Analytic Geometry and Calculus 2
Analytic Geometry and Calculus 2 MATH 2300
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This 3 page Class Notes was uploaded by Cydney Conroy on Thursday October 29, 2015. The Class Notes belongs to MATH 2300 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/231832/math-2300-university-of-colorado-at-boulder in Mathematics (M) at University of Colorado at Boulder.
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Date Created: 10/29/15
Final Review for Calculus II Here is a list of what we ve done this semester along with some exercises that you should be able to do Of course you are responsible for any homework problems we ve done so know how to do them too Exam 1 stu Section 55 amp 56 The Fundamental Theorem of Calculus Duh You d better know this Problems Section 55 2 3 7 9 10 14 18 22 25 28 29 Section 67 The Natural Logarithm as an Integral De nition of the natural log as an integral rules of logs especially the power rule for logs exponential form compared to log form derivatives and integrals of exponential functions involving bases other than e derivatives of logs involving bases other than e Problems 1 3 7 9 12 22 27 28 Section 81 Simple Equations and Models Formulas for exponential growth and decay how to solve for 1 Problems 23 33 Section 68 Inverse Trigonometric Functions De nitions of the inverse trig functions including the restricted domains necessary to de ne them evaluate trig and inverse trig functions differentiate and integrate functions involving inverse trig functions Problems 1 3 ll 17 25 3133 41 43 49 50 Section 69 Hyperbolic Functions De nitions of coshx sinhx and tanhx fundamental identity cosh2 x sinh2 x 1 differentiate and integrate functions involving hyperbolic trig functions and inverse hyperbolic trig functions Problems 1 3 7 21 43 45 Section 72 Integral Tables and Simple Substitutions Know ALL the simple integrals in Figure 721 pg492 Do simple substitutions to make integrals match formulas some miscellaneous integration techniques Problems Homework problems should suf ce Section 73 Integration by parts Integration by parts what to pick for u and dv maybe know formula for the integral of secx and how to do the integral for sec3x Problems 3 7 10 20 21 25 31 do more if you feel weak in this area Section 74 Trigonometric Integrals Integration of functions involving trig functions Know cos2 x sin2 x 1 the halfangle identities the doubleangle identities these come up in integration a lot and at least sine cosine and tangent for the popular angles Problems 9 17 19 27 30 35 Exam 2 Stu Section 75 Rational Functions and Partial Fractions De nition of a rational function amp a proper rational function how to divide polynomials with long division how to write a rational function in its partialfraction decomposition how to nd linear and quadratic factors of the partial fraction decomposition recalling de nition of irreducible including completing the square and the 2 important functions at the bottom of pg5 14 Problems 1 9 11 15 19 27 29 Section 76 Trigonometric Substitution See table at bottom of pg 517 and know how to use the triangle Problems 1 3 5 7 25 27 31 Section 77 Integrals Involving Quadratic Polynomials Completing the square again dealing with quadratic polynomials using previous techniques Problems 1 3 5 7 18 19 29 do partial fractions first Section 78 Improper Integrals Convergence and divergence of the two types of improper integrals in nite limits and in nite discontinuities on the interval of integration L Hospital s rule can be useful here Problems 5 7 13 19 23 25 33 Section 92 Polar Coordinates You should memorize the formulas for converting back and forth between polar and rectangular Recognizing polar equations for circles cardiods lemiscons and rosetype graphs would be very useful there was a handout on this Either way you should be able to sketch graphs of polar equations and be able to nd points of intersection of two different polar equations Problems lab 2ab 3 5 7 9 13 39 41 43 45 49 53 Section 93 Area Computations in Polar Coordinates Area of regions bounded by polar graphs Memorize the appropriate formulas Problems 7 19 25 29 Section 94 Parametric Curves Be able to eliminate parameter to convert to rectangular form memorize formula for 1st and 2quot 1 derivatives of smooth parametric curves p g648 and be able to evaluate these at a speci c point Problems 5 7 17 19 25 27 Section 95 Integral computations with Parametric Curves Know formulas on pg655 which includes nding area under a parametric curve volume of revolution arc length and area of a surface of revolution Problems 1 3 7 11 19 Exam 3 stu f except for Section 96 Section 96 Conic Sections and Applications Know formulas and how to nd them for the parabola ellipse and hyperbola including their general forms and how to sketch them Problems Homework problems should suffice Section 102 In nite Sequences In this section we de ned a sequence gave examples of sequences including the Fibonacci sequence gave the de nition of the limit of a sequence introduced limit laws for sequences the substitution law for sequences the squeeze law for sequences L Hopital s rule for sequences bounded monotonic sequences and the bounded monotonic sequence property Problems 3 9 l3 15 16 23 28 29 30 33 39 59 Section 103 In nite Series and Convergence De nition of an infmite series a term in a series the sum of a series the nthpartial sum the idea of convergence and divergence telescoping sums the geometric series and its sum properties of convergent series n39h term test for divergence divergence of the harmonic series series that are eventually the same Problems 1 5 9 12 18 23 24 27 39 54 Section 104 Taylor Series and Taylor Polynomials Polynomial approximations de nition of the Taylor polynomial of f x at the point x a Taylor s formula including the nth 7 degree remainder for f x at x a Taylor series including derivations of the Taylor series for ex cosx sinx the Maclaurin series Euler s formula Problems 1 2 3 6 11 19 22 25 27 28 29 32 33 37 38 Section 105 The Integral Test De nition of a positiveterm series the integral test de nition of a p series Problems 1 2 5 6 7 12 15 17 22 23 24 29 30 31 32 34 36 Section 106 Comparison tests for positive term series Idea of comparing positiveterm series to positiveterm series which are known to converge diverge mostly the geometric series and pseries Comparison test limit comparison test rearrangement and grouping Problems 1 3 4 5 6 8 11 12 14 15 16 19 22 24 27 33 35 Section 107 Alternating series and absolute convergence De nition of an alternating series Alternating Series Test Alternating series remainder estimate de nition of absolute convergence absolute convergence implies convergence difference between being absolutely convergent conditionally convergent or divergent Ratio test Root test Problems 1 4 5 7 9 10 14 15 21 22 23 24 25 26 29 31 38 41 49 Section 108 Power Series De nition of a power series convergence of a power series using ideas from Ratio test de nition of radius of convergence de nition of interval of convergence how to nd the interval of convergence of a power series de nition of and how to nd the interval of convergence of a power series in powers of x e Problems 1 2 3 5 6 7 8 11 13 15 19 21 23 New material since Exam 3 for review problems just look over homework Section 109 Power Series Computations Finding error using alternating series remainder estimate for alternating series adding amp multiplying power series to get new power series and their radii of convergence power series amp indeterminate forms Section 81 Simple Equations and Models Using separation of variables on a differential equation with one variable missing Section 83 Separable Equations and Applications Using separation of variables on differential equations in both variables Newton s law of cooling Section 1010 Series Solutions ofDifferential Equations Power series method shifting of indices of summation identity principle for power series recurrence relation radius of convergence of power series solution power series de nitions of functions Section 65 Force and Work De nition of work work as an integral
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