Use convergence tests to classify alternating series as absolutely convergent, conditionally convergent, or divergent, and satisfy hypotheses about alternating series. Use Maclaurin and Taylor polynomials to improve the accuracy of local linear approximations. Use Maclaurin and Taylor series and functions defined by power series to find the radius of convergence and the interval of convergence, and differentiate and integrate power series. Establish the convergence of Taylor series. Express given equations in polar coordinates, find equations for polar graphs, and sketch curves in polar coordinates using appropriate technology. Derive the formulas required to find slopes, tangent lines, and arc lengths of both parametric and polar curves. Find areas of regions that are bounded by polar curves. Sketch parabolas, ellipses, and hyperbolas, and identify foci, vertices, axes, and asymptotes, and use these conic sections in polar coordinates. Explain polar coordinates, analyze polar graphs, and find area enclosed by regions described by polar coordinates. Find dot products of two vectors, including finding projections, find the cross product of two vectors in space, including calculation of the scalar triple product, and use a parametric equation to find tangent lines and arc length. Use vectors to derive equations of planes in 3-space, including the use of these equations to solve geometric problems, time permitting. Find limits, derivatives, and integrals for vector-valued functions, time permitting. Course Content: Applications of the Definite Integrals (some review) Volumes using Cross-Sections/Cylindrical Shells Arc length by the differential formula Areas of Surfaces of Revolution Revolved about the x-axis Revolved about the y-axis Physical Applications Work by a constant force Work by a variable force Hooke’s Law for Springs Fluid force problems Integrals and Transcendental Functions The Logarithm Defined as an Integral e and natural logarithm rules, definitions, and domain/range Integrals leading to logarithms, trigonometric functions Exponential Change Separable Differential Equations Techniques for expressing differential equation Initial value problems Applications of differential equations
Use convergence tests to classify alternating series as absolutely convergent, conditionally convergent, or divergent, and satisfy hypotheses about alternating series. Use Maclaurin and Taylor polynomials to improve the accuracy of local linear approximations. Use Maclaurin and Taylor series and functions defined by power series to find the radius of convergence and the interval of convergence, and differentiate and integrate power series. Establish the convergence of Taylor series. Express given equations in polar coordinates, find equations for polar graphs, and sketch curves in polar coordinates using appropriate technology. Derive the formulas required to find slopes, tangent lines, and arc lengths of both parametric and polar curves. Find areas of regions that are bounded by polar curves. Sketch parabolas, ellipses, and hyperbolas, and identify foci, vertices, axes, and asymptotes, and use these conic sections in polar coordinates. Explain polar coordinates, analyze polar graphs, and find area enclosed by regions described by polar coordinates. Find dot products of two vectors, including finding projections, find the cross product of two vectors in space, including calculation of the scalar triple product, and use a parametric equation to find tangent lines and arc length. Use vectors to derive equations of planes in 3-space, including the use of these equations to solve geometric problems, time permitting. Find limits, derivatives, and integrals for vector-valued functions, time permitting. Course Content: Applications of the Definite Integrals (some review) Volumes using Cross-Sections/Cylindrical Shells Arc length by the differential formula Areas of Surfaces of Revolution Revolved about the x-axis Revolved about the y-axis Physical Applications Work by a constant force Work by a variable force Hooke’s Law for Springs Fluid force problems Integrals and Transcendental Functions The Logarithm Defined as an Integral e and natural logarithm rules, definitions, and domain/range Integrals leading to logarithms, trigonometric functions Exponential Change Separable Differential Equations Techniques for expressing differential equation Initial value problems Applications of differential equations