Catalog Search Results
1) Understanding Calculus II: Problems, Solutions, and Tips: Moments, Centers of Mass, and Centroids
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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.
5) Understanding Calculus II: Problems, Solutions, and Tips: Differential Equations - Growth and Decay
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
Use integration to find areas and volumes. Begin by trying your hand at planar regions bounded by two curves. Then review the disk method for calculating volumes. Next, focus on ellipses as well as solids obtained by rotating ellipses about an axis. Finally, see how your knowledge of ellipsoids applies to the planet Saturn.
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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.
13) Understanding Calculus II: Problems, Solutions, and Tips: Curvature and the Maximum Bend of a Curve
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
Deepen your skill with vectors by exploring the dot product method for determining the angle between two nonzero vectors. Then turn to projections of one vector onto another. Close with some typical applications of dot product and projection that involve force and work.
Author
Series
Language
English
Description
In the first of two episodes on polar coordinates, review the main properties and graphs of this specialized coordinate system. Consider the cardioids, which have a heart shape. Then look at the derivative of a function in polar coordinates, and study where the graph has horizontal and vertical tangents.
16) Understanding Calculus II: Problems, Solutions, and Tips: Indeterminate Forms and L'Hpital's Rule
Author
Series
Language
English
Description
Revisit the concept of limits from elementary calculus, focusing on expressions that are indeterminate because the limit of the function may not exist. Learn how to use L'Hopital's famous rule for evaluating indeterminate forms, applying this valuable theorem to a variety of examples.
Author
Series
Language
English
Description
Develop more convergence tests, learning how the direct comparison test for positive-term series compares a given series with a known series. The limit comparison test is similar but more powerful, since it allows analysis of a series without having a term-by-term comparison with a known series.
Author
Series
Language
English
Description
Does the celebrated harmonic series diverge or converge? Discover a proof using the integral test. Then generalize to define an entire class of series called p-series, and prove a theorem showing when they converge. Close with the sum of the harmonic series, the fascinating Euler-Mascheroni constant, which is not known to be rational or irrational.
Author
Series
Language
English
Description
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.
Author
Series
Language
English
Description
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?
In Interlibrary Loan
Didn't find what you need? Items not owned by Santa Barbara Public Library can be requested from other Interlibrary Loan libraries to be delivered to your local library for pickup.
Didn't find it?
Can't find what you are looking for? Try suggesting a title. Submit Request