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