Review Sheet - Introduction to Differential Equations | MATH 2214, Study notes of Differential Equations

Download Review Sheet - Introduction to Differential Equations | MATH 2214 and more Study notes Differential Equations in PDF only on Docsity! 1206 Review This document is meant lo serve as a review of some of the things you will be expected to know throughout the semester. GENERAL You need to be able to do baise avithmetic, algebra, geometry, and trigonometry. You should be comfortable using fractions, DO NOT be afraid of fractions! All angles in this class should be in RADIANS. GRAPHING Be able to graph with and without a calculator. You should be able to recognize when an equation is a line, a parabola, a circle etc. You will NOT be allowed a calculator on tests. You will be expected to be able to graph basic functions like: 1. Lines: y= ma+b 2, Parabolas: y = 20%, y= 2? +82, y= —Se4, a= y?, 2 = Sy? + By ete. 3. Trig functions: sin(z), cos(z), tan(a) 4, Logs and exponentials: y =e", y= a®, y= e7*, y= a", y = In(z) Note: the domain for the natural log function is 2 > 0 5. Circles and Half Circles: 2? + y® = a?, y= Va? — 27, (w@ — xo)? +(y- po) =a, y= Vat — 27 +0 6. Square Roots: y = «fa (Note: this is not the same as the graph of a = y”), y= Va $3 etc. 7. Miscellaneous: Know the basic shape of y = 2°, y = 24 etc. Be able to solve for the intersection points of two graphs, the quadratic formula is often helpful: The two x values that solve the equation ax’ + be+c = 0 are given by: ¢ = abies WHEN ALL ELSE FAILS PLOT SOME POINTS! ‘ LAWS OF EXPONENTS L. oxo = goth 2, =a? # 3. (et)b = 4. at = Yee LAWS OF LOGARITHMS 1. log, (ey) = log, (2) + Joga (y) N _ log() = log (#) — Jote(¥) ) . loge(e) = In(z) infe*)=2 loga(2”) = Plog, («) - Ine) = 1 InQl) =6 wo rnan pw  TRIGONOMETRY Kvow your lig! [tis important and you will uae it. All angles i NOT USE DEGIERS. Remember: ging = 22h ros = tte = sind = iy cos) = Foy tant = 0G nv this clags should be in RADIANS. DO oo oof = sold = voles eae = aay HC? = cesta wall = TATE You can use the formulas and figure above to remember the following: Radians | sind | cosé | tan? 1 Wid 0 You should from this table, the triangles above, or a unit circle, be able to for any special angle be able to determine. the sine, cosine etc, of it ie sin 7, cos ar etc, You should also be able to, given any value of cos@, sin @ etc. be able to determine the angle that gave you that value. IMPORTANT NOTES: 1. The sin-}(n) and the cos~*(e) functions take a number between -1 and 1 and return an ANGLE. 2, The ten+(x) function takes any number from —oo to co and returns an ANGLE 3, sin“ (e) A ma (similarly for cosine and tangent). 4, {cos(n)]? means cos(x) * ¢os(z) it is often written as cos*{x). [cos(e)|* does not mean cos(n?) HH! 5. cos? (x) + sin®(v) = 1 Be able to derive the two other similar formulas from this one. 6. sin20 = 2sinflcosd LIMITS You-need to be comfortable-taking-limits... Know..that, if you haye the quotient of two polynomials and the Swer-in-the numerator is Jargex tha the denominator and you are taking thé-liniit as Se poee to765" that tlie unit is 60; If’ the highest powers dre. the same. in the numerator’aid, denomitiator then the limit’as x.goes tO‘ o0'is the ‘ratio. of'the coefficients in front of yout highest power of x,. Ifthe ‘highest power in the denominator ‘is lar ‘an the lighest-powet in the nutaerator then ‘the liimit-as x Gods to ‘00 Is vero. REFERERCE PAGES ||| Table of Integrats Basic Forms 1. fardo= ww | odu i [ ese cot ydu = —eseu + C 2 ( udu = we +O n¥ =) 12. [anv du = tn see] + C ! A J Mal 13, [cot de = tn sin] + a {S=miulte | [sin | Vw uw f secu a = In fsecu + tan] + C : 4. fe du=e" + ‘ & . ' a 15. f eseu d= In fesew ~ cota] + C 2 5. | a" du = oo +C , , ! ! ina . 16. | a sm tee : a ‘ 4. | sin udu = —cos ue + C 7 | cos nda = sin + C 8 | sectu du = tanu + C 9. cse*w du = —colu + C 10. [sec utanudy = secu + C . 4 2. | ut fate d= qe +?) fat Fae - inte +4 ; —S : Forms Involving v wea>t : j d u : i 2. [ve Fut du ate Fat Sint +V¥e@Fr)+e i i : : 4 : i 13, [YEE y= Ya Fain a a + ie + eT ty = ~ inlet Ver FE) FC uF W : i 24, rd aw 25. | Oe = ine t Jae) + C ati « 2 2 : i 26. | Se + fare we — Sala + fetta +c a q dy 2 IN @ Til  ||}] Tegenometry Angle Measurement radians = 180° . 5 180° v= rad ind — 8 v ey 180 s=rd (8 in radians) ight Angle Trigonometry ty sin @= EP ose gE yp hyp OPP. opp a hy cos 6= oh seo 9 = DE <A d hyp adj adj dj tan @= SBR ot = adj P Trigonometeic Functions sing=> ose B= r Graphs of trigonometric Functions Foadamental Identities 1 oe = rg se T8 sin @ cos tan = 00 @ = 1 2 cot @= —— sin@ + cos? = 1 tan 8 1+ an? = sec* 1+ cota = esc78 cos(—6) = cos # sn(Z - 5) = cos } = . = _ The Law of Sines B sinA sin(— 6) = —sin @ tan(—6) = —tan @ @ The Lau of Cosines a? = b+ c* — Qe cos A b? =a +c? — QaccosB cha a? +b? ~ 2ab cos C ¥. ¥ y y= tan yesinx ye cosx , ' : om, } | S| a9 Addition and Subtraction Formulas * are i ‘a = sin(x + y) = sinx cosy + cosx siny “1 4 yo! sin(s — y) = sin x cos y ~ cos.x sin y | tot cos(x + 3) = cos x cosy — sin x siny yh yeeses ye yeseex yh ye eotx cos(x — y) = cos.x cosy + sin x sin y AR TR \ | 1 io wit | \ tanta +9) Ta any io 14 4 pod = tanly — 9) tan x — tany 7 lnk fom | Wx 7 ar —- = Jyupmy 4 | 1 4 i 1 | | 1 + tan x tan y Vv! AN 1 Y of \N eo\ HL Double-Angle Formulas sin 2x = 2 sin x cos x Irigonometric Functions of Important Angles cos 2x = cos’x — sin?x = 2 cos?x — 1 6 radians sin @ cos @ tan @ 2tan x \ tan 2. z oe 0 0 1 0 1 — tans 30° 7/6 2 3/2 3/3 45° nf4 2/2 2/2 1 Half-Angle Formulas 60° af 3/2 2 3 a _ Le cos 2x », Lt os 28 90° nf? i 9 - aa) ose 2 Tk 2 til