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- https://math.libretexts.org/Courses/Fullerton_College/Math_100%3A_Liberal_Arts_Math_(Claassen_and_Ikeda)/10%3A_Appendix/10.06%3A_Problem_Solving/10.6.03%3A_Proportions_and_Rates\( \begin{array} {ll} {20\text{ seconds } \cdot \frac{1 \text { minute }}{60 \text { seconds }} \cdot \frac{1 \text { hour }}{60 \text { minutes }}=\frac{1}{180}\text{ hour}} & {\text{Now we can multi...\( \begin{array} {ll} {20\text{ seconds } \cdot \frac{1 \text { minute }}{60 \text { seconds }} \cdot \frac{1 \text { hour }}{60 \text { minutes }}=\frac{1}{180}\text{ hour}} & {\text{Now we can multiply by the }15\text{ miles/hr}} \\ {\frac{1}{180} \text { hour } \cdot \frac{15 \text { miles }}{\text { Ihour }}=\frac{1}{12} \text { mile }} & {\text{Now we can convert to feet}} \\ {\frac{1}{12} \text { mile } \cdot \frac{5280 \text { feet }}{1 \text { mile }}=440 \text { feet}} & { } \end{array…
- https://math.libretexts.org/Courses/Fullerton_College/Math_100%3A_Liberal_Arts_Math_(Claassen_and_Ikeda)/06%3A_ProbabilityThe probability of a specified event is the chance or likelihood that it will occur. Another view would be subjective in nature, in other words an educated guess. In this course we will mostly be conc...The probability of a specified event is the chance or likelihood that it will occur. Another view would be subjective in nature, in other words an educated guess. In this course we will mostly be concerned with theoretical probability, which is defined as follows: Suppose there is a situation with \(n\) equally likely possible outcomes and that \(m\) of those \(n\) outcomes correspond to a particular event; then the probability of that event is defined as \(\dfrac{m}{n}\).
- 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/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.04%3A_Complex_and_Vector-Valued_Functions_on_(E1)Let \(f^{\prime} \geq 0\) on \(I-Q.\) Fix any \(x, y \in I(x<y)\) and define \(g(t)=0\) on \(E^{1}.\) Then \(\left|g^{\prime}\right|=0 \leq f^{\prime}\) on \(I-Q .\) Thus \(g\) and \(f\) satisfy Theor...Let \(f^{\prime} \geq 0\) on \(I-Q.\) Fix any \(x, y \in I(x<y)\) and define \(g(t)=0\) on \(E^{1}.\) Then \(\left|g^{\prime}\right|=0 \leq f^{\prime}\) on \(I-Q .\) Thus \(g\) and \(f\) satisfy Theorem 1 (with their roles reversed on \(I,\) and certainly on the subinterval \([x, y].\) Thus we have
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.05%3A_Antiderivatives_(Primitives_Integrals)If \(F^{\prime}=f\) on a set \(B \subseteq I,\) we say that \(\int f\) is exact on \(B\) and call \(F\) an exact primitive on \(B.\) Thus if \(Q=\emptyset, \int f\) is exact on all of \(I.\) Thus, set...If \(F^{\prime}=f\) on a set \(B \subseteq I,\) we say that \(\int f\) is exact on \(B\) and call \(F\) an exact primitive on \(B.\) Thus if \(Q=\emptyset, \int f\) is exact on all of \(I.\) Thus, setting \(H=f g,\) we have \(H=\int\left(f g^{\prime}+f^{\prime} g\right)\) on \(I.\) Hence by Corollary 1 if \(\int f^{\prime} g\) exists on \(I,\) so does \(\int\left(\left(f g^{\prime}+f^{\prime} g\right)-f^{\prime} g\right)=\int f g^{\prime},\) and
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.11%3A_Integral_Definitions_of_Some_Functions\[\begin{aligned} \log x y &=\int_{1}^{x y} \frac{1}{t} d t=\int_{1 / x}^{y} \frac{1}{s} d s \\ &=\int_{1 / x}^{1} \frac{1}{s} d s+\int_{1}^{y} \frac{1}{s} d s \\ &=-\log \frac{1}{x}+\log y \\ &=\log ...\[\begin{aligned} \log x y &=\int_{1}^{x y} \frac{1}{t} d t=\int_{1 / x}^{y} \frac{1}{s} d s \\ &=\int_{1 / x}^{1} \frac{1}{s} d s+\int_{1}^{y} \frac{1}{s} d s \\ &=-\log \frac{1}{x}+\log y \\ &=\log x+\log y. \end{aligned}\] The function \(F\) as redefined in Theorem 2 will be denoted by \(F_{0}.\) It is a primitive of \(f\) on the closed interval \(\overline{I}\) (exact on \(I).\) Thus \(F_{0}(x)=\int_{0}^{x} f,\) \(-1 \leq x \leq 1,\) and we may now write
- https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Real_Analysis_(Trench)/07%3A_Integrals_of_Functions_of_Several_Variables/7.03%3A_Change_of_Variables_in_Multiple_Integrals
- https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Real_Analysis_(Trench)This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an und...This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.
- https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Real_Analysis_(Trench)/05%3A_Real-Valued_Functions_of_Several_VariablesSECTION 5.1 deals with the structure of \(\R^ n\), the space of ordered \(n\)-tuples of real numbers, which we call {}. We define the sum of two vectors, the product of a vector and a real number, the...SECTION 5.1 deals with the structure of \(\R^ n\), the space of ordered \(n\)-tuples of real numbers, which we call {}. We define the sum of two vectors, the product of a vector and a real number, the length of a vector, and the inner product of two vectors. SECTION 5.2 deals with boundedness, limits, continuity, and uniform continuity of a function of \(n\) variables; that is, a function defined on a subset of \(\R^n\).
- https://math.libretexts.org/Bookshelves/Algebra/Advanced_AlgebraIntermediate Algebra is the second part of a two-part course in Algebra that builds on the basics learned in Elementary Algebra and introduces the more advanced topics required for further study of ap...Intermediate Algebra is the second part of a two-part course in Algebra that builds on the basics learned in Elementary Algebra and introduces the more advanced topics required for further study of applications found in most disciplines. Used as a standalone textbook, it offers plenty of review as well as something new to engage the student in each chapter. This textbook introduces functions early and stresses the geometry behind the algebra.
- https://math.libretexts.org/Courses/Penn_State_University_Greater_Allegheny/MATH_110%3A_Techniques_of_Calculus_I_(Gaydos)/01%3A_Review/1.03%3A_Linear_FunctionsLinear functions can always be written in the form \[f(x)=b+mx\nonumber \] or \[f(x)=mx+b\nonumber \] where \(b\) is the initial or starting value of the function (with input \(x = 0\)), and \(m\) is ...Linear functions can always be written in the form \[f(x)=b+mx\nonumber \] or \[f(x)=mx+b\nonumber \] where \(b\) is the initial or starting value of the function (with input \(x = 0\)), and \(m\) is the constant rate of change of the function.
