DARTMOUTH MATH 5 - SOME BASIC MATH REVIEW

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Math 5: Some Basic Math ReviewAlex BarnettMarch 2007. early version. more things may be added1 Symbols• < less than, > greater than• ≪ much less than, ≫ much greater than (much usually meaning at least 10 times)• ≈ approximately equal to (e.g. 10% err or)• ∼ “of order” (i.e. roughly same size as, but could be off by factor 2 or 3)• ∝ proportional to (e.g. area of circle A ∝ r2, time ∝ 1/speed when distance held constant)• ≡ a definition, as opposed to an equation to solve• ⇒ “therefore”, “it follows th at”• x · y another way of writing x × y or simply xy• |x| absolute value of x, that is, x if x > 0 but −x if x < 02 Handling large and small numbersUse powers of ten to write in exponent notation, i.e. 107= 10, 000, 000, 10−5= 0.00001. So,• Radius of Earth ≈ 6, 400, 000 m = 6.4 × 106m = 6, 400 km.• Radius of human hair ∼ 0.00005 m = 5 × 10−5m = 50 µm.• Note common scientific abbreviations: k (kilo) = 103, M (mega) = 106, ; m (milli) = 10−3, µ(micro) = 10−6, n (nano) = 10−9, p (pico) = 10−12··· Can you thin k of common words whichhave these abbreviations in them? (e.g. kilometer).• Rule for multiplying numbers in exponent notation is (a × 10n)(b × 10m) = ab × 10n+m. Fordividing, use (a × 10n)/(b × 10m) = (a/b) ×10n−m.Significant digits: e.g. 1/700 = 0.001428571428 ··· = 0.001429 (to four significant digits) or =0.00143 = 1.43 × 10−3(to three significant digits). In this course, you won’t need to quote answersto more than 3 or 4 significant digits; in fact it’s often meaningless to do so.PRACTICE1. What is the cross-sectional area of the human hair mentioned above? Express your answer inboth (µm)2and in m2.2. What is 1.999 expressed to 3 significant digits? How about 2.003? What is the maximumpercentage error you can cause by expressing an exact number to only 3 significant digits?3. What is the US annual military budget ($400 billion) expressed in exponent notation?3 Powers, logarithm, exponentialYou take take any number to the power of anything, e.g. 23= 2×2×2 = 8, or 3−2= 1/(3×3) = 1/9,or 161/2=√16 = 4. [But beware non-integer powers of negative numbers → get a complex number,yuk. . . ] The power you take it to is called the exponent. Useful things:• an· am= an+m1• (an)m= anm• a−n= 1/anLogarithm (abbreviated to log) is the inverse of taking a power, i.e. it gives you the exponent.Logs are very common, for instance: the (semitone) scale of musical notes in Western music islogarithmic (every octave is a doubling in frequency), and sound intensity is measured in decibels(dB), proportional to the log of the intensity.Base ten. If y = 10xthen the inverse of this is x = log10y.Base “e”. Defined as e = 2.7182818 ··· This is the “natural” base for taking exponents and loga-rithms. If y = exthen the inverse of this is x = log y or less ambiguously written ln y. Useful things(apply to both base e and base 10):• log(ab) = log a + log b. Useful since multiplication has become adding.• log(1/a) = −log a• log an= n log aExponential growth/decay: y(t) = eαt. (t could be time, or anything).• α = growth/decay rate = 1/τ where τ = ‘time constant’ (roughly, how long it is taking).• In time τ , y gets e times bigger/smaller; in 2τ, gets e2times bigger/smaller; etc.• α > 0 gives growth (e.g. b acteria multiplying exponentially), α < 0 gives decay (e.g. amplitudeof damped oscillator once hit).012345678y = etααα = 1/τpositiveαnegative−τ2ττ0e1/et4 Dimensions, areas, volumesWe live in a three-dimensional (3d) world (we can move up/down, left/right and back/forward, giving3 independent directions). We deal with s ituations with other numbers of dimensions:• 1d: motion along a line. Amount of s tuff = length (un its of m). Need 1 coordinate to describemotion (e.g. x).• 2d: motion on a flat plane. Amount of stuff = area (units of m2). Need 2 coordinates todescribe motion (e.g. x, y).• 3d: motion in our world. Amount of stuff = volume (units of m3). Need 3 coordinates todescribe motion (e.g. x, y, z).Area of a square (2d) of side length a is A = a2; volume of cube (3d) of side length a is V = a3.Generally, it’s adin a d-dimensional world.Area/volume of other simple geometric shapes:2• Circle (2d) radius r has perimeter 2πr and area πr2.• Sphere (3d) radius r has surface area 4π r2and volume (4/3)π r3.• Triangle (2d) has area of h alf base times height.PRACTICE1. One liter (1 l) is the volume of a cube of 10 cm side length. What is 1 l in m3? How manymm3in a liter?2. What is the surface area of the Earth (a sphere)?5 Solving equationsRoman and Greek symbols a, t, ω, etc stand for numbers, sometimes with units too. e.g. ω = 3rad s−1, in which ω is an (angular) frequency. You could call this a instead, but in physics certainsymbols are used by convention, and not others (follow by example).You want to m anipulate a given equation to get the an equation of the form “just the thing youwant = some other stuff”. The only rules are:• you must do th e same thing to both sides of the equation.• you can cancel identical things out on top and bottom of fraction, (as long as they are notequal to zero.)• you can factorize things, e.g. ab + ac = a(b + c).Dimensions check. Dimensions of a quantity means how many powers of mass (M), length (L),and time (T) it has. This is common for science but we won’t emphasize it in this math course.Mostly we’ll deal with time. Valid equations must have same dimensions on each side. In otherwords a f requency cannot be equal to a time, s uch as f = 4T , because one has dimensions 1/T, theother T, and the number 4 has n o dimensions. The correct formula f = 1/T has same dimen s ions1/T on both sides. Check your dimensions agree otherwise you’ve mad e a mistake!1. Solve the following equations for x: 3x + 4 = 5x − 6, 3x2= 0.03, 5e−x/4= 0.5, a =√4x,2 + 3 log(a/x) = y, 2x= a.2. What are the dimensions of frequency? width in frequency of a peak in a frequency responsecurve? time constant τ? decay rate α? quality factor Q?3. I have a bottle exactly twice as long as another bottle, and the same shape in every respect(i.e. it is an exact scale model). What is the ratio of the bottles’ surface areas? The ratio oftheir volumes?4. A tuning fork’s amplitude dies with exponential decay, time constant 1 second. How long doesit take for the amplitude to be 1% of the original? Does the


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