Midterm Exam I - Review

Transcription

1 Midterm Exam I - Review Chapter 8 ad Appedices B,H The followig is a list of importat cocepts will be tested o Midterm Exam. This is ot a complete list of the material that you should kow for the course, but it is a good idicatio of what will be emphasized o the midterm. A thorough uderstadig of all of the followig cocepts will help you perform well o the exam. Some places to fid problems o these topics are the followig: i the book, i the slides, i the homework, o quizzes, ad WebAssig. Sequeces (Sectio 8.) Uderstad the ituitive defiitio of a sequetial limit. What does it mea for lim a = L? The limit of a sequece behaves i oe of three ways:. The limit exists: lim a = L. The sequece evetually settles i o a umber L ad stays close to it for large. 2. The limit teds to ± : lim a = ±. I the case where a, the values of the sequece become large ad remai large for large. 3. The limit does ot exist: lim a does ot exist. The values of a have to bouce aroud i some sese. They caot settle i o a umber, but they do ot all get large for large. Thik of ( ) as a typical example. Be able to compute limits usig techiques from Calculus o limits at ifiity (icludig L Hospital s rule). Studets should also be able to describe a sequece as mootoic or bouded.. Compute the followig sequetial limits: lim l() (b) lim (c) lim 2! 00 (d) lim! 50 (e) lim π (f) lim arcta() (g) lim si() 2 ( ) (h) lim 2 + Series (Sectio 8.2) Kow the defiitio of covergece versus divergece of series. I particular, the covergece of series is defied i terms of the limit of partial sums. For a sequece a, the sequece of partial sum S N is defied S N = N a, which is itself a sequece. Kow the Test for Divergece: if a 0 as, the a diverges.

2 Be able to recogize a geometric series, determie whether it coverges or diverges, ad compute the value of coverget geometric series. x = { x x < diverges x ad x = { x x x < diverges x Be able to recogize ad compute telescopig sums.. Determie if the followig series coverge or diverge. If they coverge, calculate the value of the series. 4 + (b) arcta( 2 π 2 + ) (c) (d) π e =2 Covergece Tests (Sectios 8.3 ad 8.4) Kow the followig series tests, whe you ca apply them, ad how to apply them. Itegral Test: If a = f() where f(x) is a cotiuous, o egative, ad decreasig fuctio o [, ), the. If 2. If f(x)dx coverges, the f(x)dx diverges, the a coverges. a diverges. p-series Test: p { coverges p > diverges p Compariso Test: If a b, a 0, ad b 0, the. If 2. If b coverges, the a diverges, the a coverges. b diverges. a Limit Compariso Test: If a b, a 0, ad c = lim satisfies 0 < c <, b the either o of the followig two hold. 2. a ad a ad b both coverge. b both diverge. 2

3 Alteratig Series Test: If a is decreasig ad a 0 as, the ( ) a coverges. a + Ratio Test: Let a be a sequece ad L = lim coverges. The a. If L <, the 2. If L >, the a is absolutely coverget. a is diverget. 3. If L =, the ratio test is icoclusive. Kow the differece betwee absolutely coverget, coditioally coverget, ad diverget series. Be able to tell whether a give series is absolutely coverget, coditioally coverget, or diverget series.. Usig the tests itroduced i Chapter 8, determie if the followig series coverge ad diverge. Wheever a test is used, you must provide evidece that the series satisfies the hypothesis of the test ad you must cite the ame of the test. 2 + (b)! (c) ( ) + 2 ( ) l (d) (g) =2 l() (e) (h) (3 ) e 2 + (f) (i) 2 + e + 3 si ( ) Power Series (Sectios 8.5 ad 8.6) Kow the defiitio of a power series: a power series ceter at a is for some sequece of umbers c. c (x a) Kow the three covergece behaviors for a power series ad how it relates to the radius of covergece for the series: (i) The series coverges oly for x = a. The radius of covergece is R = 0. (ii) The series coverges for all real umbers x. The radius of covergece is R =. (iii) There exists a umber R > 0 such that the series coverges for x a < R ad diverges for x a > R. The radius of covergece is the R specified here. 3

4 Be able to use the ratio test to determie the radius of covergece for a power series. A power series with radius of covergece R. The the series coverges for x i (a R, a+r). Plug i a R ad a + R ito the series to determie the covergece behavior for the power series at the edpoits. Be able to fid the power series for variatio of x usig derivatives ad itegrals.. Fid the radius ad iterval of covergece for the followig power series: x (2x 3) (b) (2 ) 3 (c) 0 x 3 (d) ( 2) x 4 2. For the followig fuctios, fid a power series represetatio of the fuctios ad the radius ad iterval of covergece for the power series: f(x) = x (b) g(x) = 2 x x (c) h(x) = ( x) 2 (d) r(x) = x arcta(x 2 ) (e) s(x) = l(2 x 2 ) Taylor Series ad Approximatios (Sectios 8.7 ad 8.8) Kow what Taylor series ad Taylor polyomials are: If a fuctio f(x) ca be represeted as a power series f(x) = c (x a) for x satisfyig x a < R, the the Taylor series for f(x) cetered at a is f(x) = f () (x a)! The Taylor polyomial cetered at a of degree N for f(x) is T (x) = f () (x a)! Be able to iterpret Taylor polyomials T N (x) as approximatios for f(x). We have three types of approximatios: (Itegral Approximatio) If a ad f(x) satisfy the Itegral Test for Covergece ad S = a, the S s k 4 k+ f(x) dx

5 (Alteratig Approximatio) If for Covergece ad S = ( ) b satisfies the Alteratig Series Test ( ) b, the S s k b k+ (Taylor Iequality) If f (+) (x) M for x a d, the the remaider R (x) of the Taylor series satisfies the iequality R (x) M ( + )! x a + for x a d You will eed to kow the followig Taylor series x = x for x < e x = si(x) = ( ) (2 + )! x2+ for all x cos(x) =! x for all x ( ) (2)! x2 for all x arcta(x) = ( ) x for all x. Fid a Taylor Series expasio about the value a for the followig fuctios: f(x) = xe x at a = (b) g(x) = x 2 at a = 2 (c) h(x) = x at a = 2. Approximate the followig expressios to 4 decimal poits: ( ) (b) xe x2 dx Approximate the followig expressios usig 5 terms of a series. What is the maximum error of your approximatio? ( ) (b) x cos(x) dx (c) arcta(2) Fid the sum of the series l(3) + (l(3))2 2! Polar Coordiates (Appedix H) (l(3))3 3! +... You should be able to covert poits (x, y) i the rectagular coordiate system to poits (r, θ) i the polar coordiate system ad vice versa. 5

6 θ r x y x = r cos(θ) y = r si(θ) θ = arcta( y x ) x 2 + y 2 = r 2 You should be able to plot curves defied by fuctios i polar coordiates of the form r = f(θ). You should be able to fid the equatios for lies taget to curved defied i polar coordiates r = f(θ). dy dx = dy dθ dx dθ = f (θ) si(θ) + f(θ) cos(θ) f (θ) cos(θ) f(θ) si(θ) You should be able to fid the area of a regio defied i polar coordiates by r = f(θ) for a θ b, b A = 2 f(θ)2 dθ a You should be able to fid arc legths for curves defied i polar coordiates by r = f(θ) for a θ b, b ( ) 2 dr L = r 2 + dθ dθ a. For the followig polar fuctios, fid the area cotaied i the polar graph ad (b) the arc legth of the graph over a sigle period. r = 2 si(θ) (b) r = 4 cos(θ) (c) r = si(2θ) 6