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5.4: Rational Functions
https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/05%3A_Polynomial_and_Rational_Functions/5.04%3A_Rational_Functions
Structurally, we observe that $AV_{[2,2+h]}$ is a ratio of the two functions $-64h - 16h^2$ and $h\text{.}$ Moreover, both the numerator and the denominator of the expression are themselves poly...Structurally, we observe that $AV_{[2,2+h]}$ is a ratio of the two functions $-64h - 16h^2$ and $h\text{.}$ Moreover, both the numerator and the denominator of the expression are themselves polynomial functions of the variable $h\text{.}$ Note that we may be especially interested in what occurs as $h \to 0\text{,}$ as these values will tell us the average velocity of the moving ball on shorter and shorter time intervals starting at $t = 2\text{.}$ At the same time, $AV_{[2,2+h]}$ …
3.6: Modeling temperature and population
https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/03%3A_Exponential_and_Logarithmic_Functions/3.06%3A_Modeling_temperature_and_population
In Section 3.2, we learned that Newton's Law of Cooling, which states that an object's temperature changes at a rate proportional to the difference between its own temperature and the surrounding temp...In Section 3.2, we learned that Newton's Law of Cooling, which states that an object's temperature changes at a rate proportional to the difference between its own temperature and the surrounding temperature, results in the object's temperature being modeled by functions of the form $F(t) = ab^t + c\text{.}$ In light of our subsequent work in Section 3.3 with the natural base $e\text{,}$ as well as the fact that $0 \lt b \lt 1$ in this model, we know that Newton's Law of Cooling implies t…
1: Understanding the Derivative
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative
2.7: Derivatives of Functions Given Implicitly
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/02%3A_Computing_Derivatives/2.07%3A_Derivatives_of_Functions_Given_Implicitely
Implicit Differentiation is used to identfy the derivative of a y(x) function from an equation where y cannot be solved for explicitly in terms of x, but where portions of the curve can be thought of ...Implicit Differentiation is used to identfy the derivative of a y(x) function from an equation where y cannot be solved for explicitly in terms of x, but where portions of the curve can be thought of as being generated by explicit functions of x. In this case, we say that y is an implicit function of x. The process of implicit differentiation, we take the equation that generates an implicitly given curve and differentiate both sides with respect to x while treating y as a function of x.
9.7: Derivatives and Integrals of Vector-Valued Functions
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/09%3A_Multivariable_and_Vector_Functions/9.07%3A_Derivatives_and_Integrals_of_Vector-Valued_Functions
A vector-valued function determines a curve in space as the collection of terminal points of the vectors r(t). If the curve is smooth, it is natural to ask whether r(t) has a derivative. In the sa...A vector-valued function determines a curve in space as the collection of terminal points of the vectors r(t). If the curve is smooth, it is natural to ask whether r(t) has a derivative. In the same way, our experiences with integrals in single-variable calculus prompt us to wonder what the integral of a vector-valued function might be and what it might tell us. We explore both of these questions in detail in this section.
1.2: The Notion of Limit
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative/1.02%3A_The_Notion_of_Limit
Limits enable us to examine trends in function behavior near a specific point. In particular, taking a limit at a given point asks if the function values nearby tend to approach a particular fixed val...Limits enable us to examine trends in function behavior near a specific point. In particular, taking a limit at a given point asks if the function values nearby tend to approach a particular fixed value.
4.E: The Definite Integral (Exercises)
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/04%3A_The_Definite_Integral/4.E%3A_The_Definite_Integral_(Exercises)
These are homework exercises to accompany Chapter 4 of Boelkins et al. "Active Calculus" Textmap.
1.1: Changing in Tandem
https://math.libretexts.org/Bookshelves/Precalculus/Active_Prelude_to_Calculus_(Boelkins)/01%3A_Relating_Changing_Quantities/1.01%3A_Changing_in_Tandem
Mathematics is the art of making sense of patterns. One way that patterns arise is when two quantities are changing in tandem. In this setting, we may make sense of the situation by expressing the rel...Mathematics is the art of making sense of patterns. One way that patterns arise is when two quantities are changing in tandem. In this setting, we may make sense of the situation by expressing the relationship between the changing quantities through words, through images, through data, or through a formula.
3.5: Related Rates
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/03%3A_Using_Derivatives/3.05%3A_Related_Rates
When two or more related quantities are changing as implicit functions of time, their rates of change can be related by implicitly differentiating the equation that relates the quantities themselves.
2.3: The Product and Quotient Rules
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/02%3A_Computing_Derivatives/2.03%3A_The_Product_and_Quotient_Rules
If a function is a sum, product, or quotient of simpler functions, then we can use the sum, product, or quotient rules to differentiate the overall function in terms of the simpler functions and their...If a function is a sum, product, or quotient of simpler functions, then we can use the sum, product, or quotient rules to differentiate the overall function in terms of the simpler functions and their derivatives. The product and quotient rules now complement the constant multiple and sum rules and enable us to compute the derivative of any function that consists of sums, constant multiples, products, and quotients of basic functions we already know how to differentiate.
11.8: Triple Integrals in Cylindrical and Spherical Coordinates
https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/11%3A_Multiple_Integrals/11.08%3A_Triple_Integrals_in_Cylindrical_and_Spherical_Coordinates
The spherical coordinates of a point $P$ in 3-space are $\rho$ (rho), $\theta\text{,}$ and $\phi$ (phi), where $\rho$ is the distance from $P$ to the origin, $\theta$ is the angle that t...The spherical coordinates of a point $P$ in 3-space are $\rho$ (rho), $\theta\text{,}$ and $\phi$ (phi), where $\rho$ is the distance from $P$ to the origin, $\theta$ is the angle that the projection of $P$ onto the $xy$ -plane makes with the positive $x$ -axis, and $\phi$ is the angle between the positive $z$ axis and the vector from the origin to $P\text{.}$ When $P$ has Cartesian coordinates $(x,y,z)\text{,}$ the spherical coordinates are given by

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