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1.6: Integrals Involving Exponential and Logarithmic Functions
https://math.libretexts.org/Courses/Mission_College/Mission_College_MAT_003B/01%3A_Integration/1.06%3A_Integrals_Involving_Exponential_and_Logarithmic_Functions
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 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.
7.4E: Exercises for Section 11.3
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_11.3
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 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 $|θ|>100$ .
6.6: Moments and Centers of Mass
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_Mass
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 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.
3.8: Derivatives of Inverse and Logarithmic Functions
https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/03%3A_Differentiation/3.08%3A_Derivatives_of_Inverse_and_Logarithmic_Functions
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 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.
3.6: The Chain Rule
https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/03%3A_Differentiation/3.06%3A_The_Chain_Rule
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 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 $g(x)=x^3$ .
3.7: Implicit Differentiation
https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/03%3A_Differentiation/3.07%3A_Implicit_Differentiation
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 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 $x=3$ .
1.2: Basic Classes of Functions
https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/01%3A_Functions_and_Graphs_(Precalculus_Review)/1.02%3A_Basic_Classes_of_Functions
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 ...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.
9.2E: Exercises for Section 9.2
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_9.2
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 \(\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 $\displaystyle A(n)=\sum_{l=0}^mdr^{n−lN}=\sum_{l=0}^mdr^{k+(m−l)N}=\sum_{q=0}^mdr^{k+qN}=dr^k\sum_{q=0}^mr^{Nq}=dr^k\frac{1−r^{(m+1)N}}{1−r^N},\;\text{where}\,n=k+mN.$
6.1: Areas between Curves
https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/06%3A_Applications_of_Integration/6.01%3A_Areas_between_Curves
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 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.
7.5: Other Strategies for Integration
https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/07%3A_Techniques_of_Integration/7.05%3A_Other_Strategies_for_Integration
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 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.
8.1: Basics of Differential Equations
https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/08%3A_Introduction_to_Differential_Equations/8.01%3A_Basics_of_Differential_Equations
Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.

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