The concept of a limit is foundational to calculus. It describes the value that a function "approaches" as the input (variable) gets closer and closer to some specific value.

Intuitive Idea: Imagine walking towards a wall. A limit describes the position of the wall you are approaching, even if you never actually touch it.

Notation: We write the limit of a function $f(x)$ as $x$ approaches $c$ as:

$\lim_{x \to c} f(x) = L$

This means the value of $f(x)$ gets arbitrarily close to $L$ as $x$ gets sufficiently close to $c$ (but not necessarily equal to $c$).

Why are limits important? They are used to define continuity, derivatives, and integrals precisely.

The derivative of a function measures its instantaneous rate of change, or the slope of the function's graph at a specific point. It tells us how quickly a quantity is changing.

Intuitive Idea: If a function represents the position of a car over time, its derivative represents the car's instantaneous velocity (speed and direction) at any given moment.

Notation: The derivative of a function $f(x)$ with respect to $x$ can be written as $f'(x)$ (read "f prime of x") or $\frac{dy}{dx}$ (Leibniz notation).

Definition using Limits:

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

This formula calculates the slope of the secant line between two points on the curve that are infinitesimally close together.

Common Derivatives (Examples):

• If $f(x) = c$ (a constant), then $f'(x) = 0$.
• If $f(x) = x^n$, then $f'(x) = n x^{n-1}$ (Power Rule). E.g., if $f(x) = x^2$, $f'(x) = 2x$.
• If $f(x) = \sin(x)$, then $f'(x) = \cos(x)$.
• If $f(x) = e^x$, then $f'(x) = e^x$.

Applications: Finding maximum/minimum values, optimization problems, velocity and acceleration, analyzing marginal cost/revenue in economics.

Integration is essentially the reverse process of differentiation. It involves finding the accumulation of quantities or calculating the area under the curve of a function's graph.

Intuitive Idea: If a function represents the speed of a car over time, its integral represents the total distance traveled during that time.

There are two main types of integrals:

• Indefinite Integral (Antiderivative): Finding a function $F(x)$ whose derivative is the original function $f(x)$. Written as $\int f(x) dx = F(x) + C$, where $C$ is the constant of integration.
• Definite Integral: Calculating the net signed area between the function's graph and the x-axis over a specific interval $[a, b]$. Written as $\int_{a}^{b} f(x) dx$.

The Fundamental Theorem of Calculus: This crucial theorem links differentiation and integration. It states (partially) that if $F'(x) = f(x)$, then:

$\int_{a}^{b} f(x) dx = F(b) - F(a)$

Common Integrals (Examples):

• $\int k dx = kx + C$ (k is a constant)
• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
• $\int \frac{1}{x} dx = \ln|x| + C$
• $\int e^x dx = e^x + C$
• $\int \cos(x) dx = \sin(x) + C$

Applications: Calculating areas and volumes, finding total distance from velocity, work done by a variable force, probability calculations, fluid dynamics.