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2.8: Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/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 $|x|=0$ .
2.9.5: Maxima and Minima
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.09%3A_Derivatives/2.9.05%3A_Maxima_and_Minima
Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of mater...Finding the maximum and minimum values of a function has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.
2.7.24: Physical Applications of Integration
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/02%3A_Vectors_and_Math_Review_Topics/2.07%3A_Math_Review_of_Other_Topics/2.7.24%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.
2.7.23: Anti-derivatives
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/02%3A_Vectors_and_Math_Review_Topics/2.07%3A_Math_Review_of_Other_Topics/2.7.23%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 $e^x$ .
2.9.7: Optimization Problems
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.09%3A_Derivatives/2.9.07%3A_Optimization_Problems
One common application of calculus is calculating the minimum or maximum value of a function. For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it i...One common application of calculus is calculating the minimum or maximum value of a function. For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter.
2.9.6: Derivatives and the Shape of a Graph
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.09%3A_Derivatives/2.9.06%3A_Derivatives_and_the_Shape_of_a_Graph
Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the sec...Using the results from the previous section, we are now able to determine whether a critical point of a function actually corresponds to a local extreme value. In this section, we also see how the second derivative provides information about the shape of a graph by describing whether the graph of a function curves upward or curves downward.
2.8.1: Basic Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.08%3A_Functions/2.8.01%3A_Basic_Functions
We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.
2.10.3: Physical Applications of Integration-
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.10%3A_Anti_derivatives_and_integrals/2.10.03%3A_Physical_Applications_of_Integration-
In this section, we examine some physical applications of integration. Several physical applications of the definite integral are common in engineering and physics. Definite integrals can be used to d...In this section, we examine some physical applications of integration. Several physical applications of the definite integral are common in engineering and physics. Definite integrals can be used to determine the mass of an object if its density function is known. Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.
2.8.3: Exponential_and_Logarithmic_Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.08%3A_Functions/2.8.03%3A_Exponential_and_Logarithmic_Functions
The exponential function $y=b^x$ is increasing if $b>1$ and decreasing if $0<b<1$ . Its domain is $(−∞,∞)$ and its range is $(0,∞)$ . The logarithmic function $y=\log_b(x)$ is the inverse of...The exponential function $y=b^x$ is increasing if $b>1$ and decreasing if $0<b<1$ . Its domain is $(−∞,∞)$ and its range is $(0,∞)$ . The logarithmic function $y=\log_b(x)$ is the inverse of $y=b^x$ . Its domain is $(0,∞)$ and its range is $(−∞,∞)$ . The natural exponential function is $y=e^x$ and the natural logarithmic function is $y=\ln x=\log_e x$ . Given an exponential function or logarithmic function in base $a$ , we can make a change of base to convert this function to a
2.7.15: Exponential and Logarithmic Functions
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/02%3A_Vectors_and_Math_Review_Topics/2.07%3A_Math_Review_of_Other_Topics/2.7.15%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 } \\ ... $\begin{array} {l c} {\text{Suppose we want to evaluate} \log_{a}M} & {\log_{a}M} \\ {\text{Let} \:y =\log_{a}M. }&{y=\log_{a}M} \\ {\text{Rewrite the epression in exponential form. }}&{a^{y}=M } \\ {\text{Take the }\:\log_{b} \text{of each side.}}&{\log_{b}a^{y}=\log_{b}M}\\ {\text{Use the Power Property.}}&{y\log_{b}a=\log_{b}M} \\ {\text{Solve for}\:y. }&{y=\frac{\log_{b}M}{\log_{b}a}} \\ {\text{Substiture}\:y=\log_{a}M.}&{\log_{a}M=\frac{\log_{b}M}{\log_{b}a}} \end{array}$
2.7.26: Table of Integrals
https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/02%3A_Vectors_and_Math_Review_Topics/2.07%3A_Math_Review_of_Other_Topics/2.7.26%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. $\quad \begin{align*} \displaystyle ∫\sin^n u\cos^m u\,du = −\frac{\sin^{n−1}u\cos^{m+1}u}{n+m}+\frac{n−1}{n+m}∫\sin^{n−2}u\cos^m u\,du \\[4pt] =\frac{\sin^{n+1}u\cos^{m−1}u}{n+m}+\frac{m−1}{n+m}∫\sin^n u\cos^{m−2}u \,du \end{align*}$

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