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- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/12%3A_Introduction_to_Calculus/12.05%3A_DerivativesChange divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we...Change divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we could compare the rates by determining the slopes of the graphs.
- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/12%3A_Introduction_to_CalculusCalculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a li...Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behavior of a function at points ever closer to a particular point, but without ever actually reaching that point. Calculus has two basic applications: differential calculus and integral calculus.
- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/01%3A_Functions/1.03%3A_Domain_and_RangeIn creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this se...In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions.
- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/09%3A_Systems_of_Equations_and_Inequalities/9.05%3A_Partial_FractionsDecompose a ratio of polynomials by writing the partial fractions. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to eac...Decompose a ratio of polynomials by writing the partial fractions. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. The decomposition with repeated linear factors must account for the factors of the denominator in increasing powers. The decomposition with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor.
- https://math.libretexts.org/Courses/Coastline_College/Math_C170%3A_Precalculus_(Tran)/08%3A_Further_Applications_of_Trigonometry/8.E%3A_Further_Applications_of_Trigonometry_(Exercises)If the angle of elevation from the man to the balloon is 27^{\circ}, and the angle of elevation from the woman to the balloon is 41^{\circ}, find the altitude of the balloon to the nearest foo...If the angle of elevation from the man to the balloon is 27^{\circ}, and the angle of elevation from the woman to the balloon is 41^{\circ}, find the altitude of the balloon to the nearest foot. For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin.
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/02%3A_Real_Numbers_and_Fields/2.05%3A_Some_Consequences_of_the_Completeness_AxiomThis proves our last assertion and shows that n o y \in F can be a right bound of N( for y<n \in N), or a left bound of J( for y>-m \in J). \square Now, by Theorem 2 of \(§§5-6, A+...This proves our last assertion and shows that n o y \in F can be a right bound of N( for y<n \in N), or a left bound of J( for y>-m \in J). \square Now, by Theorem 2 of §§5-6, A+m has a minimum; call it p . As p is the least of all sums x+m, p-m is the least of all x \in A ; so p-m=\min A exists, as claimed.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/08%3A_The_Eigenvalue_Problem/8.05%3A_The_Eigenvalue_Problem-_Examples\begin{array}{ccc} {P_{1} = e_{1}(e_{1}^{T}e_{1})^{-1}e_{1}^{T}}&{and}&{P_{2} = e_{2}(e_{2}^{T}e_{2})^{-1}e_{2}^{T}} \end{array} \nonumber It is not the square root of the sum of squares of its co...\begin{array}{ccc} {P_{1} = e_{1}(e_{1}^{T}e_{1})^{-1}e_{1}^{T}}&{and}&{P_{2} = e_{2}(e_{2}^{T}e_{2})^{-1}e_{2}^{T}} \end{array} \nonumber It is not the square root of the sum of squares of its components but rather the square root of the sum of squares of the magnitudes of its components. \begin{array}{ccc} {P_{1} = e_{1}(e_{1}^{H}e_{1})^{-1}e_{1}^{H}}&{and}&{P_{2} = e_{2}(e_{2}^{H}e_{2})^{-1}e_{2}^{H}} \end{array} \nonumber
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/08%3A_The_Eigenvalue_Problem/8.03%3A_The_Partial_Fraction_Expansion_of_the_ResolventR_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z)(z-\lambda_{j})^{k} dz \int R(w)(w-\lambda_{j})^{l} dw \nonumber \[R_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z) (z-\lambda_{j})^{k} \int...R_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z)(z-\lambda_{j})^{k} dz \int R(w)(w-\lambda_{j})^{l} dw \nonumber R_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z) (z-\lambda_{j})^{k} \int \frac{(w-\lambda_{j})^{l}}{w-z} dw dz-\frac{1}{(2\pi i)^2} \int R(w) (w-\lambda_{j})^{k} \int \frac{(z-\lambda_{j})^{k}}{w-z} dz dw \nonumber D_{j}^{m_{j}} = R_{j, m_{j}+1} = \frac{1}{2\pi i} \int R(z)(z-\lambda_{j})^{m_{j}} dz = 0 \nonumber
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/00%3A_Front_Matter/04%3A_PrefaceOur goal in these notes is to demonstrate the role of matrices in the modeling of physical systems and the power of matrix theory in the analysis and synthesis of such systems. In short, the vector of...Our goal in these notes is to demonstrate the role of matrices in the modeling of physical systems and the power of matrix theory in the analysis and synthesis of such systems. In short, the vector of currents is a linear transformation of the vector of voltage drops which is itself a linear transformation of the vector of potentials.
- https://math.libretexts.org/Courses/Coastline_College/Math_C160%3A_Introduction_to_Statistics_(Tran)/06%3A_Continuous_Random_Variables/6.03%3A_The_Uniform_DistributionThe uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful ...The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive.
- https://math.libretexts.org/Courses/Coastline_College/Math_C160%3A_Introduction_to_Statistics_(Tran)/08%3A_The_Central_Limit_Theorem/8.02%3A_The_Central_Limit_Theorem_for_Sample_Means_(Averages)In a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample ...In a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean. The standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (n).
