Arc Length of Parametric Curves - Calculus 2 Jan 24, 2023

Earlier in Calculus 2 (Lesson 6), we saw how to use a definite integral to calculate the arc length of a curve between two values of x. Up until now, we always calculated this arc length by working in terms of x because those curves had always been defined with functions in terms of x and y, or...

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Area Under Parametric Curves - Calculus 2 Jan 19, 2023

In Calculus 1, we are introduced to the concept that a definite integral calculates the area under a curve between two values of x. Up to this point, the area that we wanted to calculate was beneath curves that were always defined with functions in terms of x and y, or rectangular equations. But...

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Concavity of Parametric Curves - Calculus 2 Jan 17, 2023

Another application of parametric derivatives is the ability to determine the concavity for plane/parametric curves. In fact, this is specifically an application of the second parametric derivative for a set of parametric equations.

You were first introduced to concavity in Calculus 1,...

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Slope & Tangent Lines of Parametric Curves - Calculus 2 Jan 12, 2023

One of the applications of parametric derivatives is the ability to determine the slope at a particular value of t (the parameter) along a plane/ parametric curve, and use that slope to find the equation of the tangent line for that value of t. This should be a familiar process, as finding slope...

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Parametric Derivatives - Calculus 2 Jan 10, 2023

When working with plane/ parametric curves, it may sometimes be of interest to know the slope at a particular point along that curve. We learn in Calc 1 that the slope at a point along a curve/function is represented by the derivative of that function, and we have a whole arsenal of...

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Intro to Plane Curves & Parametric Equations - Calculus 2 Jan 05, 2023

In the last lesson we completed the second major group of concepts in Calculus 2, which was sequences and series. With the conclusion of those concepts, we are now ready to begin the third and final major group of concepts in Calc 2, which is parametric equations and polar...

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Taylor Polynomials & Approximations - Calculus 2 Dec 21, 2022

In a previous lesson we established two special types of power series known as Taylor series and Maclaurin series. One of the applications of these series is the ability to approximate the value of a function at a particular value of x. This is done by using what are known as Taylor...

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Binomial Series - Calculus 2 Dec 16, 2022

In the last lesson we introduced two new types of power series known as Taylor series and Maclaurin series. Many functions can be represented as power series by forming Taylor series or Maclaurin series for those functions. However, there is a specific type of function that when represented as a...

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Taylor Series & Maclaurin Series - Calculus 2 Dec 14, 2022

In the past few lessons we have focused on power series, and how we can represent functions as a power series. Specifically, we were representing functions similar to the form of the sum of a geometric series as a power series. However, this is not the only way to represent a function as a power...

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Power Series - Representation By Derivatives & Integrals - Calculus 2 Dec 09, 2022

In the previous lesson we saw how functions that are similar to the form of the sum of a geometric series can be represented with a power series. However, we can extend this idea to more functions. Specifically, we can represent functions whose derivative or integral is similar to that form...

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