Math Formulas

    Precalculus Formulas

    Algebra

    • Quadratic Formula:
      \(x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
    • Special Factoring Properties:
      \(x^2-y^2=(x+y)(x-y)\)
      \((x-y)^2=x^2-2xy+y^2\)
      \((x+y)^2=x^2+2xy+y^2\)
      \(x^3-y^3=(x-y)(x^2+xy+y^2)\)
      \(x^3+y^3=(x+y)(x^2-xy+y^2)\)
    • Logarithm Properties:
      \(y=\log_a(x) => a^y = x\)
      \(\log_a(xy) = \log_a(x) + \log_a(y)\)
      \(\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\)
      \(\log_a(x^r) = r \log_a(x)\)
      \(a^{\log_a(x)} = x\)
      \(\log_a(a^x) = x\)
      \(\log_a(1) = 0\)
      \(\log_a(a) = 1\)
      \(\log(x) = \log_{10}(x)\)
      \(\ln(x) = \log_e(x)\)
      \(\log_b(u) = \frac{\log_a(u)}{\log_a(b)} \)
    • Radical Properties:
      \(a^ma^n=a^{m+n} \)
      \((a^m)^n=a^{mn}\)
      \((ab)^n=a^{n}b^n\)
      \((\frac{a}{b})^n=\frac{a^n}{b^n}\)
      \(\frac{a^m}{a^n}=a^{m-n}\)
      \(a^{-n}=\frac{1}{a^n}\)
      \(a^{\frac{1}{n}}=\sqrt[n]{a}\)
      \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)
      \(a^{\frac{m}{n}}=(\sqrt[n]{a})^m\)
      \(\sqrt[n]{ab}=\sqrt[n]a\sqrt[n]b\)
      \(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]a}{\sqrt[n]b}\)
      \(\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]a\)

    Trigonometry

    • Right Triangles:
      \(\sin (\theta) = \frac{opp}{hyp}\), \(\csc (\theta) = \frac{hyp}{opp}\)
      \(\cos (\theta) = \frac{adj}{hyp}\), \(\sec (\theta) = \frac{hyp}{adj}\)
      \(\tan (\theta) = \frac{opp}{adj}\), \(\cot (\theta) = \frac{adj}{opp}\)
    • The Law of Sines:
      \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
    • The Law of Cosines:
      \(a^2=b^2+c^2-2bc\cos A\)
      \(b^2=a^2+c^2-2ac\cos B\)
      \(c^2=a^2+b^2-2ab\cos C\)
    • Pythagorean Identity:
      \(\sin^2(x) + \cos^2(x) = 1\)
      \(1+\tan^2(x) = \sec^2(x)\)
      \(1+\cot^2(x) + \csc^2(x)\)
    • Sum and Difference Formulas:
      \begin{equation}\small\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)\end{equation}
      \begin{equation}\small\cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b)\end{equation}
    • Double Angle Formulas:
      \(\sin(2x) = 2\sin(x)\cos(x)\)
      \(\cos(2x) = \cos^2(x) - \sin^2(x)\)
    • Trigonometric Identities:
      \(\sin(x)\csc(x)=1\)
      \(\cos(x)\sec(x)=1\)
      \(\tan(x)\cot(x)=1\)
      \(\tan(x)=\frac{\sin(x)}{\cos(x)}\), \(\cot(x)=\frac{\cos(x)}{\sin(x)}\)
    • Hyperbolic Definition:
      \(\sinh (x) = \frac{e^x -e^{-x}}{2}\)
      \(\cosh (x) = \frac{e^x +e^{-x}}{2}\)
      \(\tanh (x) = \frac{\sinh (x)}{\cosh (x)}\)
      \(\coth (x) = \frac{\cosh (x)}{\sinh (x)}\)
      \(\operatorname{sech}(x) = \frac{1}{\cosh (x)}\)
      \(\operatorname{csch}(x) = \frac{1}{\sinh (x)}\)
    • Hyperbolic Identities:
      \(\sinh (-x) = -\sinh (x)\)
      \(\cosh (-x) = \cosh (x)\)
      \(\cosh^{2}(x) - \sinh^{2}(x) = 1\)
      \(1-\tanh^{2} (x) = \operatorname{sech^2}(x)\)
      \begin{equation}\small \sinh (x+y) = \sinh x\cosh y + \cosh x \sinh y\end{equation}
      \begin{equation}\small\cosh (x+y) = \cosh x\cosh y + \sinh x\sinh y\end{equation}
    • Interactive Unit Circle:
      Angle: 0°
      sin(θ) = 0
      cos(θ) = 1
      tan(θ) = 0

    Geometry

    • Pythagorean Theorem:
      \(c^2 = a^2 + b^2\)
    • Square:
      \(A = s^2\)
      \(C=4s\)
    • Rectangle:
      \(A = lw\)
      \(C=2l+2w\)
    • Triangle:
      \(A = \frac{1}{2}bh\)
      \(C=a+b+c\)
    • Circle:
      \(A = \pi r^2\)
      \(C=2\pi r\)
    • Cube:
      \(V = s^3\)
      \(S=6s^2\)
    • Rectangle Prism:
      \(V = lwh\)
      \(S=2(lw+wh+hl)\)
    • Cone:
      \(V = \frac{1}{3}\pi r^2 h\)
      \(S=\pi r(r+\sqrt{h^2+ r^2})\)
    • Sphere:
      \(V = \frac{4}{3}\pi r^3\)
      \(S=4\pi r^2\)
    • Distance Formula:
      \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
    • Equation of a Circle:
      \((x - h)^2 + (y - k)^2 = r^2\)
    • Slope-intercept form of a Line:
      \(y = mx + b\)
    • Point-slope form of a Line:
      \(y - y_1 = m(x - x_1)\)

    Calculus Formulas

    Limit

    • Ways to Solve Limit:
      • 1.Direct Substitution
      • When possible, substitute the value of x directly into the function to evaluate the limit.
      • Example: \(\lim_{x\to 2}{3x+4} = 3(2)+4=10\)
      • 2. Factoring
      • Factor the expression to simplify it, especially when you encounter indeterminate forms like \(\frac{0}{0}\).
      • Example: \begin{split}\lim_{x\to 2}{\frac{x^2 -4}{x-2}} & = lim_{x\to 2}{\frac{(x-2)(x+2)}{x-2}}\\ & =lim_{x\to 2}{x+2}=4\end{split}
      • 3. Rationalizing
      • Rationalize the numerator or denominator if there are square roots to eliminate indeterminate forms like \(\frac{0}{0}\).
      • Example: \(\lim_{x\to 0}{\frac{\sqrt{x+1} -1}{x}}\). Multiply numerator and denominator by \(\sqrt{x+1}+1\)
      • 4. L'Hopital's Rule
      • Use L'Hopital's Rule when limits result in indeterminate forms \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Differentiate the numerator and denominator separately.
      • Example: \(\lim_{x\to 0}{\frac{\sin x}{x}}\).
        Apply L'Hopital's Rule:
        \(f(x)=\sin x =>f'(x)=\cos x\)
        \(g(x)=x=>g'(x)=1\).
        \(\lim_{x\to 0}{\frac{\cos x}{1}}=1\)
      • 5. Squeeze Theorem
      • Use Squeeze Theorem when the function is "squeezed" between two other functions that have the same limit.
      • Example: \(\lim_{x\to 0}{x^2\sin {\frac{1}{x}}}=0\).
        Since \(-x^2\leq x^2 \sin {\frac{1}{x}} \leq x^2\)
      • 6. Dominant Term for Infinity Limits
      • When \(x\rightarrow\infty\), focus on the dominant term (the highest power of x) in the numerator and denominator.
      • Example: \(\lim_{x\to \infty}{\frac{3x^2+5x}{2x^2-x}}=\lim_{x\to \infty}{\frac{3+\frac{5}{x}}{2-\frac{1}{x}}}=\frac{3}{2}\)
      • 7. Trigonometric Limits
      • Key limits to memorize:
      • \(\lim_{x\to 0}{\frac{\sin x}{x}}=1\)
      • \(\lim_{x\to 0}{\frac{\cos x -1}{x}}=0\)
      • \(\lim_{x\to 0}{\frac{1-\cos x}{x^2}}=\frac{1}{2}\)
      • 8. Logarithmic and Exponential Limits
      • Use properties of logarithms and exponentials to simplify limits:
      • \(\lim_{x\to 0^+}{\ln(x)} =-\infty\)
      • \(\lim_{x\to \infty}{e^x}=\infty\)
      • \(\lim_{x\to -\infty}{e^x}=0\)
      • 9. Change of Variables
      • Use substitution to simplify complicated expressions. Often used when dealing with limits approaching infinity or zero.
      • Example: \(\lim_{x\to \infty}{\frac{1}{x}}\)
      • Let \(u =\frac{1}{x}\)
      • So \(\lim_{x\to \infty}{\frac{1}{x}}=\lim_{u\to 0}{u}=0\)
      • 10. Limits at Infinity: Horizontal Asymptotes
      • For rational functions \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials:
      • If degrees of numerator and denominator are equal: the limit is the ratio of leading coefficients.
      • If the degree of numerator is higher: the limit is \(\infty\) or \(-\infty\).
      • If the degree of denominator is higher: the limit is 0.
      • 11. Limits with Piecewise Functions
      • Check limits from both sides (left and right-hand limits). If they agree, that is the limit.
      • Example: For a piecewise function \begin{gather*}lim_{x\to a^-}{f(x)}= lim_{x\to a^+}{f(x)}=L\\ lim_{x\to a}{f(x)}=L\end{gather*}
    • Limit Definition:
      \begin{gather*}lim_{x\to a^-}{f(x)}= lim_{x\to a^+}{f(x)}=L\\ lim_{x\to a}{f(x)}=L\end{gather*}
      \begin{gather*}if\hspace{0.3cm}lim_{x\to a^-}{f(x)}\neq lim_{x\to a^+}{f(x)}\\lim_{x\to a}{f(x)}=undefined\end{gather*}
    • Sum Rule:
      \begin{equation}{\small lim_{x\to a}{f(x)+g(x)}=lim_{x\to a}f(x)+lim_{x\to a}g(x)}\end{equation}
    • Difference Rule:
      \begin{equation}{\small lim_{x\to a}{f(x)-g(x)}=lim_{x\to a}f(x)-lim_{x\to a}g(x)}\end{equation}
    • Product Rule:
      \begin{equation}{\small lim_{x\to a}{f(x).g(x)}=lim_{x\to a}f(x).lim_{x\to a}g(x)}\end{equation}
    • Quotient Rule:
      \(lim_{x\to a}{\frac{f(x)}{g(x)}}=\frac{lim_{x\to a}f(x)}{lim_{x\to a}g(x)}\)

    Derivatives

    • Basic Rule:
      \(\frac{d}{dx}(c) = 0\)
      \(\frac{d}{dx}(x) = 1\)
      \(\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))\)
      \(\frac{d}{dx}(f(x)+g(x)) \\= f'(x)+g'(x)\)
      \(\frac{d}{dx}(f(x)-g(x)) \\= f'(x)-g'(x)\)
    • Power Rule:
      \(\frac{d}{dx}(x^n) = nx^{n-1}\)
    • Product Rule:
      \(\frac{d}{dx}(f(x).g(x)) \\= f'(x)g(x)+f(x)g'(x)\)
    • Quotient Rule:
      \(\frac{d}{dx}(\frac{f(x)}{g(x)}) \\= \frac{f'(x)g(x)-f(x)g'(x)}{(g(x))^2}\)
    • Chain Rule:
      \(\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)\)
    • Derivatives of Trigonometric Functions:
      \(\frac{d}{dx}(\sin x) = \cos x\)
      \(\frac{d}{dx}(\cos x) = -\sin x\)
      \(\frac{d}{dx}(\tan x) = \sec^2 x\)
      \(\frac{d}{dx}(\cot x) = -\csc^2 x\)
      \(\frac{d}{dx}(\sec x) = \sec x\tan x\)
      \(\frac{d}{dx}(\csc x) = -\csc x\cot x\)
    • Exponential Rule:
      \(\frac{d}{dx}(e^x)=e^x\)
      \(\frac{d}{dx}(e^{f(x)})=e^{f(x)}.f'(x)\)
      \(\frac{d}{dx}(a^{f(x)})=ln(a).a^{f(x)}.f'(x)\)
    • Logarithm Rule:
      \(\frac{d}{dx}(\ln(x))=\frac{1}{x}\)
      \(\frac{d}{dx}(\ln{f(x)})=\frac{1}{f(x)}f'(x)\)
      \(\frac{d}{dx}(\log_a(x))=\frac{1}{x\ln(a)}\)
      \(\frac{d}{dx}(\log_a{f(x)})=\frac{1}{f(x)\ln(a)}f'(x)\)
    • Derivatives of Inverse Trigonometric Functions:
      \(\frac{d}{dx}(\sin^{-1} (u)) = \frac{u'}{\sqrt{1-u^2}}\)
      \(\frac{d}{dx}(\cos^{-1} (u)) = \frac{-u'}{\sqrt{1-u^2}}\)
      \(\frac{d}{dx}(\tan^{-1} (u)) = \frac{u'}{1+u^2}\)
      \(\frac{d}{dx}(\cot^{-1} (u)) = \frac{-u'}{1+u^2}\)
      \(\frac{d}{dx}(\sec^{-1} (u)) = \frac{u'}{|u|\sqrt{u^2 -1}}\)
      \(\frac{d}{dx}(\csc^{-1} (u)) = \frac{-u'}{|u|\sqrt{u^2 -1}}\)
    • Derivatives of Hyperbolic Functions:
      \(\frac{d}{dx}(\sinh (x)) = \cosh (x)\)
      \(\frac{d}{dx}(\cosh (x)) = \sinh (x)\)
      \(\frac{d}{dx}(\tanh (x)) = \operatorname{sech^2}(x)\)
      \(\frac{d}{dx}(\coth (x)) = -\operatorname{csch^2}(x)\)
      \(\frac{d}{dx}(\operatorname{sech}(x)) = -\operatorname{sech}(x)\tanh(x)\)
      \(\frac{d}{dx}(\operatorname{csch}(x)) = -\operatorname{csch}(x)\coth(x)\)

    Integrals

    • Power Rule:
      \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\)
    • Sum/Difference Rule:
      \(\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx\)
    • Integration by Substitution:
      \(\int f(g(x))g'(x) dx = \int f(u) du\)
    • Integration by Parts:
      \(\int u dv = uv - \int v du\)
    • Integrals of Trigonometric Functions:
      • \(\int \sin x dx = -\cos x + C\)
      • \(\int \cos x dx = \sin x + C\)
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