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This 7 page Study Guide was uploaded by Jasmine Ngo on Tuesday March 29, 2016. The Study Guide belongs to MATH152 at Illinois Institute of Technology taught by Dr. Shive in Spring 2016. Since its upload, it has received 37 views. For similar materials see Calculus II in Calculus and Pre Calculus at Illinois Institute of Technology.
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Date Created: 03/29/16
Calculus II Study Guide Chapters 9.5, 10.1, 10.4, 11.1-11.2 Chapter 9.5 Linear Equations 1 order Linear equations are ones that can be put into the form y’+P(x)*y=Q(x). Can be solved by multiplying both sides by I(x)=e^(integral of p(x)dx) , then integrate both sides Solution: Integral of (e^(integral of p(x)dx)*Q(x))dx all over e^(integral of p(x)dx) 10.1 Parametric Equations: Let x and y be functions of an independent variable t, given by x=x(t) and y=y(t). t is a parameter and x and y are parametric equations. t is given to be a value of a point in x and y plane, tracing out a parametric curve. Plugging in values of t will give x and y coordinates. The graph when plotted will look like a parabola 10.2 Calculus with Parametric Curves Finding the slope of the tangent lines to the parametric curve will help in sketching the parametric curve. By using the above equation, we know that: When dy/dt=0 and dx/dt is not zero, then (dy/dt)/(dx/dt)= 0, or that the tangent line is zero. When dy/dt is not 0 and dx/dt is zero, then tangent is vertical The tangents of a parametric curve can be solved by using the above equation, by taking the derivative of y and dividing it by the derivative of x. Cycloid-A cycloid has parametric equations: X=r(ϴ-sin ϴ) 10.3 Polar Coordinates: Choose a point O in a pole and draw a ray starting at O called the polar axis. Point P can be represented as (r, ϴ) where r and ϴ are polar coordinates. This is the polar coordinate system. We use the convention that angle is positive if measured in the counterclockwise direction. We define (-r, ϴ) as the reflection of (r, ϴ) through the pole. We define (r, -ϴ) as the addition of (r, ϴ) and 2 Pi. Connection between Cartesian coordinates (x,y) and polar coordinates (r, ϴ): X=rcos ϴ Y=rsin ϴ R^2=x^2+y^2 tan ϴ=y/x Tangents to Polar Curves: To find tangent of polar curve use r=f(ϴ) X=rcos ϴ=> x=f(ϴ)cos ϴ Y=rsin ϴ=>y=f(ϴ)sin ϴ 10.4 Areas with Polar Coordinates Each interval of [a,b] can be divided into n subintervals. Subinterval length can be calculated by: Delta ϴ=(b-a)/n Circular wedge has area of : Total area: 11.1 Sequences A sequence is a list of numbers written in order of a1,a2,a3,a4…an. Can be written as : 11.2 Series Geometric series is when each term is obtained from the preceding one by multiplying it by a ratio. The geometric series is convergent if |r|<1. The geometric series is divergent if |r|>=1. If we try to sum up the terms of a sequence from n=1, to n=infinity, we have an infinite series.
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