APGP

International Baccalaureate Higher Level Mathematics

IB HL Math

MF 27 Walkthrough

$(2+3x)^5 \\rightarrow (1+x)^n$ or $(a+b)^n$

$(a+b)^n = a^n + \\binom{n}{1}a^{n-1}b + \\binom{n}{2}a^{n-2}b^2 + \\binom{n}{3}a^{n-3}b^3 + ... + b^n$, where $n$ is a positive integer

$\\binom{n}{r} = \\dfrac{n!}{r!(n-r)!} = \\dfrac{n(n-1)(n-2)}{3!}$

$f(x) = f(0) + xf'(0) + \\dfrac{x^2}{2!}f''(0) + ... + \\dfrac{x^n}{n!}f^{(n)}(0) + ...$

$(1+x)^n = 1 + nx + \\dfrac{n(n-1)}{2!}x^2 + \\dfrac{n(n-1)...(n-r+1)}{r!}x^r + ...$ for $|x| < 1$

$e^x = 1 + x + \\dfrac{x^2}{2!} + \\dfrac{x^3}{3!} + ... + \\dfrac{x^r}{r!} + ...$ (all $x$)

$\\sin x = x - \\dfrac{x^3}{3!} + \\dfrac{x^5}{5!} - ... + \\dfrac{(-1)^r x^{2r+1}}{(2r+1)!} + ...$ (all $x$)

$\\cos x = 1 - \\dfrac{x^2}{2!} + \\dfrac{x^4}{4!} - ... + \\dfrac{(-1)^r x^{2r}}{(2r)!} + ...$ (all $x$)

$\\ln(1+x) = x - \\dfrac{x^2}{2} + \\dfrac{x^3}{3} - ... + \\dfrac{(-1)^{r+1} x^r}{r} + ...$ for $-1 < x \\leq 1$

From MF27: $\\tan x \\approx x$ when $x$ is small

Since $\\tan x = \\dfrac{\\sin x}{\\cos x} \\approx \\dfrac{x - \\frac{x^3}{3!}}{1 - \\frac{x^2}{2!}} \\approx x\\left(1 - \\frac{x^2}{6}\\right)\\left(1 - \\frac{x^2}{2}\\right)^{-1}$

Also: $\\sin x \\approx \\tan x \\approx x$ when $x$ is small

Non-repeated linear factors: $\\dfrac{px+q}{(ax+b)(cx+d)} = \\dfrac{A}{(ax+b)} + \\dfrac{B}{(cx+d)}$

Repeated linear factors: $\\dfrac{px^2+qx+r}{(ax+b)(cx+d)^2} = \\dfrac{A}{(ax+b)} + \\dfrac{B}{(cx+d)} + \\dfrac{C}{(cx+d)^2}$

Non-repeated quadratic: $\\dfrac{px^2+qx+r}{(ax+b)(x^2+c^2)} = \\dfrac{A}{(ax+b)} + \\dfrac{Bx+C}{(x^2+c^2)}$

$3\\cos x - 2\\sin x = R\\cos(x + \\alpha)$

$= R(\\cos x \\cos\\alpha - \\sin x \\sin\\alpha)$

Comparing: $R\\cos\\alpha = 3$, $R\\sin\\alpha = 2$

$R = \\sqrt{3^2 + 2^2} = \\sqrt{13}$, $\\tan\\alpha = \\dfrac{2}{3}$, $\\alpha = \\tan^{-1}\\left(\\dfrac{2}{3}\\right)$

$\\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B$

$\\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B$

$\\tan(A \\pm B) = \\dfrac{\\tan A \\pm \\tan B}{1 \\mp \\tan A \\tan B}$

$\\cos 2A = \\cos^2 A - \\sin^2 A = 2\\cos^2 A - 1 = 1 - 2\\sin^2 A$

$\\tan 2A = \\dfrac{2\\tan A}{1 - \\tan^2 A}$

$-\\dfrac{\\pi}{2} \\leq \\sin^{-1} x \\leq \\dfrac{\\pi}{2}$ for $|x| \\leq 1$

$0 \\leq \\cos^{-1} x \\leq \\pi$ for $|x| \\leq 1$

$-\\dfrac{\\pi}{2} < \\tan^{-1} x < \\dfrac{\\pi}{2}$

$\\dfrac{\mathrm{d}}{\mathrm{d}x}(\\sin^{-1} x) = \\dfrac{1}{\\sqrt{1-x^2}}$

$\\dfrac{\mathrm{d}}{\mathrm{d}x}(\\cos^{-1} x) = -\\dfrac{1}{\\sqrt{1-x^2}}$

$\\dfrac{\mathrm{d}}{\mathrm{d}x}(\\tan^{-1} x) = \\dfrac{1}{1+x^2}$

$\\dfrac{\mathrm{d}}{\mathrm{d}x}(\\csc x) = -\\csc x \\cot x$

$\\dfrac{\mathrm{d}}{\mathrm{d}x}(\\sec x) = \\sec x \\tan x$

$\\displaystyle\int \\dfrac{1}{x^2 + a^2} dx = \\dfrac{1}{a}\\tan^{-1}\\left(\\dfrac{x}{a}\\right)$

$\\displaystyle\int \\dfrac{1}{\\sqrt{a^2 - x^2}} dx = \\sin^{-1}\\left(\\dfrac{x}{a}\\right)$ for $|x| < a$

$\\displaystyle\int \\tan x \, \mathrm{d}x = \\ln|\\sec x|$ for $|x| < \\dfrac{\\pi}{2}$

$\\displaystyle\int \\cot x \, \mathrm{d}x = \\ln|\\sin x|$ for $0 < x < \\pi$

$\\displaystyle\int \\csc x \, \mathrm{d}x = -\\ln|\\csc x + \\cot x|$ for $0 < x < \\pi$

$\\displaystyle\int \\sec x \, \mathrm{d}x = \\ln|\\sec x + \\tan x|$ for $|x| < \\dfrac{\\pi}{2}$

The point dividing $AB$ in ratio $\\lambda : \\mu$ has position vector $\\dfrac{\\mu\\mathbf{a} + \\lambda\\mathbf{b}}{\\lambda + \\mu}$

$\\mathbf{a} \\times \\mathbf{b} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix} \\times \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} = \\begin{pmatrix} a_2 b_3 - a_3 b_2 \\\\ a_3 b_1 - a_1 b_3 \\\\ a_1 b_2 - a_2 b_1 \\end{pmatrix}$

Area of trapezium $= \\left(\\dfrac{a+b}{2}\\right)h$

Volume of pyramid $= \\dfrac{1}{3}(\\text{Base Area})(h)$

AM-GM Inequality: $\\dfrac{x_1 + x_2 + ... + x_n}{n} \\geq \\sqrt[n]{x_1 x_2 ... x_n}$

Cauchy-Schwarz: $\\left(\\sum u_i v_i\\right)^2 \\leq \\left(\\sum u_i^2\\right)\\left(\\sum v_i^2\\right)$

Triangle Inequality: $|x_1 + x_2 + ... + x_n| \\leq |x_1| + |x_2| + ... + |x_n|$

MF 27 Walkthrough

MF 27 Walkthrough

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