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G: Solutions to Exercises
https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.07%3A_G%3A_Solutions_to_Exercises
3.3: Surface Integrals
https://math.libretexts.org/Bookshelves/Calculus/CLP-4_Vector_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Surface_Integrals/3.03%3A_Surface_Integrals
We are now going to define two types of integrals over surfaces.
2.6: Using the Arithmetic of Derivatives – Examples
https://math.libretexts.org/Bookshelves/Calculus/CLP-1_Differential_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Derivatives/3.06%3A_Using_the_Arithmetic_of_Derivatives__Examples
In this section we illustrate the computation of derivatives using the arithmetic of derivatives — Theorems 2.4.2, 2.4.3 and 2.4.5. To make it clear which rules we are using during the examples we wil...In this section we illustrate the computation of derivatives using the arithmetic of derivatives — Theorems 2.4.2, 2.4.3 and 2.4.5. To make it clear which rules we are using during the examples we will note which theorem we are using:
2.2: Averages
https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Applications_of_Integration/2.02%3A_Averages
Another frequent application of integration is computing averages and other statistical quantities. We will not spend too much time on this topic — that is best left to a proper course in statistics —...Another frequent application of integration is computing averages and other statistical quantities. We will not spend too much time on this topic — that is best left to a proper course in statistics — however, we will demonstrate the application of integration to the problem of computing averages.
1.4: Substitution
https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Integration/1.04%3A_Substitution
In the previous section we explored the fundamental theorem of calculus and the link it provides between definite integrals and antiderivatives. Indeed, integrals with simple integrands are usually ev...In the previous section we explored the fundamental theorem of calculus and the link it provides between definite integrals and antiderivatives. Indeed, integrals with simple integrands are usually evaluated via this link.
A.13: Logarithms
https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A%3A_High_School_Material/4.1.13%3A_A.13%3A_Logarithms
In the following, $x$ and $y$ are arbitrary real numbers that are strictly bigger than 0, and $p$ and $q$ are arbitrary constants that are strictly bigger than one. \(q^{\log_q x}=x, \qquad \l...In the following, $x$ and $y$ are arbitrary real numbers that are strictly bigger than 0, and $p$ and $q$ are arbitrary constants that are strictly bigger than one. $q^{\log_q x}=x, \qquad \log_q \big(q^x\big)=x$ $\log_q x=\frac{\log_p x}{\log_p q}$ $\log_q 1=0, \qquad \log_q q=1$ $\log_q(xy)=\log_q x+\log_q y$ $\log_q\big(\frac{x}{y}\big)=\log_q x-\log_q y$ $\lim\limits_{x\rightarrow\infty}\log_q x=\infty, \qquad \lim\limits_{x\rightarrow0}\log_q x=-\infty$
A.3: Trigonometry — Definitions
https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A%3A_High_School_Material/4.1.03%3A_A.3%3A_Trigonometry_%E2%80%94_Definitions
\begin{array}{rlcrl} \sin\theta &= \dfrac{\text{opposite}}{\text{hypotenuse}} & \qquad & \csc \theta &= \dfrac{1}{\sin\theta} \\ \cos\theta &= \dfrac{\text{adjacent}}{\text{hypotenuse}} & \qquad & \se... $\begin{array}{rlcrl} \sin\theta &= \dfrac{\text{opposite}}{\text{hypotenuse}} & \qquad & \csc \theta &= \dfrac{1}{\sin\theta} \\ \cos\theta &= \dfrac{\text{adjacent}}{\text{hypotenuse}} & \qquad & \sec \theta &= \dfrac{1}{\cos\theta} \\ \tan\theta &= \dfrac{\text{opposite}}{\text{adjacent}} & \qquad & \cot \theta &= \dfrac{1}{\tan\theta} \end{array}$ \nonumber \]
A.4: Table of Integrals
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/04%3A_Appendices/4.01%3A_A_Appendices/4.1.04%3A_A.4%3A_Table_of_Integrals
Throughout this table, $a$ and $b$ are given constants, independent of $x$ and $C$ is an arbitrary constant. $a\int f(x)\ \mathrm{d}{x} +b\int g(x)\ \mathrm{d}{x} \ +\ C$ \(\int f(x)\ \mathr...Throughout this table, $a$ and $b$ are given constants, independent of $x$ and $C$ is an arbitrary constant. $a\int f(x)\ \mathrm{d}{x} +b\int g(x)\ \mathrm{d}{x} \ +\ C$ $\int f(x)\ \mathrm{d}{x} +\int g(x)\ \mathrm{d}{x} \ +\ C$ $\int f(x)\ \mathrm{d}{x} -\int g(x)\ \mathrm{d}{x} \ +\ C$ $a\int f(x)\ \mathrm{d}{x} \ +\ C$ $\ln |\csc x-\cot x|+C$ $\ln |\sec x+\tan x|+C$ $\frac{1}{\ln a}\ a^x+C$ $x\ln x -x+C$ $\textrm{arcsec} x+C$ \quad( $x \gt 1$ )
A.6: 3d Coordinate Systems
https://math.libretexts.org/Bookshelves/Calculus/CLP-4_Vector_Calculus_(Feldman_Rechnitzer_and_Yeager)/06%3A_Appendices/6.01%3A_A_Appendices/6.1.06%3A_A.6_3d_Coordinate_Systems
\[\begin{align*} \rho&=\text{ distance from }(0,0,0)\text{ to }(x,y,z)\\ \varphi&=\text{ angle between the $z$ axis and the line joining $(x,y,z)$ to $(0,0,0)$}\\ \theta&=\text{ angle between the $x$ ... $\begin{align*} \rho&=\text{ distance from }(0,0,0)\text{ to }(x,y,z)\\ \varphi&=\text{ angle between the $z$ axis and the line joining $(x,y,z)$ to $(0,0,0)$}\\ \theta&=\text{ angle between the $x$ axis and the line joining $(x,y,0)$ to $(0,0,0)$} \end{align*}$
1.7: Higher Order Derivatives
https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_2160%3A_Applied_Calculus_I/01%3A_The_Derivative/1.07%3A_Higher_Order_Derivatives
The operation of differentiation takes as input one function, $f(x)\text{,}$ and produces as output another function, $f'(x)\text{.}$ Now $f'(x)$ is once again a function. So we can differentiat...The operation of differentiation takes as input one function, $f(x)\text{,}$ and produces as output another function, $f'(x)\text{.}$ Now $f'(x)$ is once again a function. So we can differentiate it again, assuming that it is differentiable, to create a third function, called the second derivative of $f\text{.}$ And we can differentiate the second derivative again to create a fourth function, called the third derivative of $f\text{.}$ And so on.
2.4: The Chain Rule
https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/02%3A_Partial_Derivatives/2.04%3A_The_Chain_Rule
You already routinely use the one dimensional chain rule

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