Review of Skills
Covered in the
Summer Assignment
Reporting Numerical Answers
• Whenever reporting numbers, you MUST include
• The appropriate number of sig figs (magnitude)
• The direction (angle measurement, cardinal direction, linear
direction) if it’s a vector quantity
• The units
• Sig Figs
• Zeros to the left of the decimal and after a number are significant
• Zeros to the right of a decimal that ALSO follow a number are
significant
• Zeros in between numbers are significant
• Zeros after a number that are NOT followed by a decimal are NOT
significant.
• For numbers less than 1, zeros between the decimal and the first
digit are NOT significant
Sig Fig Examples
• For the following numbers, report the correct number of sig
figs.
Keeping Sig Figs
• When calculating, you must keep the appropriate sig figs!
• Try to keep the numbers in your calculator as much as possible
• Use the answer key
• Only round your answer to the appropriate number of sig figs
at the end of the calculation.
• When multiplying and dividing numbers, report your answer with
the same number of sig figs as the original number having the
least number of sig figs
• When adding or subtracting, report your answer to the same
decimal place as the original number having the least number of
decimals.
Applying the Sig Fig Rules
Answer the following with the correct number of significant
Figures
• Adding/Subtracting:
•
•
•
•
•
1) 9.748 - 9.67 = _______
2) 22.9 + 61.678 = _______
3) 4.285 + 1.8454 + 1.38 = _______
4) 22.8392 + 7.971 + 45.83 = _______
5) 7.5733 - 1.8 = _______
• Multiplying/Dividing:
•
•
•
•
•
1) 68 x 8 x 20 = __________
2) 6001 ÷ 79.8 = __________
3) 0.0026 x 62 = __________
4) 950 ÷ 9.73 = __________
5) 8600 ÷ 7.970 = __________
Scientific Notation
• Keeps things neat and concise.
• NO extra numbers
• Example
• Speed of light in a vacuum 300000000m/s
• Only one significant figure with many non-significant zeros
becomes….
• 3x108 m/s
• Put answers into scientific notation as often as possible
keeping the appropriate number of sig figs based on the given
information.
Orders of Magnitude
• Another way to keep numbers neat and concise
• The exponent tells you how many zeros
• Positive exponent means numbers greater than 1
• Negative exponent means numbers less than 1
• You can replace the orders of magnitude with well know
prefixes seen below
Reading the Table
1 kilometer = 1x103 meter
1 nanosecond = 1x10-9 seconds
Orders of Magnitude
• A race is 5kilometers long
• How many meters is that?
• How many centimeters?
• How many megameters?
• A flash of light lasts 0.00056 seconds
• How many milliseconds is that?
• How many microseconds?
• An object has a mass of 75kilograms
• How many grams is that?
• A molecule is 1.5x10-10 m long
• How many nanometers is it?
• How many picometers is it?
Orders of Magnitude
• A race is 5kilometers long
• How many meters is that?
• How many centimeters?
• How many megameters?
5000m
500,000cm
0.005Mm
• A flash of light lasts 0.00056 seconds
• How many milliseconds is that?
• How many microseconds?
0.56ms
560μs
• An object has a mass of 75kilograms
• How many grams is that?
75,000g
• A molecule is 1.5x10-10 m long
• How many nanometers is it?
• How many picometers is it?
15nm
0.015pm
Interpreting Graphs
• Both axes stand for actual physical quantities (check their
units)
• They y-intercept will give you the initial contitions.
• The slope of the graph usually stands for a physical quantity
also
• Divide the y-axis units by the x-axis units to determine what it is.
• The area between the graph and the x-axis also sometimes
stands for a physical quantity.
• Multiply the y-axis units by the x-axis units to determine what it is
• BEFORE ANSWERING ANY GRAPH QUESTIONS, YOU SHOULD
ADDRESS THE FOUR POINTS ABOVE.
Analyzing Graphs
• Assuming this is a graph of money in your bank account vs.
time, what can you tell me about your bank account?
money
• Remember to address the 4 points above
• talk segment by segment
• Use descriptive words like constant, rate of change, increasing,
decreasing…
time
money
time
• Initial condition: at the start of time, there is no money in
your bank account
• To A: the amount of money in your bank account increases at
a constant rate.
• A to B: the amount of money in your bank account is still
increasing, but the rate at which it is increasing is decreasing.
• At B: you stop earning money and start to lose it
• B to C: you are losing money at an increasing rate, then reach
a point where you are neither losing or gaining money
• Segment C: The amount of money in your bank account is
constant. You are not losing or gaining money.
money
time
• Segment D: You are again earning money at a constant rate
that is the same as the rate to A
• At E: you instantaneously stop earning money and start to
lose it at a constant bur faster rate as compared to how fast
you were earning money
• At F: you have the same amount of money in your account
that you did at C
• At G: you have no more money in your account at that instant
• At H: you are still losing money at a constant rate and now
you are in debt. You have negative money in your bank
account.
Translating one Graph to
Another
Rate of change of the money
Using your description of the flow of money just described,
sketch a graph that depicts the rate at which your bank account
is changing as a function of time.
time
Translating one Graph to
Another
Rate of change of the money
Address the four points to make sure this graph makes sense
time
Fraction Math 101
• When adding and subtracting
•
•
•
•
Turn mixed numbers into improper fractions
Find a common denominator
Add/subtract
Reduce
• When multiplying
•
•
•
•
Turn mixed numbers into improper fractions
Multiply all numerators
Multiply all denominators
Reduce
• When dividing
• Turn mixed numbers into improper fractions
• Keep the first, change to multiplying, reciprocate the second fraction
• Follow multiplying rules
19
25
65
56
19
16
32
17
7
22
1
15
6
18
1
6
1
19
4
1
2
2
12
Other Math Skills
• Algebra
• Solving for x
• Quadratic equation
• System of equations
• Trig
• Sohcahtoa
• Pythagorean theorem
• Graphs
• Calculating slope
• Geometry
• Area of…
• Circle
• Triangle
• rectangle
Algebra
• Solve for the indicated variable either numerically or symbolically
Solve A = bh for b
Solve P = 2l + 2w for w
Solve Q =
𝒄+𝒅
𝟐
Solve V =
𝟑𝒌
𝒕
for d
for t
Algebra
• Solve for the indicated variable either numerically or symbolically
𝑨
b=
𝒉
Solve A = bh for b
𝑷
w=
𝟐
Solve P = 2l + 2w for w
Solve Q =
𝒄+𝒅
𝟐
Solve V =
𝟑𝒌
𝒕
X=-2
−𝒍
X=-2
for d
for t
d=2Q-C
𝟑𝑲
t=
𝑽
X=-2
Using the quadratic formula
Solve for x
Using a system of equations
• Solve for x and y
Using a system of equations
• Solve for x and y
X=1
y= 1/6
X=17/9
y= -89/18
X=-3
y= 2
X=1
y= 3/2
Multiplying / Dividing with
Exponents
• Square roots
• Other exponents
• Multiplying different exponents with the same base
• Add the exponents
• Dividing different exponents with the same base
• Subtract the exponents
• Raising an exponent to a different power
• Multiply the two exponents
• Different bases
• Try to change the base by changing the exponents
• 82 x 23 = (4 x 2)2 x 23 = Answer
(22 x 2)2 xis23under
= (23)2here
x 23 = 26 x 23 = 29
Definition of a Scalar
• A quantity that has an magnitude (size or amount) and NO
direction
• Just memorize
•
•
•
•
•
•
•
Time
Mass
Temperature
Distance
Speed
Work and Energy
Charge
Definition of a Vector
• A quantity that has an magnitude (size or amount) and
direction (north, south, east, west OR up, down, left, right OR
an angle)
• MADFIVE
•
•
•
•
•
•
•
Momentum
Acceleration
Displacement
Force
Impulse
Velocity
Electric Field Strength
Vector Components
Reference line
θ
Original given direction
(North of East)
Vertical or
y – component of vector A
Ay = A sin θ
Horizontal or x – component of vector A
Ax = A cos θ
Vector Addition
• Any vectors pointing in the EXACT same direction: add the
magnitudes
• Any vectors pointing in the EXACT opposite direction: subtract
the magnitudes
• 2 vectors Pointing at EXACT right angles to each other: use
the Pythagorean theorem to find the magnitude, use trig to
find the direction
• 2 vectors pointing in any other directions
• Break the angled vector into its components
• Then follow the rules above
Same Direction
• Put them head to tail
• Draw resultant
• Add magnitudes
Magnitude: 7m+3m=10m
Direction: to the right
resultant
7m
3m
Opposite Direction
• Put them head to tail
• Draw resultant
• Subtract magnitudes
Magnitude: 7m-12m= -5m = 5m
Direction: to the left
(the negative sign means ‘to the left’)
resultant
7m
12m
Right Angles
•
•
•
•
Put them head to tail
Draw resultant
Use Pythagorean Theorem to find the magnitude
Use trig to find the direction (angle)
7m
θ
resultant
Magnitude: 7.6m
Direction: 23o south of east
3m
Other Angles
•
•
•
•
•
•
•
Put them head to tail
Break angled vector into its components
Draw resultant
7m+3.8m = 10.8m
Add/subtract all x-components
3.2m
Add/subtract all y-components
Use Pythagorean Theorem to find the magnitude of the resultant
Use trig to find the direction (angle) of the resultant
7m
Magnitude: 11.3m
Direction: 17o north of east
Ay = A sin θ
= 5m sin 40o
= 3.2m
θ
Ax = A cos θ
= 5m cos 40O
= 3.8M
40o
Translating Wave function
Equations
The y-value
at the time
t given
The amplitude
of the wave (it is
constant)
𝑦 𝑡 = 𝐴 sin 𝜔𝑡
Omega is the angular
frequency of the
wave. It can tell you
about the period of
the wave.
MUST BE IN
RADIANS!!!!
Period of a wave
2𝜋
𝑇=
𝜔
Frequency of a wave
f=
𝜔
2𝜋
Interpret this function
𝑦 𝑡 = 5 sin 2𝑡
• The amplitude of wave is…
• 5m
• Which means the maximum y-value is +5 and minimum y-value is -5
• ω is 2 which means…
• The period of the wave is π
• The frequency of the wave is 1/ π
• Period and frequency are always inverses of each other