Welcome back to Professor Baker's Math Class! In this lesson, we begin Chapter 4 by shifting our focus from calculating derivatives to applying them. Sections 4-1 and 4-2 are fundamental because they provide the tools we need to understand the shape of graphs and solve optimization problems. Let's break down the key concepts from the class notes.

4-1: Maximum and Minimum Values

We start by distinguishing between the highest and lowest points on a graph. The notes introduce two vital definitions:

  • Absolute (Global) Extrema: The highest or lowest value of a function over its entire domain.
  • Local (Relative) Extrema: High or low points relative to the immediate surrounding points (the "hills" and "valleys" of the graph).

To find these points, we rely on Fermat’s Theorem, which tells us that if a function $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.

This leads us to the definition of a Critical Number. A critical number of a function $f$ is a number $c$ in the domain such that either:

$$f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist}$$

In our class examples, we looked at polynomial functions like $f(x) = x^3 - 16x^3 + 18x^2$ to identify these local peaks and valleys. We also applied this to real-world models, such as the Hubble Space Telescope example, where we analyzed the velocity function $v(t)$ to find the maximum acceleration on the closed interval $[0, 126]$. Remember, when finding absolute extrema on a closed interval, you must check the critical numbers and the endpoints!

4-2: The Mean Value Theorem (MVT)

Section 4-2 introduces two major existence theorems. First, Rolle’s Theorem, which is a specific case where if you start and end at the same height ($f(a) = f(b)$), there must be a point in between where the tangent is horizontal ($f'(c) = 0$).

This generalizes to the Mean Value Theorem, one of the most important theorems in calculus. It states that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there is a number $c$ such that:

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Geometrically, this means there is at least one point $c$ where the tangent line is parallel to the secant line connecting the endpoints. Physically, it implies that at some instant, your instantaneous velocity equals your average velocity over the trip.

In the notes, we solved several examples finding this value $c$. For instance, with the function $f(x) = \ln x$ on the interval $[1, 4]$, we found that the value satisfying the MVT was:

$$c = \frac{3}{\ln 4}$$

Be sure to review the attached PDF to see the step-by-step algebraic solutions for finding critical numbers and applying MVT. Keep practicing these derivatives!