Math Review (15-19) Flashcards
write 100 x 10e4 as 1 x 10e?
write -50 x 10e-6 as -5.0 x 10e?
100 x 10e4 = 1.00 x 10e6
-50 x 10e-6 = -5.0 x 10e-5
write 0.000000004 in scientific notation.
write 7.88 x 10e-4 as a number.
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
What are the multiplication, division, and power rules of exponents?
- b^p x b^q = b^(p + q)
- b^p / b^q = b^(p - q)
- (b^p)^q = b^(p x q)
what is b^0?
1 unless b = 0.
t or f, (7 x 10^4)^2 = 49 x 10^8?
true
What are the three laws of radicals?
- root(ab) = root(a) x root(b)
- root(a/b) = root(a) / root(b)
- xroot(a^y) = a^(y/x)
e.g. cube-root(a) = a^(1/3)
what is the formula for dividing fractions? draw it.
a/b / c/d = ((ad) + (cb)) / bd
what are the two key rules of inequalities?
- if you multiply each side by a negative number, the inequality sign must flip
- if both sides are positive numbers, and you take both sides reciprocals, then the inequality sign must flip.
what are the two most common Pythagorean triples?
- 3, 4, 5
2. 5, 12, 13
t or f, any multiple of a Pythagorean triple, is another Pythagorean triple?
True
e.g. 3,4,5 (x2) = 6,8,10
Explain the dimensions of a 45,45,90 triangle. draw it.
both sides have a length of 1, the hypotenuse is the square-root of 2.
Explain the dimensions of a 30,60,90 triangle. draw it.
across from the 30 = 1
across from the 60 = square-root of 3
the hypotenuse = 2
t or f, when calculating sin, cos, and tan for any angle around a grid, r is always positive.
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
t or f, sin = y / r, cos = x / r
True
t or f, sin^-1 (1/2) may equal 30 OR 150 degrees.
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.
What is a radian?
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.
What is the formula for circumference? For area of a circle?
Circumference = π x 2r (or πd)
area of a circle = π x r^2
What is the radian value for the following degrees values? 360 180 90 60 45 30
360 = 2π 180 = π 90 = π / 2 60 = π / 3 45 = π / 4 30 = π / 6
What is the formula to convert between radians and degrees?
Angle in radians = 180 / π = angle in degrees
angle in degrees = π / 180 = angle in radians
unit analysis
t or f, radians have no units
true, since they divide length by length, units cancel out.
t or f, if both quantities are vectors, then A + B = B + A
True, the direction vector will be in the exact same direction (obviously the magnitudes are same)
t or f, A - B will point in the opposite direction of B - A.
True! and thus, A - B does not equal B - A
what simple rule do we use when drawing vector subtraction?
We convert A - B to -B + A and then draw the negative vector first (following tip-to-tail rules)
What occurs if we multiple a vector A by
- k
- -k
- 0
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)
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.
True, this is core to most physics concepts involving a vector (e.g. force, kinematics, etc).
Graphically, how can we identify if x and y are proportional?
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
How are I and T related in I = xT^4 ?
I = xT^4
I is proportional to T^4
if T is increased by 2, then I is increased by (2)^4 = 16
Explain the 4 log laws: multiplication, division, power, and number = 1.
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
What is the natural log?
Ln which is base e = 2.7
Approximate Log(2), Log(3), and Log(4)
Log(2) = 0.3 Log(3) = 0.5 Log(4) = 0.7
If [H+] = m x 10^-n. Explain how you find the pH value.
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
If pH = n.m, explain how you find [H+].
[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
