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Math Second Semester - Coggle Diagram
Math Second Semester
Chapter 5AB: Volume and Surface Area of 3D solids and Rules of Trigonometry
Names of Shapes
Cuboid: 6 sides with rectangular bases
Cylinder: 2 circular bases and parallel sides and a circular or oval cross section.
Prism: Shape whose two end faces are similar (can be rectangular triangular etc)
Sphere: A round shape where every point is equidistant from the center
Cone: A shape with one circular base
Pyramid: A shape with a rectangular base and triangular faces
Formulas: Volume
Cuboid: V= l x w x h
Cylinder: V = pi r^2 h
Prism: V = a h
Sphere: V = 4/3 pi r^3
Cone: V = 1/3 pi r^2 h
Pyramid: V = 1/3 (a h)
Formulas: Surface Area
Cuboid: SA = sigma (area of faces)
Cylinder: SA = h (2 pi r) + 2(pi r^2)
Prism: SA = l h + 3 l w
Sphere: SA = 4 pi r^2
Cone: SA = pi r^2 + pi r L (slant height)
Pyramid: SA = 2 b h + l w
Right Triangle Trigonometry
Pythagorean Theorum:
a^2 + b^2 = c^2
Triangle Sum Theorum:
m<a + m<b + m<c = 180
Soh Cah Toa
Sine = opposite / hypotenuse
Cosine = adjacent / hypotenuse
Tangent = opposite / adjacent
Law of Sines: use for ASA or SAA (2 angles and 1 side)
a / sinA = c / sinC = b / sinB
Ambiguous Case: for SSA
0 triangles = without error
1 triangle = angles less than 180 degrees
2 triangles = they both work
Law of Cosines: missing side or missing angle (SAS or SSSA)
c^2 = a^2 + b^2 - 2 a b cosC
Chapter 5: Exponentials and Logarithms
5.1 Exponential Growth and Decay
Formula: y = ab^t
y = a (1+r)^t for growth
y = a (1-r)^t for decay
y is the final amount, a is the initial amount, r is the rate of increase or decrease, the parenthesis turn into b, and t is the number of time periods.
Items that decrease in value over time: cars, jewelry, video games, etc
Exponential Rules:
a^b X a^c = a^b+c
a^b/a^c = a^b-c
(a/b)^2 = a^2 / b^2
(a / bc)^2 = a^2 / b^2 X c^2
(a / b+c)^2 = a^2 / (b+c)^2
When a number is raised to a negative power, it is the reciprocal of the positive number
5.2 Exponential Functions - Rational (Fractional) Exponents
Growth: add the percentage to 1
Decay: subtract percentage from 1
a "root" x^b = x^b/a
x^-1/a = 1 / a "root" x
5.3 and 5.4 Exponential Functions and Compound Interest
f(x) = ab^x where a, b > 0 and b does NOT equal 1
Exponential graph of f(x) = ab^x where b>1 --->
Domain = all real numbers, Range = y > 0, y-intercept = (0,a)
A(t) = Ao (1+r) ---> Ao is initial amount, r is rate as a decimal, and t is time in years
A(t) = Ao (1+ r/n) ^nt ---> n is number of times compounded a year
A(t) = Pe^rt ---> r is a decimal, used for continuous
5.5 Logarithmic Functions
log "base" ab = c if and only if a^c=b
y = 10^x and y = log x are inverses of eachother
log a = c if and only if 10^c = a
ln a = c if and only if e^c = a
5.6 Laws of Logarithms
If M and N are positive and real numbers, and b is a positive number other than 1, then...
log "base" b MN = log "base" b M + log "base" b N
log "base" b M/N = log "base" M - log "base" N
log "base" M = log "base" N if and only if M=N
log "base" b M^k = k log "base" b M for any real number k