AP Prep

Here's a general approach:
1. Understand the problem: Identify the quantity you need to maximise or minimise.
2. Formulate the function: Express that quantity as a function of one variable. This often requires using constraints to eliminate other variables.
3. Find critical points: Calculate the first

Integrals are essentially fancy summators. A definite integral ($\int_a^b f(x) \,dx$) calculates the exact accumulation of a function $f(x)$ from point $a$ to point $b$. An indefinite integral ($\int f(x) \,dx$) finds the family of functions whose derivative is $f(x)$, always including an arbitrary constant $C$.

The FTC Part 1 states that if $F(x) = \int_a^x f(t) \,dt$, then $F'(x) = f(x)$. This means differentiation and integration are inverse operations.

We write it like this: $\lim_{x \to c} f(x) = L$

This reads as "the limit of $f(x)$ as $x$ approaches $c$ is $L$."

Think of it like this:
* From the left: $x$ gets closer to $c$ from values less than $c$. We write: $\lim_{x \to c^-} f(x)$
* From the right: $x$ gets closer to $c$ from values greater than $c$. We write: $\lim_{x \to c^+} f(x)$
* For the overall limit to exist ($\lim_{x \to c} f(x)$), the limit from the left must equal the limit from the right. If they don't match, the limit does not exist (DNE).

When you're finding the derivative, you're essentially looking for a new function that describes the slope of the original function's tangent line at every point.

These are your building blocks. You'll use these rules constantly.