Fall 2025: Chapter 2-7 & 2-8 - Unlocking Derivatives!
Welcome to a deep dive into the world of derivatives! These notes from Professor Baker's Math Class cover some key concepts. Let's break it down:
1. The Difference Quotient: Your Derivative Toolkit
The difference quotient is the foundation for understanding derivatives. It allows us to calculate the average rate of change of a function over a small interval. Remember the formula:
$$ \frac{f(x + h) - f(x)}{h} $$And we can use this difference quotient to find the derivative at a specific number, like 5.
Here's an example of finding a derivative at a given number by simplifying the difference quotient:
Let $f(x) = \frac{-5}{x^2}$.
- Find the difference quotient $\frac{f(x) - f(5)}{x - 5}$ and simplify.
- Then, find $f'(5)$. The steps involve algebraic manipulation to eliminate the singularity and then evaluating the limit. This can be simplified to $\frac{x+5}{5x^2}$ and by plugging in 5, the final answer is $\frac{2}{25}$.
2. Tangent Lines: Connecting Functions and Derivatives
The derivative at a point gives the slope of the tangent line to the function at that point. Understanding this relationship is crucial for many applications. For example:
Suppose $y = 5x - 2$ is the equation of the tangent line to the graph of a function $y = f(x)$ at $x = \frac{1}{4}$. Find $f(\frac{1}{4})$ and $f'(\frac{1}{4})$. Since the line is tangent to the curve at that point, then $f'(\frac{1}{4})= 5$ and $f(\frac{1}{4}) = 5(\frac{1}{4}) - 2 = -\frac{3}{4}$.
3. Interpreting Graphs: Derivative Insights
The graph of a function holds valuable information about its derivative. A positive derivative indicates an increasing function, a negative derivative indicates a decreasing function, and a zero derivative indicates a horizontal tangent (a local max or min!).
If we're given the graph of $f$, we can often order values of the derivative at different points. The steeper the slope, the larger the absolute value of the derivative.
f'(-3) is very steep and positive, meaning this value should be very large. Conversely, f'(-1) is negative since the slope of f is negative at that point.
4. Rates of Change: Average vs. Instantaneous
Distinguish between average and instantaneous rates of change. The average rate of change is the slope of the secant line between two points, while the instantaneous rate of change is the limit of the average rate of change as the interval shrinks to zero (the derivative!).
5. Differentiability: When Derivatives Exist
A function is differentiable at a point if its derivative exists at that point. This means the function must be continuous and have a well-defined tangent line. Key things to watch out for:
- Corners/Cusps: The derivative doesn't exist at sharp corners or cusps.
- Discontinuities: A function must be continuous to be differentiable.
- Vertical Tangents: The derivative is undefined (infinite) at a vertical tangent.
Understanding these concepts will give you a solid foundation for mastering derivatives. Keep practicing, and you'll be a derivative pro in no time!