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Review parabolas, ellipses, and hyperbolas, focusing on how calculus deepens our understanding of these shapes. First, look at parabolas and arc length computation. Then turn to ellipses, their formulas, and the concept of eccentricity. Next, examine hyperbolas. End by looking ahead to parametric equations.
3) Understanding Calculus II: Problems, Solutions, and Tips: Differential Equations - Growth and Decay
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In the first of three episodes on differential equations, learn various techniques for solving these very useful equations, including separation of variables and Euler's method, which is the simplest numerical technique for finding approximate solutions. Then look at growth and decay models, with two intriguing applications.
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Combine parametric equations, curves, vectors, and vector-valued functions to form a model for motion in the plane. In the process, derive equations for the motion of a projectile subject to gravity. Solve several projectile problems, including whether a baseball hit at a certain velocity will be a home run.
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Use your knowledge of vectors to explore vector-valued functions, which are functions whose values are vectors. The derivative of such a function is a vector tangent to the graph that points in the direction of motion. An important application is describing the motion of a particle.
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Learn what distinguishes Calculus II from Calculus I. Then embark on a three-episode review, beginning with the top 10 student pitfalls from precalculus. Next, Professor Edwards gives a refresher on basic functions and their graphs, which are essential tools for solving calculus problems.
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Explore integrals of trigonometric functions, finding that they are often easy to evaluate if either sine or cosine occurs to an odd power. If both are raised to an even power, you must resort to half-angle trigonometric formulas. Then look at products of tangents and secants, which also divide into easy and hard cases.
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Start the first of 11 episodes on one of the most important topics in Calculus II: infinite series. The concept of an infinite series is based on sequences, which can be thought of as an infinite list of real numbers. Explore the characteristics of different sequences, including the celebrated Fibonacci sequence.
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Parametric equations consider variables such as x and y in terms of one or more additional variables, known as parameters. This adds more levels of information, especially orientation, to the graph of a parametric curve. Examine the calculus concept of slope in parametric equations, and look closely at the equation of the cycloid.
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Try out techniques for approximating a function with a polynomial. The first example shows how to construct the first-degree Maclaurin polynomial for the exponential function. These polynomials are a special case of Taylor polynomials, which you investigate along with Taylor's theorem.
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Continue your exploration of the power of integral calculus. First, review arc length computations. Then, calculate the areas of surfaces of revolution. Close by surveying the concept of work, answering questions such as, how much work does it take to lift an object from Earth's surface to 800 miles in space?
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Put your precalculus skills to use by splitting up complicated algebraic expressions to make them easier to integrate. Learn how to deal with linear factors, repeated linear factors, and irreducible quadratic factors. Finally, apply these techniques to the solution of the logistic differential equation.
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Begin a series of episodes on techniques of integration, also known as finding antiderivatives. After reviewing some basic formulas from Calculus I, learn to develop the method called integration by parts, which is based on the product rule for derivatives. Explore applications involving centers of mass and area.
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Complete your review by going over the basic facts of integration. After a simple example of integration by substitution, turn to definite integrals and the area problem. Reacquaint yourself with the fundamental theorem of calculus and the second fundamental theorem of calculus. End the episode by solving a simple differential equation.
15) Understanding Calculus II: Problems, Solutions, and Tips: Moments, Centers of Mass, and Centroids
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Study moments and centers of mass, developing formulas for finding the balancing point of a planar area, or lamina. Progress from one-dimensional examples to arbitrary planar regions. Close with the famous theorem of Pappus, using it to calculate the volume of a torus.
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Begin a series of episodes on vectors in the plane by defining vectors and their properties, and reviewing vector notation. Then learn how to express an arbitrary vector in terms of the standard unit vectors. Finally, apply what you've learned to an application involving force.
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Investigate linear differential equations, which typically cannot be solved by separation of variables. The key to their solution is what Professor Edwards calls the "magic integrating factor." Try several examples and applications. Then return to an equation involving Euler's method, which was originally considered in an earlier lesson.
18) Understanding Calculus II: Problems, Solutions, and Tips: Applications of Differential Equations
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Continue your study of differential equations by examining orthogonal trajectories, curves that intersect a given family of curves at right angles. These occur in thermodynamics and other fields. Then develop the famous logistic differential equation, which is widely used in mathematical biology.
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Finish your exploration of convergence tests with the ratio and root tests. The ratio test is particularly useful for series having factorials, whereas the root test is useful for series involving roots to a given power. Close by asking if these tests work on the p-series, introduced in an earlier episode.
20) Understanding Calculus II: Problems, Solutions, and Tips: Curvature and the Maximum Bend of a Curve
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See how the concept of curvature helps with analysis of the acceleration vector. Come full circle by using ideas from elementary calculus to determine the point of maximum curvature. Then close by looking ahead at the riches offered by the continued study of calculus.
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