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- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/02%3A_Computing_Derivatives/2.07%3A_Derivatives_of_Functions_Given_ImplicitelyImplicit 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.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative/1.02%3A_The_Notion_of_LimitLimits 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.
- 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.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/03%3A_Using_Derivatives/3.05%3A_Related_RatesWhen 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.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/02%3A_Computing_Derivatives/2.03%3A_The_Product_and_Quotient_RulesIf 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.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/05%3A_Finding_Antiderivatives_and_Evaluating_Integrals/5.01%3A_Construction_Accurate_Graphs_of_AntiderivativesGiven the graph of a function f, we can construct the graph of its antiderivative F provided that (a) we know a starting value of F, say F(a), and (b) we can evaluate the integral R b a f (x) dx exact...Given the graph of a function f, we can construct the graph of its antiderivative F provided that (a) we know a starting value of F, say F(a), and (b) we can evaluate the integral R b a f (x) dx exactly for relevant choices of a and b. Thus, any function with at least one antiderivative in fact has infinitely many, and the graphs of any two antiderivatives will differ only by a vertical translation.
- https://math.libretexts.org/Under_Construction/Purgatory/Book%3A_Active_Calculus_(Boelkins_et_al.)/08%3A_Sequences_and_Series/8.03%3A_Series_of_Real_NumbersAn infinite series is a sum of the elements in an infinite sequence. The sequence of partial sums of a series P∞ k=1 ak tells us about the convergence or divergence of the series. The series converges...An infinite series is a sum of the elements in an infinite sequence. The sequence of partial sums of a series P∞ k=1 ak tells us about the convergence or divergence of the series. The series converges if the sequence of partial sums converges.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative/1.01%3A_How_do_we_Measure_VelocityThe average velocity on [a,b] can be viewed geometrically as the slope of the line between the points (a,s(a)) and (b,s(b)) on the graph of y=s(t). The instantaneous velocity of a moving object at...The average velocity on [a,b] can be viewed geometrically as the slope of the line between the points (a,s(a)) and (b,s(b)) on the graph of y=s(t). The instantaneous velocity of a moving object at a fixed time is estimated by considering average velocities on shorter and shorter time intervals that contain the instant of interest
- https://math.libretexts.org/Under_Construction/Purgatory/Book%3A_Active_Calculus_(Boelkins_et_al.)/01%3A_Understanding_the_Derivative/1.08%3A_The_Tangent_Line_ApproximationThe principle of local linearity tells us that if we zoom in on a point where a function y = f (x) is differentiable, the function should become indistinguishable from its tangent line. That is, a dif...The principle of local linearity tells us that if we zoom in on a point where a function y = f (x) is differentiable, the function should become indistinguishable from its tangent line. That is, a differentiable function looks linear when viewed up close.
- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/02%3A_Computing_DerivativesThroughout Chapter 2, we will be working to develop shortcut derivative rules that will help us to bypass the limit definition of the derivative in order to quickly determine the formula for f'(x)...Throughout Chapter 2, we will be working to develop shortcut derivative rules that will help us to bypass the limit definition of the derivative in order to quickly determine the formula for f'(x) when we are given a formula for f (x).
