Study Guide - Test 2 (MATH-2212, Fall-2018).pdf - MATH-2212...

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MATH-2212: Calculus of One Variable II Study Guide for Test 2, Fall 2018 Study Guide for Test 2 It goes without saying that mathematics develops cumulatively, and all subsequent topics usually depend heavily on some (if not all) previous material. This study guide is intended to emphasize the new material learned in this portion of the course. Knowing and understanding everything that has been covered before — in prerequisite courses and earlier in this course — is also crucial for success on this test. Test Contents Section 7.8 – Improper Integrals Section 8.1 – Arc Length Section 10.1 – Curves Defined by Parametric Equations Section 10.2 – Calculus with Parametric Curves Section 10.3 – Polar Coordinates Section 10.4 – Areas and Lengths in Polar Coordinates Section 11.1 – Sequences Key Concepts Improper integrals; convergence and divergence of improper integrals. Parametric equations and curves; converting parametric equations to cartesian equations by eliminating the parameter; graphs of curves given parametrically. Polar coordinates; conversion between polar and cartesian coordinates; sketching graphs of equations in polar coordinates. Arc length of a curve; finding the arc length of curves given by explicit functions, parametric equations, or polar equations. Equations of tangent lines to parametrically defined curves. Area under a curve defined by parametric equations. Areas enclosed by polar curves. – 1–
MATH-2212: Calculus of One Variable II Study Guide for Test 2, Fall 2018 Sequences of real numbers; sequence notation; common term formulas for sequences; mono- tonic sequences. The limit of a sequence (intuitively); convergent and divergent sequences; relationships among convergence, boundedness, and monotonicity properties. Some special limit formulas. Some Important Definitions, Formulas, and Properties Conversion Formulas Between Polar and Cartesian Coordinates x = r cos θ r 2 = x 2 + y 2 y = r sin θ tan θ = y x Derivatives and Tangent Lines to Parametric Curves For a parametrically defined curve x = x ( t ), y = y ( t ), the derivative of y as a function of x can be found as dy dx = dy/dt dx/dt , provided dy dx 6 = 0 . An equation of the tangent line to the graph of a parametrically defined curve x = x ( t ), y = y ( t ) at a point t 0 , i.e at a point ( x 0 = x ( t 0 ) , y 0 = y ( t 0 )), can be set up as y - y 0 = dy dx t = t 0 · ( x - x 0 ) , provided dy dx t = t 0 6

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