Welcome back to class! In our session on September 27th, we tackled two pivotal sections of our Calculus journey: Section 2-7 (Derivatives and Rates of Change) and Section 2-8 (The Derivative as a Function). These concepts form the bedrock of everything we will do moving forward, linking the geometry of slopes to the algebra of limits.

1. Tangents and Rates of Change (Section 2-7)

We began by looking at how to find the slope of a tangent line to a curve at a specific point. Recall that a secant line connects two points, but a tangent line touches the curve at exactly one point. To find the slope ($m$) of that tangent line, we use a limit:

$$m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Alternatively, we often use the difference quotient definition involving $h$, where $h$ represents the distance between $x$ and $a$:

$$m = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Class Example: We applied this to the hyperbola $y = 3/x$ at the point $(3,1)$. By setting up the limit and carefully simplifying the complex fraction, we found the slope to be $m = -1/3$. This allowed us to write the equation of the tangent line as $y - 1 = -\frac{1}{3}(x - 3)$.

2. The Derivative as a Function (Section 2-8)

In Section 2-8, we generalized this concept. Instead of finding the slope at just one specific number $a$, we let the input be a variable $x$. This gives us a new function, denoted as $f'(x)$, which returns the slope of the original function at any given point.

The formal definition is:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

We practiced the algebra required to solve these limits. For example, for the function $f(x) = x^2 - 8x + 9$, we expanded $(x+h)^2$, distributed the constants, and cancelled out terms to eventually isolate $h$. The result was $f'(a) = 2a - 8$.

3. Notation and Differentiability

As we advance, you will see various notations for the derivative. While $f'(x)$ is common, we also use Leibniz notation:

  • $y'$
  • $\frac{dy}{dx}$
  • $\frac{df}{dx}$
  • $\frac{d}{dx}f(x)$

Finally, we discussed differentiability. Remember, a function is not differentiable at a point if:

  1. The graph has a corner or "kink" (slope is undefined).
  2. There is a discontinuity (a jump or hole).
  3. There is a vertical tangent (slope is infinite).

Keep practicing those algebraic expansions! Being comfortable with expanding $(x+h)^2$ and $(x+h)^3$ is crucial for these limit definitions.