Course Review

A function $f$ assigns each input $x$ (the domain) exactly one output $y=f(x)$; the outputs form the range. Passes the vertical line test.

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• Polynomial: domain $(-\infty,\infty)$.
• Radical $\sqrt{g(x)}$: require $g(x)\ge 0$.
• Rational $\dfrac{p(x)}{q(x)}$: require $q(x)\ne 0$.

Symmetry: even if $f(-x)=f(x)$ ($y$-axis); odd if $f(-x)=-f(x)$ (origin).

Difference quotient: $\dfrac{f(x+h)-f(x)}{h},\ h\ne 0$.   Average rate of change on $[a,b]$: $\dfrac{f(b)-f(a)}{b-a}$ (secant slope).

Combinations: $(f\pm g)(x)=f(x)\pm g(x)$, $(fg)(x)=f(x)g(x)$, $\left(\tfrac{f}{g}\right)(x)=\dfrac{f(x)}{g(x)}$.

Composition: $(f\circ g)(x)=f\big(g(x)\big)$ (not commutative). Domain: those $x$ in dom $g$ with $g(x)$ in dom $f$.

Inverse: exists iff $f$ is one-to-one (horizontal line test). Swap $x,y$ and solve; $f\big(f^{-1}(x)\big)=x$ and $f^{-1}\big(f(x)\big)=x$; graphs reflect over $y=x$.

Parents: $c,\ x,\ |x|,\ \sqrt{x},\ x^2,\ x^3,\ \dfrac1x,\ \lfloor x\rfloor$.

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• $+d$ up / $-d$ down; $f(x-c)$ right $c$, $f(x+c)$ left (horizontal moves oppose the sign).
• $a<0$: reflect over $x$-axis; $|a|>1$ vertical stretch, $0<|a|<1$ shrink.
• $b<0$: reflect over $y$-axis; $|b|>1$ horizontal shrink, $0<|b|<1$ stretch (factor: $f\big(b(x-\tfrac cb)\big)$).

For leading term $a_nx^n$: if $n$ even, both ends same way ($a_n>0$ up/up, $a_n<0$ down/down); if $n$ odd, ends opposite ($a_n>0$ down/up). A zero of multiplicity $m$: graph crosses if $m$ odd, touches (turns) if $m$ even. A degree-$n$ polynomial has at most $n-1$ turning points.

Division algorithm: $f(x)=d(x)q(x)+r(x)$.

Remainder Theorem: dividing $f$ by $(x-k)$ leaves remainder $f(k)$.

Factor Theorem: $(x-k)$ is a factor $\iff f(k)=0$.

Rational Zero Test: any rational zero of a polynomial with integer coefficients has the form $\dfrac{p}{q}$, $p\mid$ (constant term), $q\mid$ (leading coefficient).

Fundamental Thm of Algebra: degree $n$ ($n\ge1$) has exactly $n$ complex zeros (with multiplicity). Complex zeros of a real polynomial come in conjugate pairs $a\pm bi$.

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• Vertical asymptote at zeros of $D$ not cancelled by $N$; a cancelled factor gives a hole.
• Horizontal asymptote: deg $N<$ deg $D\Rightarrow y=0$; equal degree $\Rightarrow y=\dfrac{\text{lead }N}{\text{lead }D}$; deg $N>$ deg $D\Rightarrow$ none.
• Slant asymptote when deg $N=$ deg $D+1$: the quotient from long division.

$f(x)=a\,b^{x}$ ($b>0,\ b\ne1$): horizontal asymptote $y=0$, domain $(-\infty,\infty)$, range $(0,\infty)$; grows if $b>1$, decays if $0<b<1$. The natural base $e\approx2.71828$.

Compound interest: $A=P\!\left(1+\dfrac{r}{n}\right)^{nt}$; continuous: $A=Pe^{rt}$.

$y=\log_b x \iff b^{y}=x$ (so $\log$ is the inverse of $b^{x}$); $\ln x=\log_e x$. Domain $(0,\infty)$.

Inverse: $b^{\log_b x}=x$, $\log_b b^{x}=x$.   Change of base: $\log_b x=\dfrac{\ln x}{\ln b}=\dfrac{\log x}{\log b}$.

$180^\circ=\pi$ rad. Arc length: $s=r\theta$; area of sector: $A=\tfrac12 r^2\theta$; angular/linear speed: $v=r\omega$ ($\theta$ in radians). Coterminal: add multiples of $360^\circ$ or $2\pi$.

On the unit circle a terminal point is $(\cos\theta,\sin\theta)$; $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$, and $\csc,\sec,\cot$ are the reciprocals of $\sin,\cos,\tan$. Right triangle: $\sin=\dfrac{\text{opp}}{\text{hyp}}$, $\cos=\dfrac{\text{adj}}{\text{hyp}}$, $\tan=\dfrac{\text{opp}}{\text{adj}}$.

First-quadrant exact values:

Signs by quadrant: ASTC — All, Sine, Tangent, Cosine positive in QI--QIV.

For $y=a\sin(bx-c)+d$ (same for $\cos$): amplitude $|a|$, period $\dfrac{2\pi}{b}$, phase shift $\dfrac{c}{b}$, midline $y=d$. For $\tan$ the period is $\dfrac{\pi}{b}$.

Inverse ranges: $\arcsin x\in[-\tfrac\pi2,\tfrac\pi2]$, $\arccos x\in[0,\pi]$, $\arctan x\in(-\tfrac\pi2,\tfrac\pi2)$.

Reciprocal/Quotient: $\csc=\tfrac1{\sin}$, $\sec=\tfrac1{\cos}$, $\cot=\tfrac1{\tan}=\tfrac{\cos}{\sin}$, $\tan=\tfrac{\sin}{\cos}$.

Pythagorean: $\sin^2\theta+\cos^2\theta=1$, $1+\tan^2\theta=\sec^2\theta$, $1+\cot^2\theta=\csc^2\theta$.

Even/Odd: $\sin(-\theta)=-\sin\theta$, $\cos(-\theta)=\cos\theta$, $\tan(-\theta)=-\tan\theta$.

Cofunction: $\sin\!\big(\tfrac\pi2-\theta\big)=\cos\theta$, $\tan\!\big(\tfrac\pi2-\theta\big)=\cot\theta$.

Power-reducing: $\sin^2\theta=\dfrac{1-\cos2\theta}{2}$, $\cos^2\theta=\dfrac{1+\cos2\theta}{2}$.

Half-angle: $\sin\dfrac\theta2=\pm\sqrt{\dfrac{1-\cos\theta}{2}}$, $\cos\dfrac\theta2=\pm\sqrt{\dfrac{1+\cos\theta}{2}}$, $\tan\dfrac\theta2=\dfrac{1-\cos\theta}{\sin\theta}=\dfrac{\sin\theta}{1+\cos\theta}$.

Product-to-sum: $\sin A\sin B=\tfrac12[\cos(A-B)-\cos(A+B)]$, $\cos A\cos B=\tfrac12[\cos(A-B)+\cos(A+B)]$, $\sin A\cos B=\tfrac12[\sin(A+B)+\sin(A-B)]$.

Law of Sines: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$ (cases AAS, ASA, SSA).

Ambiguous case (SSA): given $a,b,A$ with $A$ acute and $h=b\sin A$: no triangle if $a<h$; one if $a=h$ or $a\ge b$; two if $h<a<b$.

Law of Cosines: $c^2=a^2+b^2-2ab\cos C$ (cases SAS, SSS).

Area: $\tfrac12 ab\sin C$; Heron: $\sqrt{s(s-a)(s-b)(s-c)}$ with $s=\dfrac{a+b+c}{2}$.

$\mathbf{v}=\langle v_1,v_2\rangle$, magnitude $\|\mathbf{v}\|=\sqrt{v_1^2+v_2^2}$, direction angle $\theta$ with $\mathbf{v}=\|\mathbf{v}\|\langle\cos\theta,\sin\theta\rangle$. Unit vector $\mathbf{u}=\dfrac{\mathbf{v}}{\|\mathbf{v}\|}$; standard form $v_1\mathbf{i}+v_2\mathbf{j}$.

Dot product: $\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2=\|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta$. Angle: $\cos\theta=\dfrac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\,\|\mathbf{v}\|}$. Orthogonal $\iff\mathbf{u}\cdot\mathbf{v}=0$.

Projection: $\operatorname{proj}_{\mathbf{v}}\mathbf{u}=\dfrac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{v}\|^2}\,\mathbf{v}$.

Point $(r,\theta)$. Polar $\to$ rectangular: $x=r\cos\theta,\ y=r\sin\theta$. Rectangular $\to$ polar: $r=\sqrt{x^2+y^2},\ \tan\theta=\dfrac yx$ (choose $\theta$ by the quadrant of $(x,y)$).

Common curves: circle $r=a$; line $\theta=\alpha$; rose $r=a\cos(n\theta)$ ($n$ petals if odd, $2n$ if even); limaçon $r=a\pm b\cos\theta$; cardioid when $a=b$.

$z=a+bi$, $|z|=r=\sqrt{a^2+b^2}$. Trig (polar) form: $z=r(\cos\theta+i\sin\theta)$ with $\tan\theta=\dfrac ba$.

Product/Quotient: $z_1z_2=r_1r_2\big[\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)\big]$, $\dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}\big[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\big]$.

DeMoivre: $z^n=r^n\big(\cos n\theta+i\sin n\theta\big)$.

$n$th roots: $\sqrt[n]{r}\left[\cos\dfrac{\theta+2\pi k}{n}+i\sin\dfrac{\theta+2\pi k}{n}\right]$, $k=0,1,\dots,n-1$ (equally spaced by $\dfrac{2\pi}{n}$).

Methods: substitution, elimination, or row-reduce the augmented matrix to row-echelon form (leading 1's, zeros below) then back-substitute. Row operations: swap rows, multiply a row by a nonzero constant, add a multiple of one row to another. A system is inconsistent (no solution) if a row gives $0=$ nonzero; dependent (infinitely many) if a variable is free.

Add/subtract entrywise (same size). Product $AB$: rows of $A$ dotted with columns of $B$; defined when (cols of $A$) $=$ (rows of $B$); not commutative. Identity $I$: $AI=IA=A$.

$2\times2$ inverse: for $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, $\det A=ad-bc$ and $A^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ (exists iff $\det A\ne0$).

$3\times3$ determinant by cofactor expansion. Cramer's Rule: $x_i=\dfrac{\det A_i}{\det A}$, where $A_i$ replaces column $i$ with the constants (requires $\det A\ne0$).

For a proper rational $\dfrac{N(x)}{D(x)}$ (divide first if improper), split by the factors of $D$:

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• Linear $(ax+b)$: term $\dfrac{A}{ax+b}$; repeated $(ax+b)^k$: one term per power up to $k$.
• Irreducible quadratic $(ax^2+bx+c)$: term $\dfrac{Bx+C}{ax^2+bx+c}$.

Clear denominators and match coefficients (or substitute convenient $x$-values) to solve.

Circle: $(x-h)^2+(y-k)^2=r^2$.

Parabola: $(x-h)^2=4p(y-k)$ opens up/down; $(y-k)^2=4p(x-h)$ opens left/right. Focus is $p$ from vertex; directrix is the opposite side; $|4p|$ is the latus rectum.

Ellipse: $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$, $a>b$: major axis length $2a$, minor $2b$, foci $c$ from center with $c^2=a^2-b^2$.

Hyperbola: $\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$: vertices $a$ from center, foci with $c^2=a^2+b^2$, asymptotes $y-k=\pm\dfrac ba(x-h)$.

Eccentricity $e=\dfrac ca$: circle $e=0$; ellipse $0<e<1$; parabola $e=1$; hyperbola $e>1$.

Parametric: $x=f(t),\ y=g(t)$ traces a curve with orientation as $t$ increases. Eliminate the parameter by solving for $t$ (or using $\cos^2t+\sin^2t=1$) to get a relation in $x,y$.

Arithmetic (common difference $d$): $a_n=a_1+(n-1)d$; sum $S_n=\dfrac n2(a_1+a_n)=\dfrac n2\big[2a_1+(n-1)d\big]$.

Geometric (common ratio $r$): $a_n=a_1r^{\,n-1}$; sum $S_n=a_1\dfrac{1-r^{n}}{1-r}$.

Infinite geometric: converges iff $|r|<1$, then $S=\dfrac{a_1}{1-r}$.

Sigma: $\displaystyle\sum_{i=1}^{n}a_i=a_1+a_2+\cdots+a_n$. Useful: $\displaystyle\sum_{i=1}^{n}i=\dfrac{n(n+1)}{2}$.

Mathematical induction: prove a base case ($n=1$), then assume $P(k)$ and prove $P(k+1)$; concludes $P(n)$ for all $n$.

Counting: permutations $_nP_r=\dfrac{n!}{(n-r)!}$ (order matters); combinations $_nC_r=\dbinom{n}{r}=\dfrac{n!}{r!(n-r)!}$ (order does not).

The $(k+1)$th term is $\dbinom{n}{k}a^{\,n-k}b^{k}$. Coefficients form Pascal's triangle; $\dbinom{n}{k}=\dbinom{n}{n-k}$.

$\displaystyle\lim_{x\to c}f(x)=L$ means $f(x)$ approaches $L$ as $x\to c$. It exists iff the one-sided limits agree: $\displaystyle\lim_{x\to c^-}f=\lim_{x\to c^+}f$. Limits ignore the actual value $f(c)$.

For constants/sums/products/quotients the limit distributes:

Polynomials/rationals: substitute $c$ when the denominator is nonzero.

$\tfrac00$ form: factor and cancel, rationalize, or simplify, then substitute.

Infinite limits: if $f\to\pm\infty$ near $c$ there is a vertical asymptote $x=c$.

Limits at infinity (end behavior of rationals): compare degrees like horizontal asymptotes — lower top $\to0$, equal $\to$ ratio of leading coefficients, higher top $\to\pm\infty$.

$f$ is continuous at $c$ if all hold: (1) $f(c)$ defined, (2) $\displaystyle\lim_{x\to c}f(x)$ exists, (3) $\displaystyle\lim_{x\to c}f(x)=f(c)$.

Limit definition of the derivative:

the instantaneous rate of change / slope of the tangent line (the difference quotient as $h\to0$).