Page Description Equation
85 average speed \[ \text{average speed} =\frac{\text{distance traveled}} {\text{time elapsed}} \]
86 average rate of change, where $h=x_2-x_1,\, h\neq 0$ \[ \begin{align*} \frac{\Delta y}{\Delta x} &=\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}\\ &=\frac{f\left(x_1+h\right)-f\left(x_1\right)}{h} \end{align*} \]
90 limit of the identity function \[ \lim\limits_{x\rightarrow x_0}x=x_0 \]
90 limit of the constant function \[ \lim\limits_{x\rightarrow x_0}k=k \]
91 some functions with no limit \[ \begin{align*} U\left(x\right) &= \left\{ \begin{matrix} 0 &x\lt 0\\ 1 &x\geq0\\ \end{matrix} \right.\\ g\left(x\right) &= \left\{ \begin{matrix} \frac{1}{x} &x\neq0\\ 0 &x=0\\ \end{matrix} \right.\\ f\left(x\right) &= \left\{ \begin{matrix} 0 &x\le 0\\ \sin{\frac{1}{x}} &x\gt0\\ \end{matrix} \right. \end{align*} \]
92 Formal definition of the limit \[ \lim\limits_{x\rightarrow x_0}f\left(x\right)=L\\ \text{if}\; \forall\varepsilon>0,\, \exists\delta>0,\,\forall x\\ 0\lt\abs{x-x_0}\lt\delta \Rightarrow \abs{f\left(x\right)-L}\lt\varepsilon \]
99 sum difference rule for limits \[ \lim\limits_{x\rightarrow c} \left(f\left(x\right)\pm g\left(x\right)\right) =L\pm M \]
99 product rule for limits \[ \lim\limits_{x\rightarrow c} \left(f\left(x\right)\cdot g\left(x\right)\right) =L\cdot M \]
99 constant multiple rule for limits \[ \lim\limits_{x\rightarrow c} \left(k\cdot f\left(x\right)\right) =k\cdot L \]
99 quotient rule for limits \[ \lim\limits_{x\rightarrow c} \frac{f\left(x\right)}{g\left(x\right)} =\frac{L}{M} \]
99 power rule for limits, where $r,s\in\mathbb{Ζ},$ provided $L^{r/s}\in\mathbb{R}$ \[ \lim\limits_{x\rightarrow c} \left(f\left(x\right)\right)^{r/s} =L^{r/s}\\ \]
100 limit of a polynomial \[ \lim\limits_{x\rightarrow c}P\left(x\right)=P\left(c\right) \]
100 limit of a rational function \[ \lim\limits_{x\rightarrow c} \frac{P\left(x\right)} {Q\left(x\right)} =\frac{P\left(c\right)} {Q\left(c\right)} \]
102 sandwich (squeeze, pinching) theorem \[ \left. \begin{matrix} g\left(x\right) \le f\left(x\right) \le h\left(x\right)\\ \lim\limits_{x\rightarrow c} g\left(x\right) =\lim\limits_{x\rightarrow c} h\left(x\right) =L\\ \end{matrix} \right\} \Rightarrow\lim\limits_{x\rightarrow c} f\left(x\right) =L \]
104 right-hand limit \[ \lim\limits_{x\rightarrow a^+}f\left(x\right)=L \]
104 left-hand limit \[ \lim\limits_{x\rightarrow a^-}f\left(x\right)=M \]
105 relation between one-sided and two-sided limits \[ \lim\limits_{x\rightarrow c}f\left(x\right) =L \Leftrightarrow \left\{ \begin{matrix} \lim\limits_{x\rightarrow c^-}f\left(x\right) =L\\ \lim\limits_{x\rightarrow c^+}f\left(x\right)=L \end{matrix} \right. \]
106 theorem where $\theta$ is in radians \[ \lim\limits_{\theta\rightarrow0}\frac{\sin{\theta}}{\theta}=1\\ \]
112 limit as $x$ approaches positive infinity \[ \lim\limits_{x\rightarrow\infty}f\left(x\right)=L \]
112 limit as $x$ approaches negative infinity \[ \lim\limits_{x\rightarrow-\infty}f\left(x\right)=L \]
113 sum and difference rule for limits as $x$ goes to infinity \[ \lim\limits_{x\rightarrow\pm\infty}\left(f\left(x\right)\pm g\left(x\right)\right)=L\pm M \]
113 product rule for limits as $x$ goes to infinity \[ \lim\limits_{x\rightarrow\pm\infty}\left(f\left(x\right)\cdot g\left(x\right)\right)=L\cdot M \]
113 constant multiple rule for limits as $x$ goes to infinity \[ \lim\limits_{x\rightarrow\pm\infty}\left(k\cdot f\left(x\right)\right)=k\cdot L \]
113 quotient rule for limits as $x$ goes to infinity \[ \lim\limits_{x\rightarrow\pm\infty}\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{L}{M} \]
113 power rule for limits as $x$ goes to infinity where $r,s\in\mathbb{Ζ},$ provided $L^{r/s}\in\mathbb{R}$ \[ \lim\limits_{x\rightarrow\pm\infty} \left(f\left(x\right)\right)^{r/s} =L^{r/s} \]
115 The given line $x = a$ is a vertical asymptote or infinite limit of $f$. \[ \left. \begin{matrix} \lim\limits_{x\rightarrow a^+}{f}\left(x\right) =\pm\infty\\ \text{or}\\ \lim\limits_{x\rightarrow a^-}{f}\left(x\right) =\pm\infty \end{matrix} \right\} \Rightarrow \begin{matrix} \text{Vertical}\\ \text{Asymptote} \end{matrix} (x=a) \]
115 The given line $y = b$ is a horizontal asymptote of $f$. \[ \left. \begin{matrix} \lim\limits_{x\rightarrow\infty}{f}\left(x\right) =b\\ \text{or}\\ \lim\limits_{x\rightarrow-\infty}{f}\left(x\right) =b \end{matrix} \right\} \Rightarrow \begin{matrix} \text{Horizontal}\\ \text{Asymptote} \end{matrix} \left(y=b\right) \]
119 infinite limits, positive infinity as a limit \[ \lim\limits_{x\rightarrow x_0}f\left(x\right)=\infty\\ \text{if }\; \forall B\gt0,\,\exists\delta\gt0,\,\forall x\\ 0\lt\abs{x-x_0}\lt\delta\Rightarrow f\left(x\right)\gt B \]
infinite limits, negative infinity as a limit \[ \lim\limits_{x\rightarrow x_0}f\left(x\right)=-\infty\\ \text{if }\;\forall B\gt0,\,\exists\delta>0,\,\forall x\\ 0\lt \abs{x-x_0}\lt\delta\Rightarrow f\left(x\right)\lt -B \]
120 $g$ is a right end behavior model for $f$ iff the condition holds. \[ \lim\limits_{x\rightarrow\infty}\frac{f\left(x\right)}{g\left(x\right)}=1 \]
120 $g$ is a left end behavior model for $f$ iff the condition holds. \[ \lim\limits_{x\rightarrow-\infty}\frac{f\left(x\right)}{g\left(x\right)}=1 \]
125 continuity at an interior point of domain, (both right and left continous) \[ \lim\limits_{x\rightarrow c}f\left(x\right)=f\left(c\right)\\ \]
125 continuity at a left endpoint of domain, right-continuous, continuous from the right \[ \lim\limits_{x\rightarrow a^+}f\left(x\right)=f\left(a\right) \]
125 continuity at a right endpoint of domain, left-continuous, continuous from the left \[ \lim\limits_{x\rightarrow b^-}f\left(x\right)=f\left(b\right) \]
126 continuity test at an interior point Function $f$ is continuous at a point $c$ if and only if:
  1. $c\in\text{dom}f\quad (f\left(c\right) \text{ exists})$
  2. $\lim\limits_{x\rightarrow c}f\left(x\right)\text{ exists }$
  3. $\lim\limits_{x\rightarrow c}f\left(x\right)=f\left(c\right)$
129 A function has the Intermediate Value Property if it never takes on two values without taking on all values in between. In symbols, it has this property if the condition holds: \[ \forall y_0,\,\exists c\\ f\left(a\right)\le y_0\le f\left(b\right) \Rightarrow \left\{ \begin{matrix} f\left(c\right)=y_0\\ \text{and}\\ c\in\left[a,b\right] \end{matrix} \right. \]
136 slope of a curve at a point $P\left(x_0,f\left(x_0\right)\right)$ \[ m=\lim\limits_{h\rightarrow0}\frac{f\left(x_0+h\right)-f\left(x_0\right)}{h} \]
138 difference quotient \[ \frac{f\left(x_0+h\right)-f\left(x_0\right)}{h} \]

Notes

Page Notes
87 The line passing through two points on a curve is the secant of the curve between the two points. Geometrically, the secant is the average rate of change of the function is a secant slope.
88
136		 tangent to a curve. Defined formally on p. 136
88 instantaneous rate of change. Slope of the tangent to the curve at the point
89 arbitrarily close, sufficiently close, as close as we want. See limit definition for formal notion.
94 finding deltas. See examples in book.
103 one-sided limits, two-sided limits. Left-hand, right-hand, limits
113 degree of a polynomial The largest exponent of the polynomial.
113 limit of a rational function as $x$ goes to infinity. Divide numerator and denominator by the highest power of $x$ in the denominator. What happens then depends on the degree of the polynomials involved.
114 numerator and denominator same degree. See book for examples.
114 degree of numerator less than degree of denominator. See book for examples.
114 degree of numerator greater than degree of denominator. See book for examples.
116 Asymptotes need not be two-sided. The asymptote need not be approached from both sides of the line.
118 Substituting $\frac{1}{x}$ to find limits. See book for examples.
118 Sandwich theorem for $x\rightarrow\pm\infty$. This theorem is the same for this case.
118 finding limits as $x\rightarrow 0$ or $x\rightarrow\pm\infty$. See book for examples.
121 If the degree of the numerator is one more than the degree of the denominator, the graph of the rational function $f(x)$ has an oblique (slanted) asymptote. We find an equation for the asymptote by dividing numerator by denominator to express $f$ as a linear function plus a remainder that goes to zero as $x\rightarrow\pm\infty.$
125 $f$ is discontinuous at a point or has a point of discontinuity $c$ if it is not continuous at $c.$ Point $c$ need not be in the domain of $f.$
126 one-sided continuity test. Same as that for an interior point, but replace the limit with the appropriate one-sided limit. (One-sided continuity can be used both at an interior point and an end point.)
127 removable discontinuity. See book for examples.
127 jump discontinuity. See book for examples.
127 infinite discontinuity. See book for examples.
127 oscillating discontinuity See book for examples.
127 continuous on an interval. A function $f$ is continuous on an interval if it is continuous at every point of the interval.
127 continuous function. A function $f$ is continuous if it is continuous at every point of its domain.
127 examples of continuous functions. polynomials, rationals, roots, trigonometric functions, inverse trigonometric functions, exponentials, logarithms
128 combinations of continuous functions. If functions $f$ and $g$ are continuous at point $c$, then so are their sum, difference, product, quotient, constant multiples, and composite.
130 intermediate value theorem for continuous functions. If a function is continuous then it has the intermediate value property. Note the contrapositive: if a function does not have the intermediate value property on an interval, then it is not continuous. However, a discontinuous function can have the intermediate value property. Hence continuity and the intermediate value property are not equivalent conditions. Cf. definition of continuity.
130 connected. Not given a definition. See book for context.
130 consequence for root finding. Any interval on which a function changes sign contains a zero of the function by the Intermediate Value Theorem.
130 graphical procedure for finding zeros of a continuous function.
  1. Graph the function over a large interval to see roughly where the zeros are.
  2. Zoom in on each zero to estimate is $x$-value.
136 tangent line to a curve at a point. The tangent line to a curve at point $P$ is the line through $P$ whose slope is the slope of the curve at $p.$