Pre-Calc Reference

Algebra toolkit

Exponents and radicals

Product / quotient \(x^a x^b = x^{a+b}\), \(\dfrac{x^a}{x^b} = x^{a-b}\)

Power of a power \((x^a)^b = x^{ab}\), \((xy)^a = x^a y^a\)

Negative and zero \(x^{-a} = \dfrac{1}{x^a}\), \(x^0 = 1\) for \(x \neq 0\)

Fractional \(x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\)

Rewrite before differentiating \(\dfrac{1}{\sqrt{x}} = x^{-1/2}\), \(\sqrt[3]{x^2} = x^{2/3}\) essential habit

Factoring and special products

Difference of squares \(a^2 - b^2 = (a-b)(a+b)\)

Perfect squares \((a \pm b)^2 = a^2 \pm 2ab + b^2\)

Sum / difference of cubes \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)

Grouping \(ax + ay + bx + by = (a+b)(x+y)\)

Why it matters factoring is how a \(0/0\) limit gets resolved used constantly

Quadratics

Quadratic formula \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Discriminant \(b^2 - 4ac\) positive: two real roots; zero: one; negative: none counts roots

Vertex form \(y = a(x-h)^2 + k\), vertex at \((h, k)\)

Vertex from standard form \(h = -\dfrac{b}{2a}\), then \(k = f(h)\)

Completing the square \(x^2 + bx = \left(x + \tfrac{b}{2}\right)^2 - \tfrac{b^2}{4}\)

Fractions and radicals

Common denominator \(\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}\)

Complex fraction multiply top and bottom by the inner denominators

Rationalizing multiply by the conjugate: \(\dfrac{1}{\sqrt{a}-\sqrt{b}} \cdot \dfrac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}}\)

Absolute value \(|x| = a \Rightarrow x = \pm a\); \(|x| < a \Rightarrow -a < x < a\)

Never do this \(\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}\), \(\dfrac{1}{a+b} \neq \dfrac{1}{a} + \dfrac{1}{b}\) common error

Functions

Domain and range

Denominators exclude anything making the bottom zero

Even roots need the inside \(\geq 0\)

Logarithms need the argument \(> 0\) strictly

Interval notation \((a,b)\) excludes ends, \([a,b]\) includes them, \(\infty\) is always open

Vertical line test a graph is a function if no vertical line meets it twice

Composition and inverses

Composition \((f \circ g)(x) = f(g(x))\), inner function first

Decomposing spotting \(f(g(x))\) is what the chain rule needs essential habit

Inverse swap \(x\) and \(y\), then solve for \(y\)

Inverse property \(f(f^{-1}(x)) = x\); the graphs reflect across \(y = x\)

Exists only if the function is one-to-one, so it passes the horizontal line test

Symmetry

Even \(f(-x) = f(x)\), symmetric about the \(y\)-axis (\(x^2\), \(\cos x\))

Odd \(f(-x) = -f(x)\), symmetric about the origin (\(x^3\), \(\sin x\))

Neither most functions, including \(e^x\) and \(x^2 + x\)

Quick test substitute \(-x\) and compare with the original

Transformations of \(y = f(x)\)

\(f(x) + k\) shifts up \(k\); \(f(x) - k\) shifts down

\(f(x - h)\) shifts right \(h\), opposite the sign counterintuitive

\(a\,f(x)\) stretches vertically by \(a\); \(a < 0\) also flips over the \(x\)-axis

\(f(bx)\) compresses horizontally by \(b\), again the reverse of what it looks like

Order horizontal changes act on the input first, vertical ones on the output last

Polynomials and rational functions

Lines and polynomials

Slope \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\), the quantity a derivative generalizes

Point-slope form \(y - y_1 = m(x - x_1)\) used for every tangent line

Parallel / perpendicular equal slopes; slopes multiplying to \(-1\)

End behavior set by the leading term: even degree matches at both ends, odd degree opposes

Roots and multiplicity odd multiplicity crosses the axis, even multiplicity touches and turns back

Asymptotes and holes

Vertical asymptote denominator zero that does not cancel

Hole a factor that cancels from top and bottom not an asymptote

Top degree \(<\) bottom horizontal asymptote \(y = 0\)

Degrees equal horizontal asymptote at the ratio of leading coefficients

Top degree one higher a slant asymptote, found by polynomial division

Exponentials and logarithms

Log rules

Definition \(\log_b x = y \iff b^y = x\), with \(b > 0\), \(b \neq 1\)

Product / quotient \(\log(xy) = \log x + \log y\), \(\log\dfrac{x}{y} = \log x - \log y\)

Power \(\log(x^n) = n\log x\) used for log differentiation

Change of base \(\log_b x = \dfrac{\ln x}{\ln b}\)

Never do this \(\log(x+y) \neq \log x + \log y\) common error

Working with \(e\) and \(\ln\)

Natural log \(\ln x = \log_e x\), with \(e \approx 2.71828\)

Inverse pair \(e^{\ln x} = x\) for \(x > 0\), and \(\ln(e^x) = x\) for all \(x\)

Known values \(\ln 1 = 0\), \(\ln e = 1\); \(\ln x \to -\infty\) as \(x \to 0^+\)

Solving \(b^x = c\) take \(\ln\) of both sides, then \(x = \dfrac{\ln c}{\ln b}\)

Rewriting any base \(b^x = e^{x\ln b}\), the form calculus prefers

Trigonometry

Unit circle values

\(0\) \(\sin = 0\), \(\cos = 1\), \(\tan = 0\)

\(\pi/6\) (\(30^\circ\)) \(\sin = \tfrac{1}{2}\), \(\cos = \tfrac{\sqrt{3}}{2}\), \(\tan = \tfrac{\sqrt{3}}{3}\)

\(\pi/4\) (\(45^\circ\)) \(\sin = \cos = \tfrac{\sqrt{2}}{2}\), \(\tan = 1\)

\(\pi/3\) (\(60^\circ\)) \(\sin = \tfrac{\sqrt{3}}{2}\), \(\cos = \tfrac{1}{2}\), \(\tan = \sqrt{3}\)

\(\pi/2\) (\(90^\circ\)) \(\sin = 1\), \(\cos = 0\), \(\tan\) undefined

Signs by quadrant I all, II sine, III tangent, IV cosine

Identities

Reciprocal \(\csc = \tfrac{1}{\sin}\), \(\sec = \tfrac{1}{\cos}\), \(\cot = \tfrac{1}{\tan}\)

Pythagorean \(\sin^2\theta + \cos^2\theta = 1\)

Derived forms \(1 + \tan^2\theta = \sec^2\theta\), \(1 + \cot^2\theta = \csc^2\theta\)

Double angle \(\sin 2\theta = 2\sin\theta\cos\theta\), \(\cos 2\theta = \cos^2\theta - \sin^2\theta\)

Power reduction \(\sin^2\theta = \tfrac{1 - \cos 2\theta}{2}\), \(\cos^2\theta = \tfrac{1 + \cos 2\theta}{2}\) needed for integrals

Sum \(\sin(a \pm b) = \sin a\cos b \pm \cos a\sin b\)

Graphs

\(y = a\sin(bx + c) + d\) amplitude \(|a|\), period \(\dfrac{2\pi}{|b|}\), vertical shift \(d\)

Sine and cosine range \([-1, 1]\), period \(2\pi\), defined everywhere

Tangent period \(\pi\), asymptotes at \(\tfrac{\pi}{2} + n\pi\)

Radians only every calculus formula assumes radians common error

Inverse trig

\(\arcsin x\) domain \([-1,1]\), range \(\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]\)

\(\arccos x\) domain \([-1,1]\), range \([0, \pi]\)

\(\arctan x\) domain all reals, range \(\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)\)

Why ranges are restricted trig functions repeat, so they are only invertible on one chosen cycle

Getting ready for limits

Algebraic moves that resolve \(0/0\)

Factor and cancel \(\dfrac{x^2 - 9}{x - 3} = x + 3\) for \(x \neq 3\)

Multiply by the conjugate use it whenever a square root sits in a difference

Clear inner fractions multiply through by the common denominator

Divide by the highest power the standard move for limits as \(x \to \infty\)

Things worth memorizing now

Difference quotient \(\dfrac{f(x+h) - f(x)}{h}\), the definition of the derivative

Piecewise continuity the two sides must agree in value at the joining point

Binomial expansion \((x+h)^2 = x^2 + 2xh + h^2\), \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\)

Function notation \(f(x+h)\) means substitute \(x+h\) everywhere, not add \(h\) to the output common error