to sketch accurate graphs and solve complex real-world problems.

Note: Please bring your graphing calculator to class! While we will focus on analytical methods, the calculator is a vital tool for verifying our results.

Section 4.5: Summary of Curve Sketching

In this section, we refine our ability to visualize functions. It is not enough to simply plot points; we must understand the behavior of the function $y = f(x)$ across its entire domain. Based on our lecture notes, we will follow a systematic checklist to sketch curves:

  • Domain and Intercepts: Identify the set of possible $x$-values and locate $x$-intercepts and $y$-intercepts.
  • Symmetry: Determine if the function is even ($f(-x) = f(x)$, symmetric about the $y$-axis) or odd ($f(-x) = -f(x)$, symmetric about the origin).
  • Asymptotes: We will examine vertical asymptotes (where the function is undefined) and horizontal asymptotes (analyzing $\lim_{x \to \infty} f(x)$). As mentioned in the lecture video, we also look for slant asymptotes, which occur in rational functions when the degree of the numerator is one higher than the denominator.
  • Intervals of Increase/Decrease: By analyzing the sign of the first derivative $f'(x)$, we determine where the graph rises and falls.
  • Concavity and Inflection Points: By analyzing the second derivative $f''(x)$, we determine where the graph curves upward or downward.

Section 4.6: Optimization Problems

Optimization is one of the most practical applications of calculus. Whether you are an engineer, economist, or scientist, the ability to maximize efficiency or minimize cost is essential. In class, we will break down word problems into solvable mathematical models.

For example, we will look at problems like the "box with a square base" discussed in the video, where we must maximize volume given a specific surface area constraint. Our step-by-step strategy includes:

  1. Understand the Problem: Read carefully, identify the knowns and unknowns, and draw a diagram.
  2. Introduce Notation: Assign symbols to the quantities (e.g., let $Q$ be the quantity to be maximized).
  3. Express the Function: Write an equation for the quantity to be optimized, such as $Q = f(x, y)$.
  4. Use Constraints: Identify relationships between variables to eliminate all but one variable, resulting in a function of a single variable, $f(x)$.
  5. Apply Calculus: Find the absolute maximum or minimum value using the derivative $f'(x)$. We will look for critical numbers where $f'(x) = 0$ or does not exist, and test the endpoints of the domain.

These sections represent the culmination of our work with derivatives. Come prepared to think critically and visualize the math!