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
What is an Equilateral Triangle?
A triangle were all the sides and angles are the same.
All the angles inside are 60 degrees.
60+60+60= 180
In order to convert degrees to radians, what formula is used?
degree * pi/180
In order to convert radians to degrees, what formula is used?
radians * 180/ pi
S sec^2 x dx =
tan x + C
What is a perpendicular bisector?
A line or a segment perpendicular to a segment that passes through the midpoint of that segment.
S sin x =
-cos x +C
If you are given: a point and a slope, then what formula do you use to find the equation of a line?
A(x,y) m= 1/2
y-y1 = m (x-x1)
The surface area of the prism equals the sum of the areas of the five faces:
Surface area=
top, bottom, front, right, and left
surface area: top + bottom+ front + right + left
General equation of a hyperbola:
(x-h)^2/ a^2 - (y-k)^2/ b^2 = 1
arctan x =
tan^-1 x
If f(x) = tan^-1 x, f'(x) =
f’(x) = 1/ 1+x^2
What is the volume of a cone?
V= 1/3 pi r^2 h r= radius h= height
If a triangle ABC is SSS (side, side, side) triangle (a,b, and c are known) the measure of angle A can be calculated by: ___
Law of cosine:
m<a></a>
Facts about Correlation:
- Correlation must be between _ & _
- Correlation remains unchanged when __.
- Correlation does not have ___.
- Correlation implies __, not ___
- -1 and 1
- switching x and y variables
- units. changing the units on your data will not affect the correlation
- implies association, not causation
Concavity Test:
* If f’‘(x) >0, for all x on interval, then the graph of f is __ on that interval.
- If f’‘(x) <0, for all x on an interval, then the graph of f is __ on that interval.
- f’‘(x) >0 concave upward
* f’‘(x) < 0 concave downward
Surface Area of a cube:
SA= S^3
Volume of a cube:
V= 6s^2
Complement rule for probability:
Two events are complementary if:
exactly one of the events must occur. If A is an event, then A’ is the complementary event of A, or “NOT A”
P(A’) = 1- P(A)
pi radian =
180 degrees
1 radian =
180/pi = 57.3 degrees
Facts about Inverse Functions:
- The graph of the inverse function f^-1(x) is obtained by ___ the graph of f(x) about the line __.
- The __ of f(x) is the __ of f^-1(x), whereas, the __ of f(x) is the __ of f^-1(x)
- f^-1 (x) does not equal
- reflecting the graph of f(x) about the line y=x
- The domain of f(x) is the range of f^-1(x). Whereas, the range of f(x) is the domain of f^-1(x)
- does not equal 1/f(x)
Vertex form for a parabola:
y= a(x-h)^2 + k
h,k
Standard Form for a parabola:
y= ax^2+bx+c
-b/2a, y
1+ tan^2 x =
sec^2 x
tan^2 x =
sec^2 -1
cos^2 o + sin^2 o =
1
sin 2o =
2 sin o cos o
The Mean Value Theorem:
Let f be a function that satisfies the following hypotheses:
*f is __ on the closed interval [a,b]
*f is __ on the open interval (a,b)
then there is a c in (a,b) such that f’(c) =
- continuous [a,b]
- differentiable (a,b)
- f’(c) = f(b)-f(a)/ b-a
What is a Translation?
Is a Translation isometry?
Moves every point of a figure or space by the same distance in a given direction.
(yes)
What is Volume?
Volume is the amount of space an object takes up.
Volume is a 3D measurements, so it can be measured in cm^3 or simply in liters.
Liters are usually used to measure volumes of liquids or gases.
What does parallelogram mean?
A quadrilateral with two pairs of parallel sides
What does quadrilateral mean?
A four sided polygon, the sum of whose interior angles is 360 degrees
What shapes are quadrilateral?
Parallelogram, Rectangle, Square, Rhombus, Trapezoid, Kite, Isosceles triangle
What shapes are parallelogram?
Square, Rectangle, Rhombus, and Rhomboid
Compound Interest
A= P( 1 + r/n)^nt
Circumference of a circle
C= 2 pi r
What does “circumference” mean?
it is the perimeter of a circle or ellipse.
Circumference of a semi-circle
C= pi * d / 2
What does Inequality mean?
a statement of an order relationship—greater than, greater than or equal to, less than, or less than or equal to—between two numbers or algebraic expressions. (,/>,=, not = to)
What is the Triangle Inequality Theorem?
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
example:
a+b > c
a+c >b
c+b > a
What is an Isosceles Right Triangle?
45-45-90 triangle
What does Differentiable mean?
The derivative exist at each point in its domain.
A square is a
Rhombus
What’s the formula for the area of a rhombus?
A= 1/2 d^2
d= diagonal
What’s the volume V of the rectangular box?
V= l x w x h
l= length w= width h= height
T or F: All squares are parallelograms
True
T or F: All rectangles are rhombuses
False, bc all four sides of a rectangle don’t have to be equal.
T or F: all rhombuses are squares
False, bc in an rhombuses all of its sides are congruent
T or F: All squares are rhombuses
True, bc all of its sides and angles are congruent
What is the area of a semicircle?
A= 1/2 pi r^2
How many inches are in a foot?
12 inches
If you multiply or divide by a negative number in an inequidad, what happens to the sign? ()
You have to flip the sign!!
Perpendicular lines have opposite ___ slopes.
Reciprocal
Parallel lines have the same __.
Slope
Perpendicular lines have __ reciprocal slopes
Opposite
example: m= 1/3 m= -3/1 = -3
__ lines have opposite reciprocal slopes
Perpendicular
How do you find the vertex on a parabola?
Vertex: -b/2a
What does Axis Of Symmetry mean?
The vertical line through the vertex on the graph of a quadratic function.
A graph where two lines have the same slope and the same y intersect. How many solutions does it have?
Infinitely solutions
A graph that contains two parallel lines, how many solutions does it have?
No solution, because it does not intersect.
A graph that has two intersecting lines, how many solutions does it have?
1 solution
How do we find the zeros of a function?
Look where the x intercepts crosses the x axis. In other words, where y is zero.
What’s the difference between inverse and determinant
Determinant: (ad-bc)
Inverse/ 1/ad-bc
Two triangles are SIMILAR if:
They both share the common angle A
What are corresponding angles?
Angles that are on the same corner at each intersection. (They will be set to equal to (=))
What is the formula for density:
(Mass/Volume)
Formula for the perimeter of a rectangle?
P= 2l + 2w
= 2(l + w)
L= length W= width
Formula to find the Richter Scale Value:
R= log (l x lo/lo) lo= Intensity of an earthquake that is barely felt I= Intensity R= Richter Scale Value
Def of Exponential Functions
Patterns that can get continuously multiplied by some number
Def of Exponential Growth
The base of the exponential is greater than 1 (the numbers keep getting bigger, bigger)
In exponential growth, __, while in exponential decay, __
exponential growth the base of the exponential is greater than 1 while in exponential decay the base of the exponential falls between 0 and 1.
Def: Rational Functions
Is a function that is a fraction and has the property that both its numerator and denominator are polynomials.
Is R(x)= (-2x^5+4x^2-1/x^9) a rational function?
yes, both numerator and denominator are polynomials!
Is R(x)= 1/((x-1)(x^2+3)) a rational function?
yes, both numerator and denominator are polynomials!
Is R(x)= (sqrt(x) + x^2)/(3x^2-9x+2) a rational function?
no, in order for a number to be considered “polynomial” it must be a real number. The square root is not an integers.
Def of Polynomial:
Is any function of the form f(x)=a^n
a= real numbers and the exponents of each x is a non-negative integer!
How do we find the Vertical Asymptotes using a rational function?
First make sure there is no common factors and if there isnt then set the denominator = 0 and solve for the variable.
How do we find Horizontal Asymptotes?
Look at the degrees of the polynomial.
Whats the HA? F(x)= 2x/3x^2+1
HA: y=0
Whats the HA? F(x)= 2x^2/3x^2+1
HA: y=2/3
Whats the HA? F(x)= 2x^3/3x^2+1
HA: DNE
What is the HA? F(x)= 1/(x-2)(x-3)
HA: y=0
What is a conjugate? And give an example:
A conjugate is a binomial where the sign on the second term has been switched. 2+3i -> 2-3i
Def: Composite Function
Is the use of the output of one function as the input of another. f(g(x))
Def: Finite Sequence and give an example:
Sequences that end
The alphabet starts with A ends with Z
Def: Infinite Sequence
Sequences that keep on going and going
Def: Geometric Sequence
Is a sequence of numbers where each number is found by multiplying the previous number by a constant.
What’s the Recursive Rule formula?
an = a(n-1) +1
