Praxis 5161 Flashcards
Surface Area and Volume of a Sphere
Volume = 4/3 pi r^3
Surface area= 4 pi r^2
What is the probability of selecting x item?
= # of selecting x / total x of items
S 1/x dx =
ln|x|+C
cos^2x-sin^2x =
cos 2x
f(x) = sin x
f’(x)=
f’(x) = cos x
f(x) = cos x
f’(x) =
f’(x)= -sinx
What is an Isosceles Triangle?
A triangle that has two sides of equal lengths.
S cos x dx =
sin x + C
What are supplementary angles?
add up to be 180 degrees
a+b = 180 or 2x+3+x-5 = 180
What are complementary angles?
Two angles are called complementary if their measures add to 90 degrees.
If two triangles are similar triangles, what should we set up to determine the height/base of the triangle?
Set up a proportion
ex: 4/8 =x/6
Area of a sector is given by:
*Sector is always a circle
A= pi r^2 x O/360
If A & B are __ events, the P(AnB)=0. Thus, the addition law becomes :
If A and B are mutually exclusive events, then P(AnB) =0.
Thus, the addition law becomes P(AuB) = P(A) + P(B)
i^0 = i^1 = i^2 = i^3 = i^4 =
i^0 = 1 i^1 = i i^2 = -1 i^3 = -i i^4 = 1
What are complex numbers?
numbers with real parts, imaginary numbers
ex: 1-7i or 1+7i
What are integers?
negative, positive, and zero
ex: -1, 0, 1
What are rationals?
negative, positive, 0, plus mins fractions, decimals
ex: -1/2, 1/2, 0.33, -1, 0, 1
What are real numbers?
(mom)
ex: -1, 0, 1, 1/2, -1/2, pi, e, square root for 2
everything that is not imaginary
What are irrationals?
pi, sqrt 2, e, -sqrt 5
cannot write into fraction, BUT does NOT include i
sin = cos= tan = csc = sec = cot =
sin = O/H cos = A/H tan = O/A csc = H/O sec = H/A cot = A/O
The Binomial Expansion:
(x+y)^2
(x+y)^2= x^2+2xy+y^2
If point A(x1,y1) is the initial point and point B(x2,y2) is the terminal point, a vector, AB, is defined as:
AB=
Robust in statistics:
A test is robust if it still provides in sight into a problem despite having its assumptions altered or violated.
* Range, mean, and standard deviation are very sensitive to outliers.
Since the median is the middle value of an ordered dataset, it is ROBUST to outliers.
The area of a triangle with vertices (a,b), (c,d), and (e,f) is given by:
plus/mins 1/2 |a b 1|
|c d 1|
|e f 1|
Inscribed vs Circumscribed
Inscribed: A polygon is inscribed in a circle if all of its vertices lie on the circle
Circumscribed: A polygon is circumscribed about a circle, if each of its sides is tangent to the circle.
What is an inflection point? and
What occurs in an inflection point?
Def: a point of a curve at which a change in the direction of curvature occurs.
Occurs: When the second derivative is either zero or undefined.
How do we find the inflection point if we are given a function?
find the first and second derivative. Once you find second derivative, set equal to zero and solve for x
If you are given 2 points, A(x,y) B(x,y) how do you find the slope of the line that goes through the points?
m= (y2-y1)/(x2-x1)
A(x,y)
y= mx+b to find b
What’s the period of sin ax and cos ax?
2pi/a
What’s the distance formula?
AB= sqrt (x2-x1)^2+ (y2-y1)^2
Polar Coordinates
x=
y=
(r, O)
x= r cos O
y= r sin O
When the equation contains Ax^2+Bx^2. (squared terms for both x & y)
- If A=B, the equation defines as:
- If A does not equal, the equation defines as:
A=B circle
A does not equal B ellipse
A standard deviation is larger when?
The distribution that are more spread out have a greater standard deviation
The combination (n/r) is given by:
(n/r) = n!/ (n-r)! x r!
What is the Area Diameter Circumference of a circle?
Area= pi r^2 Diameter= d=2r Circumference= c=2 pi r
Difference of surface area:
surface area of cube B - surface area of cube A
Area of a triangle:
A= 1/2 bh
base, height
Area of a rectangle:
A= b x h
Increasing and Decreasing Test:
* If f’(x) > 0, for all x on an interval, then f is ___ on that interval.
*If f’(x) <0, for all x on an interval, then f is __ on that interval.
- f’(x) >0 increasing on that interval
* f’(x)< 0 decreasing on that interval
Let A be m x n matrix and B be n x p matrix.
AB = ____ matrix
The product of two matrix m x p
What is Imaginary?
5i 10i sqrt 2i
anything with i
The vertex of a quadratic function is written in vertex form. y= ____
y= (x-h)^2+k
h,k
Law of Sine
a/sinA = b/sinB = c/sinC
The Product Rule
(f *g) = f’ (g) + f (g’)
What are the prime numbers from 1-100?
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Matrix: Reflection about the origin
identity matrix (2x2) (-1, -1)
Matrix: Reflection about the y-axis
identity matrix(2x2) (-1, 1)
Matrix: Reflection about the x-axis
identity matrix(2x2) (1,-1)
The fundamental counting principle:
If one event can occur in m ways and another event can occur in n ways, then the number of ways both events can occur is m x n
The quotient rule:
(f/g)’ = f’ (g) - f (g’) / g^2
What is a dilation?
&
Is dilation an isometry?
Enlarging or shrinking of a mathematical element using a specific factor.
Dilation changes the position and size of a figure, but not the shape.
(no)
What is Rotation?
&
Is rotation an isometry?
A shape is turned around a central point
yes
What is Reflection?
&
Is reflection an isometry?
Is a transformation that acts like a mirror: It swaps all pairs of points that are exactly opposite sides of the line of reflection.
(yes)
What is Isometry?
Is a transformation that preserves length, angle measures, parallel lines, and distance between two points.
Let (x-c)^m be a factor of a polynomial function f.
If m = odd: ___
If m = even: ____
If m = odd: graph of f crosses the x-axis at x=c
if m = even: graph of f touches the x-axis at x=c.
Area of a semi-circle:
A= 1/2 pi r^2
Circumference of a semi- circle:
C= pi r
Does a semi-circle have a volume formula?
No, bc a semi-circle is 2D, NOT 3D
