Math 141

Definitions

113 entries

amplitude (sigmoid): The amplitude $A$ is the output the graph approaches as the inputs increase without bound, minus $v$ . The right horizontal asymptote is $y = v + A$ .Lesson 9: Modeling with Sigmoid Functions
amplitude (sine): The amplitude $A$ is half the distance between the maximum and the minimum output values of the sine function. Increasing $A$ raises the peaks and deepens the troughs.Lesson 8: Modeling with Sine Functions
Assignment: Assignment is storing the value of an expression under a name. The name goes on the left of = and the expression on the right:
$\text{\rcmd{name = expression}}.$
R computes the value of the right side, stores it under the name, and prints nothing. The Environment then lists the name with its value.
Lesson 1: Introduction to R
Best-fit line: Among all lines, fitModel() finds the line that makes the sum of the squared vertical distances from the data points to the line as small as possible, and that line is the best-fit line.
Lesson 4: Modeling with Linear Functions
Better fittheorem: For linear models of the same dataset, the model with the shorter residual vector fits better. Equivalently, the model with the smaller root mean squared error (RMSE) fits better, because every model of the dataset has the same number of data points $n$ .
Lesson 23: The Method of Least Squares II
c(...): The command c(...) is the R function that returns a vector whose components are its inputs, in the order listed. The c stands for combine.Lesson 16: Introduction to Vectors II
Closest linear combinationtheorem: For a vector $\overline{\vphantom{b}v}$ and a nonzero vector $\overline{\vphantom{b}u}$ with the same number of components, the projection of $\overline{\vphantom{b}v}$ onto $\overline{\vphantom{b}u}$ is the closest linear combination of $\overline{\vphantom{b}u}$ to $\overline{\vphantom{b}v}$ . Every other linear combination of $\overline{\vphantom{b}u}$ is farther from $\overline{\vphantom{b}v}$ .
Lesson 21: Vector Projection
component: A component is an entry in the column of a vector. The first entry $v_{1}$ is the first component, the second entry $v_{2}$ is the second component, and so on. The vector $\overline{\vphantom{b}v} = \begin{pmatrix*}[r] 1 \\ 3 \end{pmatrix*}$ has components $v_{1} = 1$ and $v_{2} = 3$ .Lesson 15: Introduction to Vectors I
component functions: The two output components of a 2-D vector field $\overline{\vphantom{b}F}$ are functions of $x$ and $y$ , the component functions $g$ and $h$ of $\overline{\vphantom{b}F}$ :Lesson 16: Introduction to Vectors II
concave down: A function is concave down on an interval if, for every two points on its graph over that interval, the straight segment joining them lies below the graph between those points.Lesson 5: Modeling with Exponential Functions I
concave up: A function is concave up on an interval if, for every two points on its graph over that interval, the straight segment joining them lies above the graph between those points.Lesson 5: Modeling with Exponential Functions I
Console: The Console is the RStudio pane where R reads a command typed at the > prompt, runs it, and prints its result beneath it when there is one.
Lesson 1: Introduction to R
Derived dimension: A derived dimension is a dimension other than a fundamental dimension that can be expressed as a product of integer powers of fundamental dimensions.
Lesson 12: Dimension Analysis
dimension (of a quantity): A dimension is an inherent way of measuring a quantity.
Lesson 12: Dimension Analysis
dimension (of a vector): The dimension of a vector is the number of components it has. The vector $\overline{\vphantom{b}v} = \begin{pmatrix*}[r] 1 \\ 3 \end{pmatrix*}$ has dimension 2. The dimension counts components, and the word length is kept for the magnitude of a vector, defined in section 3.Lesson 15: Introduction to Vectors I
Dimension arithmetictheorem: Adding, subtracting, and comparing. When adding, subtracting, or comparing (for example, with $=$ , $\geq$ , or $<$ ) two quantities, they must have the same dimension, although their units may be different. The resulting dimension is equal to that of the quantities being added, subtracted, or compared.
Multiplying and dividing. When multiplying and dividing quantities, they can have any dimensions and any units. The resulting dimension is equal to the product or quotient of the dimensions of the quantities being multiplied or divided.
Exponents. In an exponential expression $a^{r}$ , the exponent $r$ must be dimensionless.
Lesson 12: Dimension Analysis
dimensionless quantity: A dimensionless quantity is a pure number that does not have a dimension. The symbol “1” denotes a dimensionless quantity and is written as $[\text{number}] = 1$ . For example, $[6] = 1$ .Lesson 12: Dimension Analysis
Domain: The domain is the set of input values of a function $y = f(x)$ .
Lesson 3: Introduction to Functions
Dot product: For two vectors $\overline{\vphantom{b}u}$ and $\overline{\vphantom{b}v}$ with the same number of components, the dot product $\overline{\vphantom{b}u}\cdot\overline{\vphantom{b}v}$ is the scalar obtained by multiplying each component of $\overline{\vphantom{b}u}$ by the corresponding component of $\overline{\vphantom{b}v}$ and summing the products:
$\overline{\vphantom{b}u}\cdot\overline{\vphantom{b}v} = \begin{pmatrix*}[r] u_1 \\ u_2 \\ \vdots \\ u_n \end{pmatrix*}\cdot \begin{pmatrix*}[r] v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix*} = u_1v_1 + u_2v_2 + \dots + u_nv_n.$ Lesson 20: Dot Product
Doubling time and halving time: For $y = Ce^{kx}$ with $k > 0$ , the doubling time is the width $\Delta x = x_2 - x_1$ of an input interval with $x_2 > x_1$ and $y(x_2) = 2y(x_1)$ ; with $k < 0$ and $y(x_2) = \frac{1}{2}y(x_1)$ , the width is the halving time. In science contexts the halving time is called the half-life.
Lesson 5: Modeling with Exponential Functions I
Environment: The Environment is the RStudio pane that lists every name stored in the current session, with its value. A value is stored under a name by assignment (section 3).
Lesson 1: Introduction to R
error (of a model): The error is the observed output at a data point minus the model’s output at that point’s input.Lesson 7: Modeling with Power Functions
Exponent rulestheorem: For every base $a$ with $a > 0$ and exponents $r$ and $s$ ,
$a^{r}a^{s} = a^{r+s}, \qquad \frac{a^{r}}{a^{s}} = a^{r-s}, \qquad (a^{r})^{s} = a^{rs}, \qquad a^{-r} = \frac{1}{a^{r}}, \qquad a^{0} = 1.$ Lesson 12: Dimension Analysis
extrapolation: An extrapolation is a prediction at an input that lies outside the input interval used to create the model. The prediction for 2030 is an extrapolation, because the data end in 2015. An extrapolation can be far off, because the data give no evidence about outputs at inputs outside its input interval.Lesson 4: Modeling with Linear Functions
Files: Files is the RStudio pane that shows the folders and files of the Posit Cloud project.Lesson 1: Introduction to R
fitted model function: The fitted model function is the R function that fitModel() returns. Given an input $x$ , it returns the fitted model’s output $y$ , including when the formula is fitted to log(y). In the command above, the fitted model function is bfit.Lesson 6: Modeling with Exponential Functions II
five-sixths level: The five-sixths level is the output five-sixths of the way from $v$ to $v + A$ , namely $v + \frac{5A}{6}$ .Lesson 9: Modeling with Sigmoid Functions
Function: A function is a rule assigning every input to exactly one output.

In symbols, $y = f(x)$ means the input $x$ is assigned to the output $y$ .
Lesson 3: Introduction to Functions
Fundamental dimension: A fundamental dimension is a dimension that cannot be expressed in terms of other dimensions.
Lesson 12: Dimension Analysis
General exponential function: An exponential function is a function of the form
$y = Ce^{k(x-h)},$
where $C$ , $k$ , and $h$ are constants with $C \neq 0$ and $k \neq 0$ , and $e = 2.71828\ldots$ . Its parameters are $C$ , the output at $x = h$ ; the steepness $k$ ; and the horizontal shift $h$ . With $h = 0$ the form is $y = Ce^{kx}$ .
Lesson 5: Modeling with Exponential Functions I
General logarithmic function: A general logarithmic function is a function of the form $y = C\ln(x - h) + v$ , where $C$ , $h$ , and $v$ are constants with $C \neq 0$ , and its domain is $(h, \infty)$ .
Lesson 5: Modeling with Exponential Functions I
Graphically identifying reasonable modelstheorem: If the standard plot $(x, y)$ of a given data set is linear, then a linear model $y = mx + b$ is reasonable.
If the semi-log plot $(x, \ln(y))$ of a given data set is nonconstant linear, then an exponential model $y = Ce^{k(x-h)}$ is reasonable.
If the log-log plot $(\ln(x), \ln(y))$ of a given data set is nonconstant linear, then a power model $y = Cx^{k}$ is reasonable.
Lesson 7: Modeling with Power Functions
Help: Help is the RStudio pane that displays the documentation page for an R function.
Lesson 1: Introduction to R
History: The History is the RStudio pane that lists the commands run in the Console, in the order run.
Lesson 1: Introduction to R
horizontal asymptote: A horizontal asymptote is a horizontal line $y = L$ that the outputs of a function approach as $x$ increases without bound or as $x$ decreases without bound.Lesson 5: Modeling with Exponential Functions I
Horizontal line testtheorem: A function $y = f(x)$ has an inverse function if and only if every horizontal line intersects its graph at most once.
Lesson 3: Introduction to Functions
horizontal shift: The horizontal shift $h$ is the first input $x = h$ at or to the right of $x = 0$ at which the sine function is increasing and its output equals the vertical shift $v$ . Increasing $h$ slides the whole curve to the right.Lesson 8: Modeling with Sine Functions
Independence from the starting outputtheorem: For $k > 0$ , the doubling time of $y = Ce^{kx}$ is the same from every starting output. For $k < 0$ , the halving time is the same from every starting output.
Lesson 5: Modeling with Exponential Functions I
Inf: R uses the numeric values Inf and -Inf for positive and negative infinity.Lesson 1: Introduction to R
Inference: Inference is using the values of a fitted model’s parameters to describe the relationship between the input and the output. Reading the slope in Example 6 is an inference: according to the best-fit model, the world population grew by about 0.0749 billion people, or 74.9 million, per year from 1950 to 2015.Lesson 4: Modeling with Linear Functions
Input variable: The input variable is the variable whose values are the inputs of a function.
Lesson 3: Introduction to Functions
intercept vector: The intercept vector is the vector $\overline{\vphantom{b}1}_n$ whose $n$ components are all 1.Lesson 19: Existence of Linear Combinations
interpolation: An interpolation is a prediction at an input that lies inside the input interval used to create the model, from the smallest data input to the largest. The prediction for 1984 is an interpolation, because the data run from 1950 to 2015.Lesson 4: Modeling with Linear Functions
Inverse function: The inverse function of $f(x)$ maps each output from $f(x)$ back to its corresponding input.
Lesson 3: Introduction to Functions
Least squares RMSEtheorem: When the entries of $\overline{\vphantom{b}x}$ are not all equal, the best-fit linear model has the smallest RMSE of all linear models $y = mx + b$ for the same data.
Lesson 22: The Method of Least Squares I
Length (magnitude): The length (or magnitude) of a vector $\overline{\vphantom{b}v}$ with $n$ components is the number written $\left\|\,\overline{\vphantom{b}v}\,\right\|$ and defined by
$\left\|\,\overline{\vphantom{b}v}\,\right\| = \sqrt{v_{1}^{2} + v_{2}^{2} + \cdots + v_{n}^{2}}.$ Lesson 15: Introduction to Vectors I
Length as a dot producttheorem: The length of a vector $\overline{\vphantom{b}u}$ is
$\|\overline{\vphantom{b}u}\| = \sqrt{\overline{\vphantom{b}u}\cdot\overline{\vphantom{b}u}}.$ Lesson 20: Dot Product
linear combination (of vectors): If $\overline{\vphantom{b}u}_1, \overline{\vphantom{b}u}_2, \dots, \overline{\vphantom{b}u}_n$ are vectors and $x_1, x_2, \dots, x_n$ are scalars, then $x_1\overline{\vphantom{b}u}_1 + x_2\overline{\vphantom{b}u}_2 + \dots + x_n\overline{\vphantom{b}u}_n$ is a linear combination of $\overline{\vphantom{b}u}_1, \overline{\vphantom{b}u}_2, \dots, \overline{\vphantom{b}u}_n$ .
Lesson 17: Linear Combinations
Linear function: For constants $m$ and $b$ , a linear function is a function of the form
$y = mx + b.$
The slope $m$ of the line through the points $(x_1, y_1)$ and $(x_2, y_2)$ , with $x_1 \neq x_2$ , is
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x} = \frac{\text{rise}}{\text{run}}.$
The vertical intercept $b$ is the $y$ -coordinate of the point $(0, b)$ where the line intersects the $y$ -axis.
Lesson 4: Modeling with Linear Functions
Log and exponent rulestheorem: For $u > 0$ , $w > 0$ , and real numbers $r$ and $s$ , with $s$ a positive integer in $\sqrt[s]{u^{r}} = u^{r/s}$ , the rules below hold.

Inverse pairs Log algebra Exponents
$e^{\ln(u)} = u$ $\ln(uw) = \ln(u) + \ln(w)$ $u^{r}u^{s} = u^{r+s}$
$\ln(e^{r}) = r$ $\ln\!\left(\tfrac{u}{w}\right) = \ln(u) - \ln(w)$ $\sqrt[s]{u^{r}} = u^{r/s}$
$\ln(1) = 0$ $\ln(u^{r}) = r\ln(u)$ $\tfrac{1}{u^{s}} = u^{-s}$
$\ln(e) = 1$ $\ln\!\left(\tfrac{1}{w}\right) = -\ln(w)$
Lesson 5: Modeling with Exponential Functions I
Log-log form: For $x > 0$ , the log-log form of a power function $y = Cx^{k}$ with $C > 0$ is the equation $\ln(y) = m\ln(x) + b$ with $m = k$ and $b = \ln(C)$ . It gives $\ln(y)$ as a linear function of $\ln(x)$ .
Lesson 7: Modeling with Power Functions
Log-log form and conversiontheorem: For $x > 0$ , the power function $y = Cx^{k}$ with $C > 0$ has the log-log form
$\ln(y) = m\ln(x) + b,$
where $m = k$ is the slope and $b = \ln(C)$ is the vertical intercept. A log-log line $\ln(y) = m\ln(x) + b$ with $m \neq 0$ converts back to the power function with
$k = m, \qquad C = e^{b}.$ Lesson 7: Modeling with Power Functions
Log-log plot: If $(x_0, y_0)$ is a point from a given data set or on a curve, with $x_0 > 0$ and $y_0 > 0$ , then its log-log plot includes the point $(\ln(x_0), \ln(y_0))$ .
Lesson 7: Modeling with Power Functions
Matrix: An $m \times n$ matrix is a rectangular array of real numbers arranged in $m$ rows and $n$ columns.
Lesson 17: Linear Combinations
matrix equation: A matrix equation $A\overline{\vphantom{b}x} = \overline{\vphantom{b}w}$ is a vector equation with its vectors as the columns of $A$ and its scalars as the components of $\overline{\vphantom{b}x}$ .Lesson 17: Linear Combinations
Matrix-vector product: For a matrix $A$ with columns $\overline{\vphantom{b}u}_1, \overline{\vphantom{b}u}_2, \dots, \overline{\vphantom{b}u}_n$ and a vector $\overline{\vphantom{b}x}$ with components $x_1, x_2, \dots, x_n$ , the matrix-vector product is
$A\overline{\vphantom{b}x} = x_1\overline{\vphantom{b}u}_1 + x_2\overline{\vphantom{b}u}_2 + \dots + x_n\overline{\vphantom{b}u}_n.$ Lesson 17: Linear Combinations
matrix(c(...), nrow = m, ncol = n): The command matrix(c(...), nrow = m, ncol = n) defines a matrix with $m$ rows and $n$ columns from a list of exactly $m \cdot n$ entries. R fills the columns in order and each column from top to bottom.Lesson 18: Matrix Equations
mean: The mean $m$ is the input at which the output is $v + \frac{A}{2}$ , halfway between the two horizontal asymptotes. The curve is steepest at $m$ .Lesson 9: Modeling with Sigmoid Functions
Method of least squares: The method of least squares is the method that selects the linear combination $m\overline{\vphantom{b}x} + b\overline{\vphantom{b}1}_n$ closest to the target vector $\overline{\vphantom{b}y}$ , in the sense that the corresponding residual vector $\overline{\vphantom{b}r} = \overline{\vphantom{b}y} - (m\overline{\vphantom{b}x} + b\overline{\vphantom{b}1}_n)$ has the shortest length possible. When the entries of $\overline{\vphantom{b}x}$ are not all equal, it selects one pair of scalars $m$ and $b$ , the slope and the vertical intercept of the best-fit linear model $y = mx + b$ .
Lesson 22: The Method of Least Squares I
Model: A model is a mathematical function whose input and output correspond to observations of some phenomenon of interest.
Lesson 4: Modeling with Linear Functions
model vector: The linear combination $m\overline{\vphantom{b}x} + b\overline{\vphantom{b}1}_n$ that the method of least squares selects is the model vector, the vector of the model’s predictions.Lesson 22: The Method of Least Squares I
monotonic: A function $f(x)$ is monotonic if $f(c) < f(d)$ for every pair of inputs $c$ and $d$ with $c < d$ , or if $f(c) > f(d)$ for every such pair.Lesson 3: Introduction to Functions
NaN: R returns NaN, “not a number”, for an undefined calculation whose result is neither Inf nor -Inf.Lesson 1: Introduction to R
natural logarithm: The natural logarithm is $\ln(x)$ , the inverse of the natural exponential function $e^{x}$ , with domain $(0, \infty)$ .Lesson 5: Modeling with Exponential Functions I
Number of solutionstheorem: A system of linear equations has exactly one solution, no solution, or infinitely many solutions.
Lesson 19: Existence of Linear Combinations
Order and scalarstheorem: Let $\overline{\vphantom{b}u}$ and $\overline{\vphantom{b}v}$ be vectors with the same number of components, and let $m$ be a scalar. The dot product gives the same scalar in either order of the two vectors, and a scalar multiple of either vector can be moved outside the product:
$\overline{\vphantom{b}u}\cdot\overline{\vphantom{b}v} = \overline{\vphantom{b}v}\cdot\overline{\vphantom{b}u}, \qquad\qquad (m\overline{\vphantom{b}u})\cdot\overline{\vphantom{b}v} = \overline{\vphantom{b}u}\cdot(m\overline{\vphantom{b}v}) = m(\overline{\vphantom{b}u}\cdot\overline{\vphantom{b}v}).$ Lesson 20: Dot Product
Orthogonal: Two vectors $\overline{\vphantom{b}u}$ and $\overline{\vphantom{b}v}$ are orthogonal when $\overline{\vphantom{b}u}\cdot\overline{\vphantom{b}v} = 0$ , where “orthogonal” is an extension of the two-dimensional idea of “perpendicular” to any number of dimensions.
Lesson 20: Dot Product
Output variable: The output variable is the variable whose values are the outputs of a function.
Lesson 3: Introduction to Functions
parameter: A parameter is a constant in a model’s formula that can take on various values. Each choice of values for the parameters gives one function.Lesson 4: Modeling with Linear Functions
Perfect fit: A linear model $y = mx + b$ fits a dataset perfectly when every data point satisfies it.
Lesson 19: Existence of Linear Combinations
period: The period $P$ is the length of the smallest input interval needed for the sine function to complete an entire oscillation. Increasing $P$ stretches the curve horizontally.Lesson 8: Modeling with Sine Functions
Plots: Plots is the RStudio pane that displays the graphs R draws.
Lesson 1: Introduction to R
Power function: A power function is of the form $y = Cx^{k}$ , where $C$ and $k$ are nonzero constants.
Lesson 7: Modeling with Power Functions
Power model: A power function used to describe a dataset is a power model.
Lesson 7: Modeling with Power Functions
Prediction: Prediction is using a model’s output at an input to estimate an output that was not observed. The value best(1984) in Example 6 is a prediction.Lesson 4: Modeling with Linear Functions
Projection: For a vector $\overline{\vphantom{b}v}$ and a nonzero vector $\overline{\vphantom{b}u}$ with the same number of components, the projection of $\overline{\vphantom{b}v}$ onto $\overline{\vphantom{b}u}$ is the vector $x\overline{\vphantom{b}u}$ , where
$x = \frac{\overline{\vphantom{b}v}\cdot\overline{\vphantom{b}u}}{\overline{\vphantom{b}u}\cdot\overline{\vphantom{b}u}}.$
The vector $x\overline{\vphantom{b}u}$ is the projection vector, and the number $x$ is the projection scalar.
Lesson 21: Vector Projection
Projection and residualtheorem: For the projection $x\overline{\vphantom{b}u}$ of $\overline{\vphantom{b}v}$ onto a nonzero vector $\overline{\vphantom{b}u}$ , the residual vector $\overline{\vphantom{b}r} = \overline{\vphantom{b}v} - x\overline{\vphantom{b}u}$ is orthogonal to $\overline{\vphantom{b}u}$ and to the projection $x\overline{\vphantom{b}u}$ , and the projection and the residual vector add to the target vector:
$x\overline{\vphantom{b}u} + \overline{\vphantom{b}r} = \overline{\vphantom{b}v}.$ Lesson 21: Vector Projection
Quantity: A quantity is the numerical value of a measurement together with its units, if it has any.
Lesson 12: Dimension Analysis
range: The range is the set of output values of a function $y = f(x)$ .Lesson 3: Introduction to Functions
reasonable: A linear model is reasonable for a data set when the data points lie on approximately the same line. In a plot, the criterion is that one straight line passes close to every point, with no bend in the pattern. Points that bend along a curve call for another type of function.Lesson 4: Modeling with Linear Functions
Residual vector: For vectors $\overline{\vphantom{b}v}$ and $\overline{\vphantom{b}u}$ and a scalar $t$ , the residual vector for the linear combination $t\overline{\vphantom{b}u}$ and the target vector $\overline{\vphantom{b}v}$ is $\overline{\vphantom{b}r} = \overline{\vphantom{b}v} - t\overline{\vphantom{b}u}$ .
Lesson 21: Vector Projection
resultant: The resultant is the single vector equal to the sum of two or more vectors. With the vectors placed tip to tail, the resultant is the arrow from the tail of the first vector to the tip of the last.Lesson 15: Introduction to Vectors I
Root mean squared error (RMSE): For a model fit to $n$ data points, the root mean squared error (RMSE) is the square root of the mean of the squared components of its residual vector $\overline{\vphantom{b}r}$ :
$\text{RMSE} = \frac{\|\overline{\vphantom{b}r}\|}{\sqrt{n}} = \sqrt{\frac{\|\overline{\vphantom{b}r}\|^2}{n}}.$ Lesson 22: The Method of Least Squares I
RScript: An RScript is a plain-text file of R commands, saved in the project, whose lines R runs in the Console, one line or the whole file at a time.
Lesson 1: Introduction to R
Scalar: A scalar is a real number.
Lesson 15: Introduction to Vectors I
Scalar multiplication: The result of scalar multiplication, for a scalar $c$ and a vector $\overline{\vphantom{b}u}$ , is the vector whose components are the components of $\overline{\vphantom{b}u}$ , each multiplied by $c$ :
$c\,\overline{\vphantom{b}u} = \begin{pmatrix*}[r] c\,u_{1} \\ \vdots \\ c\,u_{n} \end{pmatrix*}.$ Lesson 15: Introduction to Vectors I
Semi-log form: For an exponential function $y = Ce^{kx}$ with $C > 0$ , the semi-log form is the equation $\ln(y) = mx + b$ with $m = k$ and $b = \ln(C)$ . It gives $\ln(y)$ as a linear function of $x$ .
Lesson 5: Modeling with Exponential Functions I
Semi-log plot: If $(x_0, y_0)$ is a point from a given data set or curve with $y_0 > 0$ , then its semi-log plot includes the point $(x_0, \ln(y_0))$ .
Lesson 5: Modeling with Exponential Functions I
Shifted exponential model and its semi-log linetheorem: The exponential model $y = Ce^{k(x-h)}$ , with $C > 0$ and horizontal shift $h$ , has the semi-log line
$\ln(y) = m(x-h) + b, \qquad m = k, \quad b = \ln(C).$
To convert a semi-log line $\ln(y) = m(x-h) + b$ with $m \neq 0$ back to the exponential model, set
$k = m, \qquad C = e^{b}.$ Lesson 6: Modeling with Exponential Functions II
Sigmoid function: A sigmoid function is a function whose graph has an elongated S shape: it approaches one horizontal asymptote on the left and a different one on the right, and changes steeply between them.
Lesson 9: Modeling with Sigmoid Functions
Sigmoid model: A sigmoid model gives the output $y$ as a stretched and shifted pnorm:
$y = A\cdot\text{\rcmd{pnorm}}(x, m, s) + v.$
The four parameters are the amplitude $A$ , the mean $m$ , the spread $s$ , and the vertical shift $v$ . The amplitude is nonzero, $A \ne 0$ , and the spread is positive, $s > 0$ .
Lesson 9: Modeling with Sigmoid Functions
Sigmoidal data: Sigmoidal data are data whose outputs stay near one level for small inputs, rise or fall steeply through a middle stretch, and level off at a different value for large inputs.
Lesson 9: Modeling with Sigmoid Functions
Sine model: A sine model has the general form
$y = A\sin\!\left(\frac{2\pi}{P}(x - h)\right) + v,$
with four parameters: the amplitude $A$ , the period $P$ , the horizontal shift $h$ , and the vertical shift $v$ . The amplitude and the period are positive: $A > 0$ and $P > 0$ .
Lesson 8: Modeling with Sine Functions
Sinusoidal data: Sinusoidal data are data whose outputs rise and fall in the shape of a sine curve, between two roughly constant extremes, repeating at a roughly constant input spacing.
Lesson 8: Modeling with Sine Functions
Size of a matrix: The size of a matrix is written rows first, then columns, as in $m \times n$ .
Lesson 17: Linear Combinations
solution of the vector equation: The pair $x = a$ and $y = b$ is a solution of the vector equation $\overline{\vphantom{b}w} = x\overline{\vphantom{b}u} + y\overline{\vphantom{b}v}$ when $\overline{\vphantom{b}w}$ is equal to $a\overline{\vphantom{b}u} + b\overline{\vphantom{b}v}$ .Lesson 17: Linear Combinations
solve(U, v): The command solve(U, v) takes the matrix first and the target vector second. When the matrix equation $\overline{\vphantom{b}v} = U\overline{\vphantom{b}x}$ has exactly one solution and the matrix is not close to singular (Lesson 19 sheet), it returns that solution as the vector of unknowns $\overline{\vphantom{b}x}$ .Lesson 18: Matrix Equations
spread: The spread $s$ is the positive input distance from $m$ to the input where the output has covered pnorm(1) $\approx 0.8413$ of its change from $v$ to $v + A$ .Lesson 9: Modeling with Sigmoid Functions
start = list(): The start = list() argument of fitModel() gives a starting value for each parameter, such as start = list(A = 28, P = 12, h = 2.5, v = 60). The fit searches for the best-fit values beginning from these starting values.Lesson 8: Modeling with Sine Functions
Target problem: For $n$ data points $(x_1, y_1), \dots, (x_n, y_n)$ , the components of the input vector $\overline{\vphantom{b}x}$ are the inputs, and the components of the target vector $\overline{\vphantom{b}y}$ are the outputs. The intercept vector $\overline{\vphantom{b}1}_{n}$ has $n$ components, each equal to 1 (Lesson 19 Notes, section 3). The target problem is the problem of finding the scalars $m$ and $b$ that make the linear combination $m\overline{\vphantom{b}x} + b\overline{\vphantom{b}1}_{n}$ equal to the target vector $\overline{\vphantom{b}y}$ . As a vector equation and as a matrix equation, the target problem is
$m\overline{\vphantom{b}x} + b\overline{\vphantom{b}1}_{n} = \overline{\vphantom{b}y}, \qquad\qquad \begin{pmatrix*}[r] \overline{\vphantom{b}x} & \overline{\vphantom{b}1}_{n} \end{pmatrix*}\begin{pmatrix*}[r] m \\ b \end{pmatrix*} = \overline{\vphantom{b}y}.$ Lesson 23: The Method of Least Squares II
Unique solution: A system of linear equations has a unique solution when it has exactly one solution.
Lesson 19: Existence of Linear Combinations
Unit: A unit is a way to assign a numerical value to the measurement of a quantity.
Lesson 12: Dimension Analysis
Vector: A vector is an ordered list of real numbers written as a column.
Lesson 15: Introduction to Vectors I
Vector addition: The result of vector addition, for two vectors with the same number of components, is the vector whose components are the sums of the matching components:
$\overline{\vphantom{b}u} + \overline{\vphantom{b}w} = \begin{pmatrix*}[r] u_{1} + w_{1} \\ \vdots \\ u_{n} + w_{n} \end{pmatrix*}.$ Lesson 15: Introduction to Vectors I
Vector equation: A vector equation is an equality between linear combinations of vectors.
Lesson 17: Linear Combinations
Vector field: A vector field is a function whose input is a point and whose output is a vector of the same dimension as the input.
Lesson 16: Introduction to Vectors II
Vector subtraction: The result of vector subtraction, for two vectors with the same number of components, is the vector whose components are the differences of the matching components:
$\overline{\vphantom{b}u} - \overline{\vphantom{b}w} = \begin{pmatrix*}[r] u_{1} - w_{1} \\ \vdots \\ u_{n} - w_{n} \end{pmatrix*}.$ Lesson 15: Introduction to Vectors I
vertical asymptote: A vertical asymptote is a vertical line $x = a$ such that the outputs of a function grow in size without bound as $x$ approaches $a$ . The graph of $y = C\ln(x - h) + v$ has the vertical asymptote $x = h$ , approached from the right.Lesson 5: Modeling with Exponential Functions I
Vertical line testtheorem: A curve in the $xy$ -plane is a function if and only if every vertical line intersects the curve at most once.
Lesson 3: Introduction to Functions
vertical shift (sigmoid): The vertical shift $v$ is the output the graph approaches as the inputs decrease without bound. The left horizontal asymptote is $y = v$ .Lesson 9: Modeling with Sigmoid Functions
vertical shift (sine): The vertical shift $v$ is the height of the output that is halfway between the maximum and the minimum output values of the sine function. The curve oscillates about the center line $y = v$ . Increasing $v$ slides the whole curve up.Lesson 8: Modeling with Sine Functions
Viewer: The Viewer is the RStudio pane that displays web content R produces.Lesson 1: Introduction to R
window: The window of a plot is the range of $x$ values and $y$ values it shows. A curve added with add = TRUE keeps the window of the first plot and shows only the part that falls inside it. To fit every curve, set a wide enough window with xlim and ylim on the first call.Lesson 2: Computing and Graphing in R

Inverse pairs	Log algebra	Exponents
$e^{\ln(u)} = u$	$\ln(uw) = \ln(u) + \ln(w)$	$u^{r}u^{s} = u^{r+s}$
$\ln(e^{r}) = r$	$\ln\!\left(\tfrac{u}{w}\right) = \ln(u) - \ln(w)$	$\sqrt[s]{u^{r}} = u^{r/s}$
$\ln(1) = 0$	$\ln(u^{r}) = r\ln(u)$	$\tfrac{1}{u^{s}} = u^{-s}$
$\ln(e) = 1$	$\ln\!\left(\tfrac{1}{w}\right) = -\ln(w)$