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Single-Variable Calculus II (

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This course is the second installment of Single-Variable Calculus.

In Part I (Single-Variable Calculus I), we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions. In this course, we will extend our differentiation and integration abilities and apply the techniques we have learned.

Additional integration techniques, in particular, are a major part of the course. In Part I, we learned how to integrate by various formulas and by reversing the chain rule through the technique of substitution. In Part II (Single-Variable Calculus II), we will learn some clever uses of substitution, how to reverse the product rule for differentiation through a technique called integration by parts, and how to rewrite trigonometric and rational integrands that look impossible into simpler forms. Series, while a major topic in their own right, also serve to extend our integration reach: they culminate in an application that lets you integrate almost any function you’d like.

Integration allows us to calculate physical quantities for complicated objects: the length of a squiggly line, the volume of clay used to make a decorative vase, or the center of mass of a tray with variable thickness. The techniques and applications in this course also set the stage for more complicated physics concepts related to flow, whether of liquid or energy, addressed in "Multivariable Calculus".

Part I covered several applications of differentiation, including related rates. In Part II, we introduce differential equations, wherein various rates of change have a relationship to each other given by an equation. Unlike with related rates, the rates of change in a differential equation are various-degree derivatives of a function, including the function itself. For example, acceleration is the derivative of velocity, but the effect of air resistance on acceleration is a function of velocity: the faster you move, the more the air pushes back to slow you down. That relationship is a differential equation.

Upon successful completion of this course, the student will be able to:

- Define and describe the indefinite integral.

- Compute elementary definite and indefinite integrals.

- Explain the relationship between the area problem and the indefinite integral.

- Use the midpoint, trapezoidal, and Simpson’s rule to approximate the area under a curve.

- State the fundamental theorem of calculus.

- Use change of variables to compute more complicated integrals.

- Integrate transcendental, logarithmic, hyperbolic, and trigonometric functions.

- Find the area between two curves.

- Find the volumes of solids using ideas from geometry.

- Find the volumes of solids of revolution using disks, washers, and shells.

- Find the surface area of a solid of revolution.

- Compute the average value of a function.

- Use integrals to compute displacement, total distance traveled, moments, centers of mass, and work.

- Use integration by parts to compute definite and indefinite integrals.

- Use trigonometric substitution to compute definite and indefinite integrals.

- Use the natural logarithm in substitutions to compute integrals.

- Integrate rational functions using the method of partial fractions.

- Compute improper integrals of both types.

- Graph and differentiate parametric equations.

- Convert between Cartesian and polar coordinates.

- Graph and differentiate equations in polar coordinates.

- Write and interpret a parameterization for a curve.

- Find the length of a curve described in Cartesian coordinates, described in polar coordinates, or described by a parameterization.

- Compute areas under curves described by polar coordinates.

- Define convergence and limits in the context of sequences and series.

- Find the limits of sequences and series.

- Discuss the convergence of the geometric and binomial series.

- Show the convergence of positive series using the comparison, integral, limit comparison, ratio, and root tests.

- Show the divergence of a positive series using the divergence test.

- Show the convergence of alternating series.

- Define absolute and conditional convergence.

- Show the absolute convergence of a series using the comparison, integral, limit comparison, ratio, and root tests.

- Manipulate power series algebraically.

- Differentiate and integrate power series.

- Compute Taylor and MacLaurin series.

- Recognize a first order differential equation.

- Recognize an initial value problem.

- Solve a first order ODE/IVP using separation of variables.

- Draw a slope field given an ODE.

- Use Euler’s method to approximate solutions to basic ODE.

- Apply basic solution techniques for linear, first order ODE to problems involving exponential growth and decay, logistic growth, radioactive decay, compound interest, epidemiology, and Newton’s Law of Cooling.

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