MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx

Size: px

Start display at page:

Download "MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx"

Kellie Cameron Hensley
5 years ago
Views:

1 MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of ths (N). You do ot hav to valuat th itgrals. d d a. b. cos d c. + 8 Itgrat:. d. k ( 5 + ) d. ( ) 5si d 5. cos d. d 7. arcta d 8. si cos d 9. cos d 0. - si + + d. ( ) d Itgrat th followig usig th tabl of itgrals o th isid rar book covr. d cos d. d cos d. Writ ( + )( ) as th sum of two partial fractios. 7. Th vlocity V at tim t of a poit movig alog a coordiat li is of t = 0, fid a formula for its positio s at tim t. V = t t ft/sc. If th poit is at th origi 8. Food is placd i a frzr. Aftr t hours, th tmpratur of th food is chagig at a rat of whr R is i dgrs F/hr. How much has th tmpratur droppd i th first two hours? R= 0 0.t

2 9. a. Usig th tabl blow, show how to us = subitrvals ad trapzoids to approimat whr r(t) is th populatio rat i thousads pr yar at a tim t yars aftr Ja., r() t dt t 5 r(t) b. What dos th approimatio i part (a) tll about th populatio? B spcific ad iclud th corrct uits. 0. a. Giv th corrct four-dcimal plac approimatio from a calculator program for with 0 midpoit rctagls (M0). b. Fid th corrct aswr to four dcimal placs for d with your calculator. c. What is th rror i th approimatio for M0 to four dcimal placs i part a? d. Solv dy t(si( t ) = assumig y 0. Eprss y i trms of t. dt y dy. Solv = y d whr y (0) = Eprss y i trms of. Evaluat usig L'Hopital's Rul or othr aalytical mthods. si cos. lim. 0 5 lim +. a. Tll why you caot us L'Hopital's Rul to fid si 5. lim cos lim. + 0 b. Evaluat th limit i part a ad giv vidc to support your aswr. Evaluat ach limit blow ad giv vidc to support your aswr. 7. lim ( ) 8. lim Dtrmi th covrgc or divrgc of ach itgral. If covrgt, fid th valu. 9. d 0. l d. 0 si d. Fid th volum of th solid formd wh th rgio boudd by th -ais ad th curv about th -ais. y = is rvolvd

3 . a. Writ a itgral qual to th arc lgth of y = from (,) to (,9). b. Approimat th arc lgth with a calculator program.. Fid th ara of th rgio closd by 5 5. If g ( ) d = fid th avrag valu of ( ) 0 y = ad y = +. g o [0,5].. A sprig has a atural lgth of ft. A forc of 80 lb strtchs it to a lgth of ft. Fid th work do i strtchig it from a lgth of 5 ft. to a lgth of ft. 7. A tak i th shap of a ivrtd co cotais som watr. Th tak has diamtr 0 ft at th top ad is 5 ft dp (S figur). Th watr is 8 ft dp ad has dsity p =.5 lb/ft. Writ a itgral that rprsts th work rquird to pump all th watr ovr th top rim. B sur to draw a pictur showig how you ar sttig th problm up. Do ot valuat th itgral. 0 ft 5 ft 8 ft 8. Lt dy = + y d. Draw th dirctio fild tagts at th poits (-,) ad (,) o a y-graph. dy = + with = 0 by first drawig th slop fild o your calculator. Th sktch a d solutio curv usig th slop fild. 9. Sktch a solutio to y y ( ) dy 0. Cosidr th diffrtial quatio = y. Show how to us Eulr's Mthod (without a calculator d program) to approimat y(.) by startig at (,) ad usig stps of = 0... Lt + a= a. Writ th first trms of th squc { } = b. Dos { a} c. Dos = a hav a limit? If so, fid it. a = covrg or divrg? Giv a raso for your aswr. 00 g of a radioactiv substac dcays. Aftr 0 days, 80 g rmai. How much would rmai aftr days?

4 . A ball is droppd from a hight of 0 ft. Each tim it hits th floor, th ball rbouds to hight. Fid th total distac th ball travls.. Dtrmi th sum of + = of its prvious 5. Dtrmi th covrgc or divrgc of th followig sris. Justify your aswr. + a. b.! = + c. = + = l + ( ) ( ). Dtrmi if th sris = + covrgs or divrgs. Justify your aswr. 7. Fid a powr sris rprstatio of + 8 ad stat its itrval of covrgc. 8. ( ) ( + ) Dtrmi th radius ad itrval of covrgc of. = 9. Fid th MacLauri Sris for l ( + ) ad dtrmi how may trms must b usd i this sris to fid l.0 accurat to 7 dcimal placs. 50. Lt f ( ) = l ( + ). Fid th Taylor Sris for ar 0 by takig drivativs. Writ th first ozro trms of th sris. 5. a) Fid th Taylor polyomial of dgr for y = cos() with ctr a = 0. b) Us th rsult to approimat cos(0.). c) Estimat th magitud of th rror i your approimatio i part b. 5. Fid th Taylor Sris for ctrd at a = ad us th Ratio Tst to show that this sris covrgs for all. 5. Covrt to polar coordiats. Us r > 0 ad 0 θ < π. a. (, -) b. (, ) 5. Covrt th polar quatio c. (,) r = to a quatio i ad y. si θ 55. Sktch th graph of r = cosθ without usig a graphig calculator. 5 a. Fid th ara isid of th rgio isid r si ( θ ) b. Fid th ara of th rgio that is isid r si ( θ ) =. = ad outsid r =.

5 REVIEW PROBLEMS - Aswrs. 5. a. PF b. N c. U. k + C k. - l - l + + l - + C 5 cos C 7.. ( ) + C 5. ( ). ta + C 8. arcta l + + C cos 5 + C l si + C 0. + l + C C arcta + C 0 5. cos si + cos si + + C l + C 5. si + cos si + C t t S = o F = b. Th populatio icrass by about thousad from Ja., 009 to Ja., a. ( ) 0. a. 0. b. 0.9 c y =.. cos( y = t ) + C 5.. a. It is ot i th form 0 0 or. b. (look at th graph) Covrgs to 0. Divrgs. Covrgs to. 5π 5

6 . a. + d b ft - lb 8 8 W y y dy y y dy 9 7. = π ( 5 )(.5) = 7.78π ( 5 ) Hit: slop at (-,) is - ad slop at (,) is y a.,,, 7 b. ys, c. divrgs by th Tst for Divrgc. 7. g. 80 ft a. C (Ratio Tst) b. D (compar to ) c. D (itgral tst). C ( ) = ( 8 ) for = ad first trms must b usd. < < , ad R = a) + b) c) approimatly (. ) 8 = 8.9 0!

7 5. 5. a. = 0 ( )! 7π, b. π, 7 c. 5π, 5. y = 55. a circl with radius ad ctr at (,0) 5. a. π b. π + or.9 8rviw.last rvisd summr0

Review Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2

Review Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2 MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm