Limits
A limit describes the behavior of a function as its input approaches a certain value. It's fundamental to understanding derivatives and integrals.
Limit Properties
If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:
$\lim_{x \to a} [f(x) + g(x)] = L + M$
$\lim_{x \to a} [f(x) - g(x)] = L - M$
$\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
$\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{L}{M}$, provided $M \neq 0$
$\lim_{x \to a} [f(x)]^n = L^n$
Important Limit Theorems
Squeeze Theorem: If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $a$ (except possibly at $a$) and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.
L'Hpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ results in an indeterminate form (e.g., 0/0 or $\infty/\infty$), then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the limit on the right exists.
Derivatives
The derivative measures the rate of change of a function. It's the slope of the tangent line to the graph of a function at a particular point.
Basic Derivative Rules
Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$
Constant Rule: $\frac{d}{dx}[c] = 0$
Constant Multiple Rule: $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$
Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
Common Derivatives
| Function | Derivative |
|---|---|
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\log_a x$ | $\frac{1}{x \ln a}$ |
| $\arcsin x$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\arccos x$ | $-\frac{1}{\sqrt{1-x^2}}$ |
| $\arctan x$ | $\frac{1}{1+x^2}$ |
Integration
Integration is the process of finding the integral of a function, which is the reverse operation of differentiation. It has applications in calculating areas, volumes, and solving differential equations.
Basic Integration Rules
Constant Rule: $\int k \,dx = kx + C$, where $k$ is a constant
Power Rule: $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$
Sum/Difference Rule: $\int [f(x) \pm g(x)] \,dx = \int f(x) \,dx \pm \int g(x) \,dx$
Constant Multiple Rule: $\int k \cdot f(x) \,dx = k \cdot \int f(x) \,dx$
Common Integrals
| Function | Integral |
|---|---|
| $\int \sin x \,dx$ | $-\cos x + C$ |
| $\int \cos x \,dx$ | $\sin x + C$ |
| $\int \tan x \,dx$ | $-\ln|\cos x| + C$ |
| $\int \sec x \,dx$ | $\ln|\sec x + \tan x| + C$ |
| $\int \sec^2 x \,dx$ | $\tan x + C$ |
| $\int \csc^2 x \,dx$ | $-\cot x + C$ |
| $\int e^x \,dx$ | $e^x + C$ |
| $\int a^x \,dx$ | $\frac{a^x}{\ln a} + C$ |
| $\int \frac{1}{x} \,dx$ | $\ln|x| + C$ |
| $\int \frac{1}{1+x^2} \,dx$ | $\arctan x + C$ |
| $\int \frac{1}{\sqrt{1-x^2}} \,dx$ | $\arcsin x + C$ |
Integration Techniques
u-Substitution
If $u = g(x)$, then $\int f(g(x)) \cdot g'(x) \,dx = \int f(u) \,du$
Example: Evaluate $\int 2x \cdot \cos(x^2) \,dx$
Let $u = x^2$, then $du = 2x \,dx$
$\int 2x \cdot \cos(x^2) \,dx = \int \cos(u) \,du = \sin(u) + C = \sin(x^2) + C$
Let $u = x^2$, then $du = 2x \,dx$
$\int 2x \cdot \cos(x^2) \,dx = \int \cos(u) \,du = \sin(u) + C = \sin(x^2) + C$
Integration by Parts
$\int u \,dv = uv - \int v \,du$
Sequences and Series
Sequences are ordered lists of numbers, while series are the sum of terms in a sequence.
Arithmetic Sequences and Series
Arithmetic sequence: $a_n = a_1 + (n-1)d$, where $d$ is the common difference
Arithmetic series sum: $S_n = \frac{n}{2}(a_1 + a_n)$
Geometric Sequences and Series
Geometric sequence: $a_n = a_1 \cdot r^{n-1}$, where $r$ is the common ratio
Finite geometric series sum: $S_n = \frac{a_1(1-r^n)}{1-r}$, for $r \neq 1$
Infinite geometric series sum: $S_\infty = \frac{a_1}{1-r}$, for $|r| < 1$
Convergence Tests
Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges.
Ratio Test: If $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L$, then:
- If $L < 1$, the series converges absolutely
- If $L > 1$ or $L = \infty$, the series diverges
- If $L = 1$, the test is inconclusive
Important Calculus Theorems
Rolle's Theorem: If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $c \in (a,b)$ such that $f'(c) = 0$.
First Fundamental Theorem of Calculus: If $f$ is continuous on $[a,b]$, then the function $g$ defined by $g(x) = \int_a^x f(t) \,dt$ is continuous on $[a,b]$ and differentiable on $(a,b)$, with $g'(x) = f(x)$.
Second Fundamental Theorem of Calculus: If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$, then $\int_a^b f(x) \,dx = F(b) - F(a)$.
Partial Derivatives (Multivariable Calculus)
For a function $f(x_1, x_2, \ldots, x_n)$ of multiple variables, the partial derivative with respect to $x_i$ is denoted $\frac{\partial f}{\partial x_i}$ and means we differentiate with respect to $x_i$ while treating all other variables as constants.
First Partial Derivatives: $f_x(x,y) = \frac{\partial f}{\partial x}, f_y(x,y) = \frac{\partial f}{\partial y}$
Second Partial Derivatives: $f_{xx} = \frac{\partial^2 f}{\partial x^2}, f_{yy} = \frac{\partial^2 f}{\partial y^2}, f_{xy} = \frac{\partial^2 f}{\partial x \partial y}, f_{yx} = \frac{\partial^2 f}{\partial y \partial x}$
Clairaut's Theorem: If $f_{xy}$ and $f_{yx}$ are both continuous, then $f_{xy} = f_{yx}$.
Gradient: $\nabla f = \langle f_x, f_y, f_z \rangle$
Directional Derivative: $D_{\vec{u}}f = \nabla f \cdot \vec{u}$
Differential Equations
Differential equations involve derivatives of an unknown function.
First-Order Differential Equations
Separable: $M(x) \,dx + N(y) \,dy = 0$
Linear: $y' + P(x)y = Q(x)$
Exact: $M(x,y) \,dx + N(x,y) \,dy = 0$ if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$
Second-Order Linear Homogeneous Differential Equations
With constant coefficients: $ay'' + by' + cy = 0$
Characteristic equation: $ar^2 + br + c = 0$
Solutions based on roots:
- Distinct real roots $r_1, r_2$: $y = c_1 e^{r_1 x} + c_2 e^{r_2 x}$
- Repeated real root $r$: $y = c_1 e^{rx} + c_2 xe^{rx}$
- Complex roots $\alpha \pm i\beta$: $y = e^{\alpha x}(c_1 \cos(\beta x) + c_2 \sin(\beta x))$
