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# Math_1221_Study_Guide_2.pdf Math 1221

GWU

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## Popular in Calculus with PreCalculus II

## Popular in Mathematics (M)

This 5 page Study Guide was uploaded by Jamie Elliott on Monday May 2, 2016. The Study Guide belongs to Math 1221 at George Washington University taught by Roosevelt in Spring 2016. Since its upload, it has received 23 views. For similar materials see Calculus with PreCalculus II in Mathematics (M) at George Washington University.

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Date Created: 05/02/16

Math 1221: Calculus with Pre-Calculus II Study Guide Specified Objective: Chapter 4 4.1 through 4.5 Chapter 4: 4.1 To begin Chapter 4 off, we discussed the areas under the curve. This specific section covers the approximation of rectangles. Section 4.2 will cover f(x) above and below the x-axis. Section 4.1: find the area under the curve y = f(x) and bounded below the x-axis, and vertical lines x = a and x = b are a rectangle equation of rectangle: ; width = = (x2-x1) = b-a/n, where n = the number of rectangles; a ≤ x ≤ b. Where f(x) is greater than zero. Examples: E.g. 1 Use four rectangles, and use the right end points. What we now have to do, is find the area. In order to find the area, we must multiply the known Delta x (Δ x) by the quantity of every f(Point1 2 3 4, etc.) or the f(length); all f(lengths) are added together first since they are in parentheses, or brackets Area = Δx [f(R1) + f(R2) + f(R3) + … + f(Rn)]. The area for this problem is such: A(Right end points) or A =RΔx [f(R1) + f(R2) + f(R3) + f(R4)] = ½ [f(2) + f(3/2) + f(1) + f(1/2)] Because the function states y = x , every f(length) will be squared. ½ [(2) + (3/2) + 2 2 (1) + (1/2) ] = ½ [4 + 9/4 + 1 + ¼] = ½ [5 + 10/4] = ½ [20/4 + 10/4] = ½ [30/4] = ½ [15/2] = 15/4, or 3.75. The answer we come to is an overestimate since it can be assumed the area underneath the curve is not that large, for this example. Once we have gone through the left end points, and the mid points, we find out that the approximation using the left points are an underestimate (1.75), and the midpoint approximation (2.625) is closest to the actual approximation, 2.67. For approximation we find that as f(x) is increasing, the right end point approximation is an overestimate, and the left end point approximation is an underestimate. When f(x) is decreasing, the right end point approximation underestimates, and the left end point approximation overestimates. Chapter 4: 4.2 In this section, we discussed the Definite Integral Underneath a Curve. Within this subject, we get into estimating the area under a curve by approximating rectangles, similarly to Section 4.1. We use such equation to approximate the rectangles: ∑ ????=1????(???????? ∗) Δx = [f(x1*) + f(x2*) + f(x3*) + … + f(xn*)] * Δx ; on an interval [a, b]. Xi* talks about the right end points, the left end points, and the midpoints. When using ‘n’, it means the number of rectangles, or subintervals, that are being included. Because we’re not going to have Riemann Sums on the exam, they will not be included within the study guide, despite the given equation above (). Related to the given equation, ???? integrals were then discussed. ∑ ????=1 ???? ???????? ∗ * Δx = ∫ ???? ???? ???????? , where b is your interval, ???? and the ‘x’ in the dx is the independent variable, and the f(x) is equal to the integral. The [a, b] are your limits of integration; the ‘a’ is your lower limit, the ‘b’ is your upper limit. ∫???? ???? ???? ???????? is your definite integral; where your answer will be a number with ‘i’, which has ???? limits of integration. 1) ∫???? ???? ???? ???????? = actual area ???? If f(x) is above the x-axis ???? 2) ∫???? ???? ???? ???????? = area under the Is negative. The area itself is always positive. 3) ∫???? ???? ???? ???????? = difference between the two areas ; ∫???? ???? ???? ???????? = A –1A ,2where A is1 ???? ???? greater than zero, and 2 is greater than zero as well. If A = A , then ∫???? ???? ???? ???????? = 0 . 1 2 ???? ???? ∫ ???? ???? ???????? = A –1A =20 ???? The area under the f(x) and bounded by the x-axis equals to 2A. In general: Area = Area Above + Area Below Integral = Area Above – Area Below Chapter 4: 4.3 This section discussed the Definite Integral. ∫???? ???? ???? ???????? = F(b) – F(a) , the F antiderivative ???? The final answer will be always be a number. -No variable, or no “x” Chapter 4: 4.4 This section discussed the Indefinite Integral, the Antiderivative. ∫ ???? ???? ???????? = ???? ???? + ???? a) No limits of integration b) Always have a “+ C” c) Function of x ∫ sin ???? ???????? = -cos(x) + C sec^2 ???? ???????? = tan(x) + C ∫ ( ) ∫ csc ???? co ????)???????? = -csc(x) +C 2 Ex: ∫(???? + 4 2???? + 1 ???????? = ∫(2???? + 9???? + 4 ???????? (Term by Term) = 2/3 u^3 + 9/2 u^2 + 4u + C Net Change Theorem: ???????? ???? ???????? = g(b) – g(a); The definite integral of the rate of ∫???? change of a function is equal to the net change in the original function. Chapter 4: 4.5 2 ∫1 ????√ ???? − 1???????? (where the x-1 underneath the square root is the inner function) Let u = x-1; where x du = x dx∫????√???? , where in x you solve for u. u = x-1 (add 1) u + 1 = x 1 3 1 ∫ (???? + 1)√ ???????????? = ∫ (???? + 1 ∗ ???? ???????? = ∫???? + ???? )???????? Lower limit: x = 1 u = 1-1 = 0. Upper limits: x = 2 u = 2-1 = 0 5/2 3/2 5/2 3/2 u /5/2 + u /3/2 |01 = 2/5 u + 2/3 u |01 = (2/5 + 2/3 – 0) = (6/15 + 10/15 – 0) = 16/15, which is your final answer.

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