Math Review - University of Virginia

Math Review - University of Virginia

Math Review Units, Scientific Notation, Significant Figures, and Dimensional analysis Algebra Vectors Per Cent Change

Solving simultaneous equations Cramers Rule Quadratic equation Coversion to radians Unit vectors Adding, subtracting, finding components Dot product Cross product Examples Derivatives Rules Examples Integrals Examples The system of units we will use is the Standard International (SI) system; the units of the fundamental quantities are: Length meter

Mass kilogram Time second Charge - Coulomb Fundamental Physical Quantities and Their Units Unit prefixes for powers of 10, used in the SI system: Scientific Notation Scientific notation: use powers of 10 for numbers that are not between 1 and 10 (or, often, between 0.1 and 100); exponents add if multiplying and subtract if dividing: Accuracy and Significant Figures If numbers are written in scientific notation, it is clear how many significant figures there are: 6 1024 has one 6.1 1024 has two 6.14 1024 has three and so on. Calculators typically show many more digits

than are significant. It is important to know which are accurate and which are meaningless. Other systems of units: cgs, which uses the centimeter, gram, and second as basic units British, which uses the foot for length, the second for time, and the pound for force or weight all of these units are now defined relative to the SI system. Accuracy and Significant Figures The number of significant figures represents the accuracy with which a number is known. Terminal zeroes after a decimal point are significant figures: 2.0 has 2 significant figures 2.00 has 3 significant figures The number of significant figures represents the accuracy with which a number is known. Trailing zeroes with no decimal point are not significant. A number like 1200 has only 2

significant figures whereas 1200. has 4 significant figures. Dimensional Analysis The dimension of a quantity is the particular combination that characterizes it (the brackets indicate that we are talking about dimensions): [v] = [L]/[T] Note that we are not specifying units here velocity could be measured in meters per second, miles per hour, inches per year, or whatever. Problems Involving Percent Change

A cart is traveling along a track. As it passes through a photogate its speed is measured to be 3.40 m/s. Later, at a second photogate, the speed of the cart is measured to be 3.52 m/s. Find the percent change in the speed of the cart. new original %Change = 100% original 3.52 %Change = m m 3.40 s s 100% m 3.40 s %Change =3.5%

Simultaneous Equations 2x + 5 y=11 x 4 y=14 FIND X AND Y x =14 + 4 y 2(14 + 4 y) + 5 y=11 28 + 8 y+ 5 y=11 13y=39 y=3 x =14 + 4(3) =2 Cramers Rule a1 x + b1 y=c1 a2 x + b2 y=c2 c1 b1 c2 b2 c1b2 c2b1 x= =

a1 b1 a1b2 a2b1 a2 b2 = (11)(4) (14)(5) 44 70 26 = = =2 (2)(4) (1)(5) 8 5 13 a1 c1 a c a c a2 c1 y= 2 2 = 1 2 a1 b1 a1b2 a2b1 a2 b2 = (2)(14) (1)(11) 28 + 11 39 = =

=3 (2)(4) (1)(5) 8 5 13 2x + 5 y=11 x 4 y=14 Quadratic Formula EQUATION: 2 ax + bx + c =0 SOLVE FOR X: 2 b b 4 ac x= 2a SEE EXAMPLE NEXT PAGE Example 2

2x + x 1 =0 a =2 b =1 c =1 1 12 4(2)(1) x= 2(2) 1 9 1 3 x= = 4 4 1 3 x= =1 4 1 + 3 1 x= = 4 2

Derivation ax 2 + bx + c =0 b c 2 x + ( )x + ( ) =0 a a 2 b b 2 c x + ( 2 a) ( 2 a) + ( a) =0 2 2 b c b

x + ( ) =( ) + ( 2 ) 2a a 4a 2 c b 2 2 (2 ax + b) =4 a ( ) + ( 2 ) 4a a (2 ax + b)2 =b2 4 ac 2 ax + b = b2 4 ac

b b2 4 ac x= 2a Complete the Square Arc Length and Radians r =radius D =diam eter C =circum france 2r = D C = =3.14159 D C = 2r C =2r C =r 2

C S = =r 2 r S S =r is measured in radians =2 S = r2 = C 2 rad = 360 o 360 o 1rad = = 57.3 deg rad 2 Small Angle Approximation Small-angle approximation is a useful simplification of the laws of trigonometry which is only approximately true for finite angles. o

10 FOR sin ; EXAMPLE o sin(10 ) =0.173648178 o 10 =0.174532925 radians Scalars and Vectors Vectors and Unit Vectors Representation of a vector : has magnitude and direction. In 2 dimensions only two numbers are needed to describe the vector i and j are unit vectors angle and magnitude x and y components

Example of vectors Addition and subtraction Scalar or dot product Vectors r A j i r A =2i + 4 j Red arrows are the i and j unit vectors. Magnitude = A = 2 2 + 4 2 = 20 =4.47 Angle between A and x axis =

tan = y/ x =4 / 2 =2 =63.4 deg Adding Two Vectors r A =2i + 4 j r B =5i + 2 j r A r B Create a Parallelogram with The two vectors You wish you add. Adding Two Vectors r A

r r A+B r B r A =2i + 4 j r B =5i + 2 j r r A + B =7i + 6 j Note you add . x and y components Vector components in terms of sine and y cosine r r

y j x i x cos = x r sin = yr x =rcos y=rsin r = xi + yj r =(rcos )i + (rsin ) j tan = y/ x r r

Scalar product = A B = Ax Bx + AyBy r A =2i + 4 j r B =5i + 2 j r r A B =(2)(5) + (4)(2) =18 A B AB Also r r A B = A B cos 18 cos = =0.748 20 29 =41.63deg

AB is the perpendicular projection of A on B. Important later. A B 90 deg. AB r A =2i + 4 j r B =5i + 2 j r r A B =(2)(5) + (4)(2) =18 r r A B AB = B 18

AB = =3.34 29 Also AB = A cos AB = 20(0.748) AB =(4.472)(0.748) =3.34 Vectors in 3 Dimensions For a Right Handed 3D-Coordinate Systems y j i k x

z i j= k Right handed rule. Also called cross product r r =3i + 2 j+ 5 k Magnitude of r r = 32 + 2 2 + 5 2 Suppose we have two vectors in 3D and we want to add them y r1 =3i + 2 j+ 5 k 2 j 5

i x k r1 7 r2 z 1 r2 =4 i + 1 j+ 7 k Adding vectors Now add all 3 components y r r r

r =r1 + r2 r r1 =3i + 2 j+ 5 k r r2 =4 i + 1 j+ 7 k r r =1i + 3 j+ 12 k j i k r r2 r1 z x r r

Scalar product = r1 r2 r r1 =3i + 2 j+ 5 k r r2 =4 i + 1 j+ 7 k r r r1 r2 =(3)(4) + (2)(1) + (5)(7) =25 Cross Product See your textbook Chapter 3 for more information on vectors Later on we will need to talk about cross products. Cross products come up in the force on a moving charge in E/M and in torque in rotations. Differential Calculus Definition of Velocity when it is smoothly changing Define the instantaneous velocity Recall (x2 x1 ) Dx v=

= (t2 t1 ) Dt Dx as Dt v =lim Dt Example x = 12 at 2 x = f (t) (average) 0 = dx/dt (instantaneous) DISTANCE-TIME GRAPH FOR UNIFORM ACCELERATION v = Dx /Dt dx/dt = lim Dx /Dt as Dt 0 x + Dx = f(t + Dt)

x x = 12 at 2 x = f (t) Dx = f(t + Dt) x, t f(t) x = f(t) . t (t+Dt) t Differential Calculus: an example of a derivative

x = 12 at 2 f (t) = 12 at 2 x = f (t) f (t + t) = 12 a(t + t) 2 dx/dt = lim Dx /Dt as Dt = 12 a(t 2 + 2tt + (t) 2 ) 0 2 2 2 1 1 f (t + t) f (t) a(t +

2tt + (t) ) at 2 = =2 t t a(2tt + (t) 2 ) = t 1 2 dx = at dt

v =at = 12 a(2t + t) at t 0 velocity in the x direction Three Important Rules of Differentiation Power Rule Product Rule y =cxn dy/ dx =ncxn 1

y(x) = f(x)g(x) dy df dg = g(x) + f(x) dx dx dx y(x) = y(g(x)) Chain Rule dy dy dg = dx dg dx y =30 x5 dy =5(30)x4 =150 x4 dx y =3x2 (ln x) dy 1 =2(3)x(ln x) + 3x2 ( ) =6 x ln x + 3x

dx x dy =3x(2 ln x + 1) dx y =(5 x2 1)3 = g3 where g=5x 2 1 dy 2 dg =3g =3(5 x2 1)2 (10 x) dx dx dy =30 x(5 x2 1)2 dx Problem 4-7 The position of an electron is given by the following r 2 displacement vector r =3ti 4 t j+ 2 k , where t is in s and r is in m. What is the electrons velocity v(t)?

r r dr v = =3i 8tj +vy dt What is the electrons velocity at t= 2 s? vx =3m / s r dr v = =3i 16 j dt vy =16 m / s 3 +vx f -16 What is the magnitude of the velocity or speed?

v = 3 + 16 =16.28 m / s 2 2 What is the angle relative to the positive direction of the x axis? 16 j =tan ( ) = tan 1 (5.33) = 79.3deg 3 1 r v Integral Calculus How far does it go?

v=dx/dt v= at vi Dti t N N N i=1 i=1 i=1 x = x i = v it i = at it i Distance equals area under speed graph regardless of its shape

Area = x = 1/2(base)(height) = 1/2(t)(at) = 1/2at2 Integration:anti-derivative N at t = i i i=1 tf 0 tf 0 atdt where t i 0 and N 1 2 atdt = at

2 x = 12 at 2 tf 0 1 1 2 = a (tf 0) = a tf2 2 2

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