AP Calculus BC : Sequences and First Order Differential Equations
This free online course examines finite sequences, convergence tests and solving first order differential equations.Publisher: ADU
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AP Calculus BC: sequences and first-order differential equations is the fourth in a series of calculus courses. It begins by introducing the learner to the concept of finite and infinite sequences. In mathematics, a sequence is usually meant to be a progression of numbers with a clear starting point. Some important kinds of sequences are defined recursively while some sequences also stop at a certain number. In other words, they have a first term and the last term and all the terms follow a specific order. This type of sequence is called a finite sequence. This free online course describes the arithmetic sequence, geometric sequence, binary sequence and power sequence. It also describes the triangular and harmonic sequence. Infinite sequences are an endless progression of discrete objects, especially numbers. You will learn about the monotonic behaviour of sequences and what it means when a sequence is monotonically increasing or decreasing as well as when a bounded monotonic sequence converges. The cartesian graph is a wonderful way to visualize a given sequence and this course explains in detail how to graphically represent sequences.
Infinite series are very useful concepts that are not limited to mathematics. It plays roles in disciplines such as physics, chemistry, biology, and engineering. However, for the purpose of this course, we will focus on its application in mathematics. We will simply look at how to convert to a closed-form, an infinite series presented using enumeration. Convergence tests are methods of testing for the convergence, absolute convergence, conditional convergence, the interval of convergence, or divergence of an infinite series. It covers methods for determining whether or not an infinite series converges. It states the divergence theorem and explains the divergence test, comparison test, the p – test as well as ratio test. The integral test is a method of testing an infinite series of nonnegative terms for convergence by comparing them to an improper integral. When testing the convergence of a series, you should remember that there is no convergence test that works for all series. You will have to determine the right test for a given series. If you want to learn how to make the right pick, this course comes highly recommended.
Furthermore, In the last century, many algorithms were developed by numerical analysts to solve higher-order differential (and partial differential) equations. We will look at some of the theories behind first-order differential equations as well as some applications of first-order differential equations. It explores solving the first-order differential equations by integration, separable, linear, exact and numerical methods. A differential equation is a challenge. To solve an ordinary differential equation, we will have to determine the function or functions that satisfy this differential equation. How can a differential equation be classified? This course analyzes the Maclaurin and Taylor series. It explains the radius of convergence for each power series. If you want to learn about shortcuts for calculating series, or how to calculate coefficients, you should complete this course. If you are an engineer, economist, computer scientist, researcher, or a student with an interest in an excellent working knowledge of AP Calculus. Register today, to get a deeper insight on very useful techniques for approaching calculus problems.Start Course Now
Finite Sequences - Learning Outcomes
Types of Finite Sequence
More on Finite Sequence
Bounded and Monotonic Sequence
Finite Sequences - Lesson Summary
Tests for Convergence
Tests for Convergence - Learning Outcomes
Tests for Convergence
More Tests for Convergence
Tests for Convergence - Lesson Summary
First Order Differential Equations
First Order Differential Equations - Learning Outcomes
Maclaurin and Taylor Series
Differential Equations - Examples
First Order Differential Equations
First Order Differential Equations - Lesson Summary
Upon the successful completion of this course, you should be able to:
- Classify a differential equation.
- Recall how to solve a first-order differential equation using integration.
- Describe Maclaurin and Taylor series.
- Recall how to find the interval and radius of convergence of a power series.
- Identify a geometric series.
- State the divergence theorem.
- Recall how to solve the p - test for series.
- Discuss alternating series test (AST).
- Distinguish limit comparison test and ratio test.
- List the different types of finite sequences.
- Define a harmonic sequence.
- Evaluate arithmetic sequences.
- Recall how to solve for geometric sequences.
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