}]]}]}]}] 30:Ta52a,

### Mathematics: Calculus #### 1. Differential Calculus - **Limits:** - $\lim_{x \to a} f(x)$ exists if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$ - Indeterminate forms: $\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 1^\infty, 0^0, \infty^0$ - L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ - **Continuity & Differentiability:** - **Continuity at $x=a$:** $\lim_{x \to a} f(x) = f(a)$ - **Differentiability at $x=a$:** $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists. Implies continuity. - **Differentiation Rules:** - Product Rule: $(uv)' = u'v + uv'$ - Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$ - Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ - **Applications of Derivatives:** - **Tangents & Normals:** Slope of tangent $m = f'(x_1, y_1)$. Equation: $y - y_1 = m(x - x_1)$. Normal slope $= -\frac{1}{m}$. - **Increasing/Decreasing Functions:** $f'(x) > 0 \implies$ increasing; $f'(x) < 0 \implies$ decreasing. - **Maxima & Minima:** - First derivative test: Change of sign of $f'(x)$. - Second derivative test: $f''(c) < 0 \implies$ local max; $f''(c) > 0 \implies$ local min. - **Monotonicity:** Function is monotonic if it is always increasing or always decreasing. - **Mean Value Theorems:** - **Rolle's Theorem:** If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c \in (a,b)$ such that $f'(c)=0$. - **Lagrange's MVT:** If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$. #### 2. Integral Calculus - **Indefinite Integrals:** - Basic formulas: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $\int \frac{1}{x} dx = \ln|x| + C$, $\int \sin x dx = -\cos x + C$, etc. - Integration by Parts: $\int u dv = uv - \int v du$ - Partial Fractions: For rational functions. - **Definite Integrals:** - Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$. - Properties of Definite Integrals: - $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$ - $\int_0^a f(x) dx = \int_0^a f(a-x) dx$ - $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$ if $f(x)$ is even; $0$ if $f(x)$ is odd. - **Area Under Curves:** - Area between curve $y=f(x)$ and x-axis from $x=a$ to $x=b$: $\int_a^b |f(x)| dx$. - Area between two curves $y=f(x)$ and $y=g(x)$: $\int_a^b |f(x) - g(x)| dx$. - **Differential Equations:** - **Order & Degree:** Order is highest derivative. Degree is power of highest derivative after making polynomial in derivatives. - **Formation of DE:** Eliminate arbitrary constants. - **Methods of Solving:** - Variable Separable: $\int f(x) dx = \int g(y) dy$ - Homogeneous: Substitute $y=vx$, then variable separable. - Linear DE: $\frac{dy}{dx} + P(x)y = Q(x)$. Integrating Factor (IF) $= e^{\int P(x) dx}$. Solution: $y \cdot IF = \int Q(x) \cdot IF dx + C$. ### Mathematics: Algebra #### 1. Complex Numbers - **Definition:** $z = x + iy$, where $i = \sqrt{-1}$. - **Conjugate:** $\bar{z} = x - iy$. - **Modulus:** $|z| = \sqrt{x^2 + y^2}$. - **Argand Plane:** Representation of complex numbers. - **Polar Form:** $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ and $\theta = \arg(z)$. - **Euler's Form:** $z = re^{i\theta}$. - **De Moivre's Theorem:** $(cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$. - **Roots of Unity:** $z^n = 1 \implies z = e^{i\frac{2k\pi}{n}}$ for $k=0,1,...,n-1$. #### 2. Quadratic Equations - **Form:** $ax^2 + bx + c = 0