Math Review (15-19)

Question 1

write 100 x 10e4 as 1 x 10e?

write -50 x 10e-6 as -5.0 x 10e?

Answer 1

100 x 10e4 = 1.00 x 10e6

-50 x 10e-6 = -5.0 x 10e-5

Question 2

write 0.000000004 in scientific notation.

write 7.88 x 10e-4 as a number.

Answer 2

4 x 10e-9 (nano prefix)

7.88 x 10e-4 = 0.0007888
note its not 4 zeros b/c the 7 also comes before decimal place

Question 3

What are the multiplication, division, and power rules of exponents?

Answer 3

1. b^p x b^q = b^(p + q)
2. b^p / b^q = b^(p - q)
3. (b^p)^q = b^(p x q)

Question 4

what is b^0?

Answer 4

1 unless b = 0.

Question 5

t or f, (7 x 10^4)^2 = 49 x 10^8?

Answer 5

true

Question 6

What are the three laws of radicals?

Answer 6

1. root(ab) = root(a) x root(b)
2. root(a/b) = root(a) / root(b)
3. xroot(a^y) = a^(y/x)

e.g. cube-root(a) = a^(1/3)

Question 7

what is the formula for dividing fractions? draw it.

Answer 7

a/b / c/d = ((ad) + (cb)) / bd

Question 8

what are the two key rules of inequalities?

Answer 8

1. if you multiply each side by a negative number, the inequality sign must flip
2. if both sides are positive numbers, and you take both sides reciprocals, then the inequality sign must flip.

Question 9

what are the two most common Pythagorean triples?

Answer 9

1. 3, 4, 5

2. 5, 12, 13

Question 10

t or f, any multiple of a Pythagorean triple, is another Pythagorean triple?

Answer 10

True

e.g. 3,4,5 (x2) = 6,8,10

Question 11

Explain the dimensions of a 45,45,90 triangle. draw it.

Answer 11

both sides have a length of 1, the hypotenuse is the square-root of 2.

Question 12

Explain the dimensions of a 30,60,90 triangle. draw it.

Answer 12

across from the 30 = 1
across from the 60 = square-root of 3
the hypotenuse = 2

Question 13

t or f, when calculating sin, cos, and tan for any angle around a grid, r is always positive.

Answer 13

true. Recall that a^2 + b^2 = c^2 (thus r is always positive)
then x and y depend on which quadrant they lie in

Question 14

t or f, sin = y / r, cos = x / r

Answer 14

True

Question 15

t or f, sin^-1 (1/2) may equal 30 OR 150 degrees.

Answer 15

true! For any any angle, multiple different rotations around a grid give the same answer (e.g. -330 is also 1/2).
So if asked for the inverse to find an angle, apply the context of the question.

Question 16

What is a radian?

Answer 16

If you draw circle and then cut a pie slice of that circle, 1 radian is the angle of the pie where the radius equals the the arc-length.

Question 17

What is the formula for circumference? For area of a circle?

Answer 17

Circumference = π x 2r (or πd)

area of a circle = π x r^2

Question 18

What is the radian value for the following degrees values?
360 
180 
90 
60
45
30

Answer 18

360 = 2π
180 = π
90 = π / 2 
60 = π / 3 
45 =  π / 4
30 = π / 6

Question 19

What is the formula to convert between radians and degrees?

Answer 19

Angle in radians = 180 / π = angle in degrees

angle in degrees = π / 180 = angle in radians

unit analysis

Question 20

t or f, radians have no units

Answer 20

true, since they divide length by length, units cancel out.

Question 21

t or f, if both quantities are vectors, then A + B = B + A

Answer 21

True, the direction vector will be in the exact same direction (obviously the magnitudes are same)

Question 22

t or f, A - B will point in the opposite direction of B - A.

Answer 22

True! and thus, A - B does not equal B - A

Question 23

what simple rule do we use when drawing vector subtraction?

Answer 23

We convert A - B to -B + A and then draw the negative vector first (following tip-to-tail rules)

Question 24

What occurs if we multiple a vector A by

1. k
2. -k
3. 0

Answer 24

A x k = changes magnitude of the vector by the factor of k but does not change direction
A x -k = changes magnitude of the vector by the factor of k AND changes the direction
A x 0 = 0 magnitude 0 direction (but still technically a vector)

Question 25

t or f, the magnitude of any vector can be found by breaking it down into its x and y components, and then applying Pythagorean’s theorem or trig.

Answer 25

True, this is core to most physics concepts involving a vector (e.g. force, kinematics, etc).

Question 26

Graphically, how can we identify if x and y are proportional?

Answer 26

If x and y are linear AND their line goes through the graphs origin (0,0) then they are proportional. Note that if they do NOT go through the origin, then they have been translated (addition / subtraction is now involved not just multiplication).

inverse proportions have a different graph altogether

Question 27

How are I and T related in I = xT^4 ?

Answer 27

I = xT^4

I is proportional to T^4
if T is increased by 2, then I is increased by (2)^4 = 16

Question 28

Explain the 4 log laws: multiplication, division, power, and number = 1.

Answer 28

log (xy) = log(x) + log(y)

log(x/y) = log(x) - log(y)

log(x)^y = y x log(x)

log(1) = 0
since b^0 = 1

Question 29

What is the natural log?

Answer 29

Ln which is base e = 2.7

Question 30

Approximate Log(2), Log(3), and Log(4)

Answer 30

Log(2) =  0.3
Log(3) = 0.5
Log(4) = 0.7

Question 31

If [H+] = m x 10^-n. Explain how you find the pH value.

Answer 31

pH = -log[H+] 
pH = -[log(m) + log(10^-n)] factor in negative sign 
pH = -log(m) + n 

We approximate log(m)

the pH will be closer to (n-1) if m > 3
the pH will be closer to n if m < 3

Question 32

If pH = n.m, explain how you find [H+].

Answer 32

[H+] = 10^-pH 
[H+] = 10^-n.m 

[H+] = y x 10^-(n+1)

e.g. if it was -11.5 we split into 0.5 and -12.

if m > 3, then y is closer to 1
if m < 3, then y is closer to 9