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1.9.17: Derivatives and the Shape of a Graph
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.17%3A_Derivatives_and_the_Shape_of_a_Graph
let $f$ be a continuous function over an interval $I$ containing a critical point $c$ such that $f$ is differentiable over $I$ except possibly at $c$ ; if $f'$ changes sign from positive ...let $f$ be a continuous function over an interval $I$ containing a critical point $c$ such that $f$ is differentiable over $I$ except possibly at $c$ ; if $f'$ changes sign from positive to negative as $x$ increases through $c$ , then $f$ has a local maximum at $c$ ; if $f'$ changes sign from negative to positive as $x$ increases through $c$ , then $f$ has a local minimum at $c$ ; if $f'$ does not change sign as $x$ increases through $c$ , then $f$ does not hav…
1.9.23: Physical Applications of Integration
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.23%3A_Physical_Applications_of_Integration
In addition, instead of being concerned about the work done to move a single mass, we are looking at the work done to move a volume of water, and it takes more work to move the water from the bottom o...In addition, instead of being concerned about the work done to move a single mass, we are looking at the work done to move a volume of water, and it takes more work to move the water from the bottom of the tank than it does to move the water from the top of the tank. In pumping problems, the force required to lift the water to the top of the tank is the force required to overcome gravity, so it is equal to the weight of the water.
1.9.22: Anti-derivatives
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.22%3A_Anti-derivatives
Therefore, every antiderivative of $\cos x$ is of the form $\sin x+C$ for some constant $C$ and every function of the form $\sin x+C$ is an antiderivative of $\cos x$ . Therefore, every antid...Therefore, every antiderivative of $\cos x$ is of the form $\sin x+C$ for some constant $C$ and every function of the form $\sin x+C$ is an antiderivative of $\cos x$ . Therefore, every antiderivative of $e^x$ is of the form $e^x+C$ for some constant $C$ and every function of the form $e^x+C$ is an antiderivative of $e^x$ .
1.9.25: Table of Integrals
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.25%3A_Table_of_Integrals
39. $\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du$ 40. $\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du$ 41. \(\quad \begin{align*} \displaystyle ∫\sin^n u\co...39. $\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du$ 40. $\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du$ 41. $\quad \begin{align*} \displaystyle ∫\sin^n u\cos^m u\,du = −\frac{\sin^{n−1}u\cos^{m+1}u}{n+m}+\frac{n−1}{n+m}∫\sin^{n−2}u\cos^m u\,du \\[4pt] =\frac{\sin^{n+1}u\cos^{m−1}u}{n+m}+\frac{m−1}{n+m}∫\sin^n u\cos^{m−2}u \,du \end{align*}$
1.9.14: Exponential and Logarithmic Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.14%3A_Exponential_and_Logarithmic_Functions
\(\begin{array} {l c} {\text{Suppose we want to evaluate} \log_{a}M} & {\log_{a}M} \\ {\text{Let} \:y =\log_{a}M. }&{y=\log_{a}M} \\ {\text{Rewrite the epression in exponential form. }}&{a^{y}=M } \\ ... $\begin{array} {l c} {\text{Suppose we want to evaluate} \log_{a}M} & {\log_{a}M} \\ {\text{Let} \:y =\log_{a}M. }&{y=\log_{a}M} \\ {\text{Rewrite the epression in exponential form. }}&{a^{y}=M } \\ {\text{Take the }\:\log_{b} \text{of each side.}}&{\log_{b}a^{y}=\log_{b}M}\\ {\text{Use the Power Property.}}&{y\log_{b}a=\log_{b}M} \\ {\text{Solve for}\:y. }&{y=\frac{\log_{b}M}{\log_{b}a}} \\ {\text{Substiture}\:y=\log_{a}M.}&{\log_{a}M=\frac{\log_{b}M}{\log_{b}a}} \end{array}$
1.9.25: Table of Integrals
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.25%3A_Table_of_Integrals
39. $\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du$ 40. $\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du$ 41. \(\quad \begin{align*} \displaystyle ∫\sin^n u\co...39. $\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du$ 40. $\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du$ 41. $\quad \begin{align*} \displaystyle ∫\sin^n u\cos^m u\,du = −\frac{\sin^{n−1}u\cos^{m+1}u}{n+m}+\frac{n−1}{n+m}∫\sin^{n−2}u\cos^m u\,du \\[4pt] =\frac{\sin^{n+1}u\cos^{m−1}u}{n+m}+\frac{m−1}{n+m}∫\sin^n u\cos^{m−2}u \,du \end{align*}$
1.9.14: Exponential and Logarithmic Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.14%3A_Exponential_and_Logarithmic_Functions
\(\begin{array} {l c} {\text{Suppose we want to evaluate} \log_{a}M} & {\log_{a}M} \\ {\text{Let} \:y =\log_{a}M. }&{y=\log_{a}M} \\ {\text{Rewrite the epression in exponential form. }}&{a^{y}=M } \\ ... $\begin{array} {l c} {\text{Suppose we want to evaluate} \log_{a}M} & {\log_{a}M} \\ {\text{Let} \:y =\log_{a}M. }&{y=\log_{a}M} \\ {\text{Rewrite the epression in exponential form. }}&{a^{y}=M } \\ {\text{Take the }\:\log_{b} \text{of each side.}}&{\log_{b}a^{y}=\log_{b}M}\\ {\text{Use the Power Property.}}&{y\log_{b}a=\log_{b}M} \\ {\text{Solve for}\:y. }&{y=\frac{\log_{b}M}{\log_{b}a}} \\ {\text{Substiture}\:y=\log_{a}M.}&{\log_{a}M=\frac{\log_{b}M}{\log_{b}a}} \end{array}$
2.8: Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Introductory_Physics_II_(1112)/02%3A_Math_Review/2.08%3A_Functions
Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if $x<0$ , then $|x|=−x>0,$ and if $x>0$ , then \...Some students describe this function by stating that it “makes everything positive.” By the definition of the absolute value function, we see that if $x<0$ , then $|x|=−x>0,$ and if $x>0$ , then $|x|=x>0.$ However, for $x=0,$ $|x|=0.$ Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if $x=0$ , the output $|x|=0$ .
2.8.2: Trigonometric Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Introductory_Physics_II_(1112)/02%3A_Math_Review/2.08%3A_Functions/2.8.02%3A_Trigonometric_Functions
Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or ...Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions.
1.9.12: Basic Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.12%3A_Basic_Functions
For $c>0$ , the graph of $f(x+c)$ is a shift of the graph of $f(x)$ to the left $c$ units; the graph of $f(x−c)$ is a shift of the graph of $f(x)$ to the right $c$ units. For example, the...For $c>0$ , the graph of $f(x+c)$ is a shift of the graph of $f(x)$ to the left $c$ units; the graph of $f(x−c)$ is a shift of the graph of $f(x)$ to the right $c$ units. For example, the graph of the function $f(x)=3x^2$ is the graph of $y=x^2$ stretched vertically by a factor of 3, whereas the graph of $f(x)=x^2/3$ is the graph of $y=x^2$ compressed vertically by a factor of $3$ (Figure $\PageIndex{11b}$ ).
1.9.22: Anti-derivatives
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/01%3A_Introduction_to_Physics_Measurements_and_Mathematics_Tools/1.09%3A_Math_Review_of_Other_Topics/1.9.22%3A_Anti-derivatives
Therefore, every antiderivative of $\cos x$ is of the form $\sin x+C$ for some constant $C$ and every function of the form $\sin x+C$ is an antiderivative of $\cos x$ . Therefore, every antid...Therefore, every antiderivative of $\cos x$ is of the form $\sin x+C$ for some constant $C$ and every function of the form $\sin x+C$ is an antiderivative of $\cos x$ . Therefore, every antiderivative of $e^x$ is of the form $e^x+C$ for some constant $C$ and every function of the form $e^x+C$ is an antiderivative of $e^x$ .

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