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- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_400%3A_Calculus_I_-_Differential_Calculus/01%3A_Critical_Concepts_for_Calculus/1.08%3A_Chapter_1_Review_ExercisesFor the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain. Find the equation \(C=f(x)\) that describes the total cost as a functi...For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain. Find the equation \(C=f(x)\) that describes the total cost as a function of number of shirts and Carbon dating is implemented to determine how old the skeleton is by using the equation \(y=e^{rt}\), where \(y\) is the percentage of radiocarbon still present in the material, \(t\) is the number of years passed, and \(r=−0.0001210\) is the decay rate of radiocarbon.
- https://math.libretexts.org/Courses/Mission_College/Mission_College_MAT_003B/01%3A_Integration/1.06%3A_Integrals_Involving_Exponential_and_Logarithmic_FunctionsExponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functio...Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functions or logarithms.
- https://math.libretexts.org/Courses/Mission_College/Mission_College_MAT_003B/07%3A_Parametric_Equations_and_Polar_Coordinates/7.04%3A_Polar_Coordinates/7.4E%3A_Exercises_for_Section_11.3In exercises 1 - 7, plot the point whose polar coordinates are given by first constructing the angle \(θ\) and then marking off the distance \(r\) along the ray. In exercises 25 - 29, determine whethe...In exercises 1 - 7, plot the point whose polar coordinates are given by first constructing the angle \(θ\) and then marking off the distance \(r\) along the ray. In exercises 25 - 29, determine whether the graphs of the polar equation are symmetric with respect to the \(x\)-axis, the \(y\) -axis, or the origin. 64) [T] Use the results of the preceding two problems to explore the graphs of \(r=e^{−0.001θ}\) and \(r=e^{−0.0001θ}\) for \(|θ|>100\).
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/06%3A_Applications_of_Integration/6.06%3A_Moments_and_Centers_of_MassIn this section, we consider centers of mass (also called centroids, under certain conditions) and moments. The basic idea of the center of mass is the notion of a balancing point. Many of us have see...In this section, we consider centers of mass (also called centroids, under certain conditions) and moments. The basic idea of the center of mass is the notion of a balancing point. Many of us have seen performers who spin plates on the ends of sticks. The performers try to keep several of them spinning without allowing any of them to drop. Mathematically, that sweet spot is called the center of mass of the plate.
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/03%3A_Differentiation/3.08%3A_Derivatives_of_Inverse_and_Logarithmic_FunctionsFor \(E(x)=e^x, \,E′(0)=1.\) Thus, we have \(E′(x)=e^x\). (The value of \(B′(0)\) for an arbitrary function of the form \(B(x)=b^x, \,b>0,\) will be derived later.) We can use a formula to find the de...For \(E(x)=e^x, \,E′(0)=1.\) Thus, we have \(E′(x)=e^x\). (The value of \(B′(0)\) for an arbitrary function of the form \(B(x)=b^x, \,b>0,\) will be derived later.) We can use a formula to find the derivative of \(y=\ln x\), and the relationship \(\log_b x=\dfrac{\ln x}{\ln b}\) allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/03%3A_Differentiation/3.06%3A_The_Chain_RuleThus, if we think of \(h(x)=\sin(x^3)\) as the composition \((f∘g)(x)=f\big(g(x)\big)\) where \(f(x)= \sin x\) and \(g(x)=x^3\), then the derivative of \(h(x)=\sin(x^3)\) is the product of the derivat...Thus, if we think of \(h(x)=\sin(x^3)\) as the composition \((f∘g)(x)=f\big(g(x)\big)\) where \(f(x)= \sin x\) and \(g(x)=x^3\), then the derivative of \(h(x)=\sin(x^3)\) is the product of the derivative of \(g(x)=x^3\) and the derivative of the function \(f(x)=\sin x\) evaluated at the function \(g(x)=x^3\).
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/03%3A_Differentiation/3.07%3A_Implicit_DifferentiationOn the other hand, if the relationship between the function \(y\) and the variable \(x\) is expressed by an equation where \(y\) is not expressed entirely in terms of \(x\), we say that the equation d...On the other hand, if the relationship between the function \(y\) and the variable \(x\) is expressed by an equation where \(y\) is not expressed entirely in terms of \(x\), we say that the equation defines \(y\) implicitly in terms of \(x\). If we want to find the slope of the line tangent to the graph of \(x^2+y^2=25\) at the point \((3,4)\), we could evaluate the derivative of the function \(y=\sqrt{25−x^2}\) at \(x=3\).
- https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/01%3A_Functions_and_Graphs_(Precalculus_Review)/1.02%3A_Basic_Classes_of_FunctionsFigure \(\PageIndex{9}\): (a) For \(c>0\), the graph of \(y=f(x)+c\) is a vertical shift up \(c\) units of the graph of \(y=f(x)\). (b) For \(c>0\), the graph of \(y=f(x)−c\) is a vertical shift down ...Figure \(\PageIndex{9}\): (a) For \(c>0\), the graph of \(y=f(x)+c\) is a vertical shift up \(c\) units of the graph of \(y=f(x)\). (b) For \(c>0\), the graph of \(y=f(x)−c\) is a vertical shift down c units of the graph of \(y=f(x)\). 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.
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/09%3A_Sequences_and_Series/9.02%3A_Infinite_Series/9.2E%3A_Exercises_for_Section_9.2The part of the first dose after \( n\) hours is \( dr^n\), the part of the second dose is \( dr^{n−N}\), and, in general, the part remaining of the \( m^{\text{th}}\) dose is \( dr^{n−mN}\), so \(\di...The part of the first dose after \( n\) hours is \( dr^n\), the part of the second dose is \( dr^{n−N}\), and, in general, the part remaining of the \( m^{\text{th}}\) dose is \( dr^{n−mN}\), so \(\displaystyle A(n)=\sum_{l=0}^mdr^{n−lN}=\sum_{l=0}^mdr^{k+(m−l)N}=\sum_{q=0}^mdr^{k+qN}=dr^k\sum_{q=0}^mr^{Nq}=dr^k\frac{1−r^{(m+1)N}}{1−r^N},\;\text{where}\,n=k+mN.\)
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/06%3A_Applications_of_Integration/6.01%3A_Areas_between_CurvesJust as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. To find the area between two curves defined by functions, integrate th...Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves. To find the area between two curves defined by functions, integrate the difference of the functions. If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals.
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/07%3A_Techniques_of_Integration/7.05%3A_Other_Strategies_for_IntegrationIn addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are integration tables, which a...In addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are integration tables, which are readily available in many books, including the appendices to this one. Also widely available are computer algebra systems (CAS), which are found on calculators and in many campus computer labs, and are free online.
