AP Calculus AB/BC: Applications of Derivatives — Optimisation, Related Rates, Linearisation

TL;DR

Derivatives help you find maximums/minimums (optimisation), relate changing quantities (related rates), and approximate functions with lines (linearisation). These are practical applications where calculus moves beyond just calculating slopes. Mastering these will significantly boost your AP score.

1. The Mental Model

Think of derivatives as a powerful tool for understanding how things change. Optimisation uses them to find the "best" outcome, related rates uses them to track intertwined changes, and linearisation uses them to make complex functions behave like simple lines near a specific point.

2. The Core Material

Optimisation: Finding Maxima and Minima

Optimisation problems ask you to find the absolute maximum or minimum value of a function over a given interval. This usually involves finding critical points (where the derivative is zero or undefined) and checking endpoints.

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

AP Calculus AB/BC: Integrals — Fundamental Theorem, U-Substitution, Integration by Parts (BC)

TL;DR

Integrals help you find the accumulation of a rate of change, often representing area under a curve. The Fundamental Theorem of Calculus links derivatives and integrals, providing a way to evaluate definite integrals. U-substitution simplifies integrals by transforming them, while integration by parts (BC topic) handles product functions.

1. The Mental Model

Think of integration as going backward from a rate of change to the total amount that has accumulated. If you know how fast something is changing, integration tells you how much of that "something" you have now. It's like finding the total distance traveled if you know your speed at every moment.

2. The Core Material

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 Fundamental Theorem of Calculus (FTC)

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.

The FTC Part 2 is what you'll use most often for definite integrals:
$\int_a^b f(x) \,dx = F(b) - F(a)$, where $F(x)$ is any antiderivative of $f(x)$ (meaning $F'(x) = f(x)$).

To use FTC Part 2, you:
1. Find the antiderivative $F(x)$ of $f(x)$.
2. Evaluate $F(b)$.
3. Evaluate $F(a)$.
4. Subtract $F(a)$ from $F(b)$.

Example: $\int_1^3 2x \,dx$
1. Antiderivative of $2x$ is $x^2$. So $F(x) = x^2$.
2. $F(3) = 3^2 = 9$.
3. $F(1) = 1^2 = 1$.
4. $F(3) - F(1) = 9 - 1 = 8$.

U-Substitution (The "Reverse Chain Rule")

U-substitution helps integrate functions that look like they came from a chain rule derivative. The goal is to simplify the integrand by replacing a complex inner function ($u$) and its derivative ($du$).

Steps:
1. Choose a part of the integrand to be $u$, usually the "inside" function of a composition.
2. Find $du$ by differentiating $u$ with respect to the original variable (e.g., $dx$).
3. Rewrite the integral entirely in terms of $u$ and $du$. You might need to adjust constants.
4. Integrate with respect to $u$.
5. Substitute the original expression back in for $u$.
6. For definite integrals, either change the limits of integration to be in terms of

Limits, Continuity and the Intermediate Value Theorem

TL;DR

Limits describe what a function approaches, even if it doesn't actually reach that value, and they're fundamental to calculus. Continuity means a function's graph has no breaks, jumps, or holes. The Intermediate Value Theorem (IVT) uses continuity to guarantee a function hits every value between any two points.

1. The Mental Model

Imagine you're driving towards a destination. The limit is where you're headed, even if you never quite get there (maybe there's a roadblock right at the end). Continuity means your road is smooth, with no surprise gaps or sudden drops. The Intermediate Value Theorem basically says if your road is continuous, and you start at one elevation and end at another, you must have passed through all the elevations in between.

2. The Core Material

What is a Limit?

A limit tells us the value that a function "approaches" as the input (x-value) gets closer and closer to some number. It doesn't care what the function actually does at that exact x-value, only what it's headed towards.

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).

What is Continuity?

A function $f(x)$ is continuous at a point $x=c$ if its graph has no breaks, jumps, or holes at that point. If you can draw it without lifting your pencil, it's continuous!

Mathematically, a function $f(x)$ is continuous at $x=c$ if all three of these conditions are met:
1. $f(c)$ is defined (there's a point there).
2. $\lim_{x \to c} f(x)$ exists (the limit approaches a specific value).
3. $\lim_{x \to c} f(x) = f(c)$ (the limit is the point).

If any of these conditions fail, the function is discontinuous at $x=c$.

Here's how to think about different types of discontinuities:

```mermaid
graph TD
A["Function Discontinuous at x=c"] --> B["Jump Discontinuity"]
A --> C["Removable Discontinuity (Hole)"]
A --> D["Infinite Discontinuity (Vertical Asymptote)"]

B -- "Left and Right Limit

AP Calculus AB/BC: Derivatives — Rules, Chain Rule, Implicit Differentiation

TL;DR

Derivatives tell you how a function's output changes relative to its input, representing instantaneous rates of change or slopes of tangent lines. You'll use basic rules for common functions, apply the Chain Rule for composite functions, and use implicit differentiation when 'y' isn't explicitly defined. Mastering these skills is crucial for understanding function behavior and solving calculus problems.

1. The Mental Model

Think of differentiation as finding the "speedometer reading" of a function at any given point. It tells you exactly how fast one quantity is changing with respect to another right at that moment.

2. The Core Material

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.

Basic Derivative Rules

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

• Constant Rule: If $f(x) = c$ (where $c$ is any number), then $f'(x) = 0$. The slope of a horizontal line is always zero.
• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Just bring the exponent down as a coefficient and subtract 1 from the exponent.
  • Example: If $f(x) = x^4$, then $f'(x) = 4x^3$.
  • Example: If $f(x) = \frac{1}{x} = x^{-1}$, then $f'(x) = -1x^{-2} = -\frac{1}{x^2}$.
• Constant Multiple Rule: If $g(x) = c \cdot f(x)$, then $g'(x) = c \cdot f'(x)$. You can pull constants out.
  • Example: If $f(x) = 5x^3$, then $f'(x) = 5 \cdot (3x^2) = 15x^2$.
• Sum/Difference Rule: If $h(x) = f(x) \pm g(x)$, then $h'(x) = f'(x) \pm g'(x)$. You can differentiate term by term.
  • Example: If $f(x) = 3x^2 + 2x - 7$, then $f'(x) = 6x + 2 - 0 = 6x + 2$.
• Product Rule: If $h(x) = f(x) \cdot g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$.
  • Mnemonic: first D second + second D first.
• Quotient Rule: If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.
  • Mnemonic: low D high minus high D low, all over low squared.

Derivative of Common Functions

You'll need to memorize these:

• $\frac{d}{dx}(\sin x) = \cos x$
• $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$
• $\frac{d}{dx}(\sec x) = \sec x \tan x$
• $\frac{d}{dx}(\csc x) = -\csc x \cot x$
• $\frac{d}{dx}(\cot x) = -\csc^2 x$
• $\frac{d}{dx}(e^x) = e^x$
• $\frac{d}{dx}(