Math Review• Units, Scientific Notation, Significant Figures, and Dimensional analysis • Algebra -
– Per Cent Change– Solving simultaneous equations– Cramers Rule– Quadratic equation
• Trigonometry and geometry– sin, cos, and tan, Pythagorean Theorem, 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 is between 1.95 and 2.05, whereas 2.00 is between 1.995 and 2.005.
The number of significant figures represents
the accuracy with which a number is known.
Trailing zeroes with no decimal point are not
significant. This has only 2 significant figures.
1200 is between 1150 and 1250,
whereas
1200. is between 1199.5 and 1200.5.
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
Quadratic FormulaEQUATION:
ax2 +bx+ c=0
SOLVE FOR X:
x =−b± b2 −4ac
2a
SEE EXAMPLE NEXT PAGE
Example2x2 + x−1=0
a =2b=1c=−1
x =−1± 12 −4(2)(−1)
2(2)
x=−1± 9
4=−1±34
x=−1−34
=−1
x=−1+ 34
=12
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
Scalars and Vectors
Vectors and Unit Vectors
• Representation of a vector : has magnitude and direction– i and j 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
θ
tanθ =y/ x=4 / 2 =2θ =63.4deg
Angle between A and x axis =
j
i
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
Vectors in 3 Dimensions
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 vectorsWhen we get to rotations we will need to talk about cross products. Also in E/M.
Differential Calculus
Define the instantaneous velocity
Recall
(average)
as t 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)
=xt
v =limxt
DISTANCE-TIME GRAPH FOR UNIFORM ACCELERATION
x
t
(t+t)t
v x /t
x = f(t)
x + x = f(t + t)
dx/dt = lim x /t as t 0
.x, t
€
x = 12 at
2
x = f (t)
x = f(t + t) - f(t)
Differential Calculus: an example of a derivative
€
x = 12 at
2
x = f (t)
dx/dt = lim x /t as t 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
Integral Calculus
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
ti
Integration:anti-derivative
€
atiΔtii=1
N
∑ = at0
t f∫ dt where Δt i → 0 and N → ∞
x = 12 at 2
at0
t f
∫ dt= 12
at20
tf = 12
a (tf2 −0) =
12
a tf2
Vector Questions 1. Consider three vectors:
jiCjiBjiA35523203+−=+=+=rrr
a. Draw the three vectors. b. What is the length or magnitude ofAr, BrandCr? c. What is the angle between ArandCr, ArandBr, BrandCr?
2. Consider three vectors:
kjiCkjiBkjiA530172264++=−+=−+=rrr
a. What is the length or magnitude ofAr, also written asAr?
b. Write the expression for 2Ar. c. What is BArr+ ? d. What is ACrr− ? e. What is ACrr× ? f. What is the magnitude of ACrr× ? g. What is CBrr⋅? h. What is the angle between Ar and Cr ? i. Does CBrr⋅equal BCrr⋅? j. How is ACrr× and CArr×related? k. Give an example of the use of dot product in Physics and explain. l. Give an example of the use of cross product in Physics and explain. m. Imagine that the vector Ar is a force whose units are given in Newtons. Imagine vector Bris a radius vector through which the force acts in meters. What is the value of the
torque )(Frrrr×=τ, in this case?
n. Now imagine that Arcontinues to be a force vector and Cr is a displacement vector whose units are meters. What is the work done in applying force Ar through a displacementCr?
o. What is the vector sum of a vector Dr given by 40 m, 30 degrees and a vector Ergiven by 12 m, 225 degrees? Use the method of resolving vectors into their components and then adding the components
Differentiation PracticeQUESTION: Differentiate the following values with respect to x, t, or z. And let a and b be
constants.
1. nxy= 2. 5xy= 3. ay=
4. 3333xxy+=
5. )12)(5( 32−=xaxy 6. xysin= 7. xaycos= 8. )()(xgxfy= 9. xxysin3= 10. ))((xgfy= 11. axysin= 12. xey=
13. axey+−=2 14. xyln=
15. 21xxy+=
16. xxxy−=ln 17. 32zy= 18. 106021223 −+−=ttty
19. xxyln=
Integration Practice1. ∫dxxn2. ∫dx3. ∫xdx4. ∫adx5. dybyay∫±)( 236. ∫−dzz37. ∫drrc28. ∫dxx19. ∫dxeax10. ∫+dtatv)(011. ∫θθdcos12. ∫btdtsin13. dxx∫31214. dxdxdgf∫⎟⎠⎞⎜⎝⎛, wherefandgare both the functions of x.15. ∫xdxxcos16. ∫xdxxsin17. ∫axdx2sin18. ∫±22axdx