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- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.08%3A_FunctionsSome 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 \(|x|=0\).
- 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_MinimaFinding 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.
- 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_IntegrationIn 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.
- 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-derivativesTherefore, 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 \(e^x\).
- 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/2.7.19%3A_Differentiation_RulesAs we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives; rather, it is the derivative of the function in the numerator times the function in the denom...As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives; rather, it is the derivative of the function in the numerator times the function in the denominator minus the derivative of the function in the denominator times the function in the numerator, all divided by the square of the function in the denominator.
- 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_ProblemsOne 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.
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.09%3A_Derivatives/2.9.02%3A_Differentiation_RulesThe derivative of a constant function is zero. The derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on x in the derivative decrea...The derivative of a constant function is zero. The derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on x in the derivative decreases by 1. The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative. The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.
- 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/2.7.17%3A_The_Derivative_as_a_FunctionAs we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. The derivative function gives the derivative of a...As we have seen, the derivative of a function at a given point gives us the rate of change or slope of the tangent line to the function at that point. The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line.
- 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_GraphUsing 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.
- 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_FunctionsWe 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.
- 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.
