math review
DESCRIPTION
Math Review. Units, Scientific Notation, Significant Figures, and Dimensional analysis Algebra - Per Cent Change Solving simultaneous equations Cramers Rule Quadratic equation Coversion to radians Vectors Unit vectors Adding, subtracting, finding components Dot product Cross product - PowerPoint PPT PresentationTRANSCRIPT
Math Review• Units, Scientific Notation, Significant Figures, and Dimensional analysis • Algebra -
– Per Cent Change– Solving simultaneous equations– Cramers Rule– Quadratic equation– Coversion to radians
• Vectors– 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: 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:
Scientific Notation
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.
%Change=new−original
original100%
%Change=3.52
ms−3.40
ms
3.40ms
100%
%Change=3.5%
Simultaneous Equations2x + 5y=−11x−4y=14
FIND X AND Y
x =14 + 4y2(14 + 4y) + 5y=−1128 + 8y+ 5y=−1113y=−39y=−3x=14 + 4(−3) =2
Cramer’s Rule a1x +b1y=c1a2x+b2y=c2
x =
c1 b1c2 b2
a1 b1a2 b2
=c1b2 −c2b1a1b2 −a2b1
=(−11)(−4)−(14)(5)(2)(−4)−(1)(5)
=44 −70−8 −5
=−26−13
=2
y =
a1 c1a2 c2a1 b1a2 b2
=a1c2 −a2c1a1b2 −a2b1
=(2)(14)−(1)(−11)(2)(−4)−(1)(5)
=28 +11−8 −5
=39−13
=−3
2x + 5y=−11x−4y=14
Derivationax2 +bx+ c=0
x2 + (ba)x+ (
ca) =0
x+ (b2a
)⎡⎣⎢
⎤⎦⎥
2
−(b2a
)2 + (ca) =0
x+ (b2a
)⎡⎣⎢
⎤⎦⎥
2
=−(ca) + (
b2
4a2 )
(2ax+b)2 =4a2 −(ca) + (
b2
4a2 )⎡
⎣⎢
⎤
⎦⎥
(2ax+b)2 =b2 −4ac
2ax+b=± b2 −4ac
x=−b± b2 −4ac
2a
Complete the Square
Arc Length and Radians
r
2r =D
r =radiusD =diameterC =circumfrance
C
D=π =3.14159
C2r
=π
C =2πrC2π
=r
C2π
=Sθ=r
S =rθθ is measured in radians
θ =2π
S = r2π = C
2π rad = 360o
1rad =360o
2π= 57.3deg
rad
Sθ
Small Angle ApproximationSmall-angle approximation is a useful simplification of the laws of trigonometry
which is only approximately true for finite angles.
FOR θ ≤10o
10o =0.174532925 radians
sinθ ; θ
sin(10o ) =0.173648178
EXAMPLE
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
rA =2i + 4 j
Red arrows are the iand j unit vectors.
Magnitude =
A = 22 + 42 = 20 =4.47
rA
tanθ =y/ x=4 / 2 =2θ =63.4deg
Angle between A and x axis = θ
j
i
θ
Adding Two Vectors
rA =2i + 4 jrB=5i + 2 j
rA
rB Create a
Parallelogram withThe two vectors
You wish you add.
Adding Two Vectors
rA =2i + 4 jrB=5i + 2 jrA+
rB=7i + 6 j
rA
rB
rA +
rB
.Note you add x and y components
Vector components in terms of sine and cosiney
xθ
r
x
y
i
j
rcosθ =x
r
sinθ =yr
x =rcosθy=rsinθ
r =xi + yj
r =(rcosθ)i + (rsinθ) jtanθ =y/ x
Scalar product =
A
Bθ
AB
rA •
rB=AxBx + AyBy
rA =2i + 4 jrB=5i + 2 jrA•
rB=(2)(5) + (4)(2) =18
rA •
rB= A B cosθ
cosθ =18
20 29=0.748
θ =41.63deg
Also
AB is the perpendicular projection of A on B. Important later.
A
Bθ
AB
rA =2i + 4 jrB=5i + 2 jrA•
rB=(2)(5) + (4)(2) =18
AB =rA•
rB
B
AB =1829
=3.34
90 deg.
Also AB = A cosθ
AB = 20(0.748)AB =(4.472)(0.748) =3.34
For a Right Handed 3D-Coordinate Systems
x
y
ij
k
Magnitude of
Right handed rule.Also called cross product
z
i × j =k rr =−3i + 2 j + 5k
rr = 32 + 22 + 52
Suppose we have two vectors in 3D and we want to add them
x
y
z
ij
kr1
r2
25 1
7
r1 =−3i + 2 j + 5k
r2 =4i +1 j + 7k
Adding vectors
Now add all 3 components
r2
r
r1
ij
k
x
y
z
rr =
rr1 +
rr2
rr1 =−3i + 2 j + 5krr2 =4i +1 j + 7krr =1i + 3 j +12k
Scalar product =
rr1 •
rr2 =(−3)(4) + (2)(1) + (5)(7) =25
rr1 •
rr2
rr1 =−3i + 2 j + 5krr2 =4i +1 j + 7k
Cross Product See your textbook Chapter 3 for more information on vectorsLater on we will need to talk about cross products. Crossproducts come up in the force on a moving charge in E/Mand in torque in rotations.
Define the instantaneous velocity
Recall
(average)
as Δt 0 = dx/dt (instantaneous)
Example
Definition of Velocity when it is smoothly changing
€
x = 12 at
2
x = f (t)
v =(x2 −x1)(t2 −t1)
=ΔxΔt
v =limΔxΔt
DISTANCE-TIME GRAPH FOR UNIFORM ACCELERATION
x
t
(t+Δt)t
v Δx /Δt
x = f(t)
x + Δx = f(t + Δt)
dx/dt = lim Δx /Δt as Δt 0
.x, t
€
x = 12 at
2
x = f (t)
Δx = f(t + Δt) - f(t)
Differential Calculus: an example of a derivative
€
x = 12 at
2
x = f (t)
dx/dt = lim Δx /Δt as Δt 0
€
=f (t + Δt) − f (t)
Δt
€
f (t) = 12 at
2
€
f (t + Δt) = 12 a(t + Δt)2
= 12 a(t
2 + 2tΔt + (Δt)2)
€
=12 a(t
2 + 2tΔt + (Δt)2) − 12 at
2
Δt
€
=12 a(2tΔt + (Δt)2)
Δt
€
12 a(2t + Δt)
€
→ at
Δt → 0
€
dx
dt= at velocity in the x direction
v =at
y =cxn
dy/ dx=ncxn−1Power Rule
Chain Rule
Product Ruley(x) = f(x)g(x)dydx
=dfdx
g(x) + f(x)dgdx
y(x) =y(g(x))dydx
=dydg
dgdx
y =30x5
dydx
=5(30)x4 =150x4
y =3x2 (lnx)dydx
=2(3)x(lnx) + 3x2 (1x) =6xlnx+ 3x
dydx
=3x(2 lnx+1)
y (5x2 −1)3 g3 where g5x2 −1dydx
3g2 dgdx
3(5x2 −1)2(10x)
dydx
30x(5x2 −1)2
Three Important Rules of Differentiation
Problem 4-7 The position of an electron is given by the following displacement vector , where t is in s and r is in m.
What is the electron’s velocity v(t)?
What is the electron’s velocity at t= 2 s?
What is the magnitude of the velocity or speed?
What is the angle relative to the positive direction of the x axis?
+vx
+vy
-16
3
rr =3ti −4t2 j + 2k
rv =
drr
dt=3i −8tj
rv =
drdt
=3i −16 jvx =3m/ svy =−16m/ s
v = 32 +162 =16.28m/ s
φ =tan−1(−16
3) = tan−1(−5.33) = −79.3deg
rv
How far does it go?
Distance equals area under speed graph regardless of its shape
Area = x = 1/2(base)(height) = 1/2(t)(at) = 1/2at2
v=dx/dt
t
v= at
€
x = Δx ii=1
N
∑ = v iΔti = atiΔtii=1
N
∑i=1
N
∑
vi
Δti