Today we will help you with statistics problems online
Statistics problems online study is the learning about the mean, mode, median and also the range. The statistics problems online study is the term of the design of the survey and also the experiments.
Also get help with free statistics answers. Statistics problems online study is the process that the tutor explains the procedure in the online by the step by step procedure for the mathematical all problems.
Statistics problems online study refers the quantities find for the data. Statistics problems online study includes the collection, analysis and also the interpretation.
In our next post we will help you with free statistics homework help
Assess your student by charting progress on weekly tests, collecting work samples and monitoring homework performance. You may wish to consult with a specialist or with the school psychologist at this point to discuss a potential learning disability.
Meet with the child's family to discuss the observed learning disability in math. Look at options for increasing the student's success, and set up a meeting at a future date to re-evaluate. A referral to for a special education assessment may be recommended if learning problems continue. Also get help with volume of a cube formula
With the student's parents, decide if a learning disability assessment is necessary. If the student has not shown improvement at the time of the second meeting, the assessment should be carried out with the assistance of the school psychologist.
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Notation : , where a is known as base and b is known as exponent which must be a positive and integral value. Explanation of : The standard exponential form, can mathematically be expressed as multiplied by itself times. That is to say . This yeilds a single number if we know the value of a and b. eg. Assuming a = 2 and b = 3 , we can express = = = 8(as b = 3 so we multiply a = 2 with itself 3 times). Note that the answer 8 results in single number. You can also get more details on cross multiplication
Sample Examples of the Exponential Form
1. = = (3 multiplied by itself 2 times) 2. = = (4 multiplied by itself 3 times) [Note: Here in standard exponential form, if we assume a and/or b are variables, they do not yeild any number. Instead they remain in the variable format. like 3. = ( multiplied by itself 3 times) 4. = ( multiplied by itself a times) ] [ Notes 2. If we take as a decimal value such as 0.3, it comes under the section of nth root of a number, while here we are discussing how to compute standard exponential form ]
We have already defined the derivative of a function f(x) at a particular point 'a' and derivative of f(x) in general for the variable x as f(a) and f(x) respectively. The restriction in both the cases is that 'the limit must exist' differentiability of a function.
If
does not exist, then we say that the function is not differentiable.
If the above limit exists, we say the function f(x) is differentiable.
Set Theory: A set theory is a group of objects. Each object is known as a member of the set. A set can be represented using curly brackets. So a set containing the numbers 2, 4, 6, 8, 10, ... is: {2, 4, 6, 8, 10, ... } . Sets are often also represented by letters, so this set might be E = {2, 4, 6, 8, 10, ...} . Alternatively, E = {even numbers} .
Common Sets
Some sets are commonly used and so have special notation:
Other Notation
Subsets
If A is a subset of B, then all of the elements of A are also in B. For example, if A = {1, 2, 3} and B = {1, 2, 3, 4, 5} then A Í B (Í means is a subset of).
Number of Members
If A = {1, 2, 4, 8}, then n(A) = 4. This is because n(A) means the number of members in set A.
The Universal Set
The universal set is the set of all sets. All sets are therefore subsets of the universal set.
Venn Diagrams
Venn diagrams are used to represent sets. Here, the set A{1, 2, 4, 8} is shown using a circle. In Venn diagrams, sets are usually represented using circles. The universal set is the rectangle. The set A is a subset of the universal set and so it is within the rectangle. The complement of A, written A', contains all events in the sample space which are not members of A. A and A' together cover every possible eventuality. A ÈB means the union of sets A and B and contains all of the elements of both A and B. This can be represented on a Venn Diagram as follows: AÇB means the intersection of sets A and B. This contains all of the elements which are in both A and B. AÇB is shown on the Venn Diagram below: An important result connecting the number of members in sets and their unions and intersections is: n(A) + n(B) - n(AÇB) = n(AÈB)
Conic sections are the curves which can be derived from taking slices of a "double-napped" cone. (A double-napped cone, in regular English, is two cones "nose to nose", with the one cone balanced perfectly on the other.) "Section" here is used in a sense similar to that in medicine or science, where a sample (from a biopsy, for instance) is frozen or suffused with a hardening resin, and then extremely thin slices ("sections") are shaved off for viewing under a microscope. If you think of the double-napped cones as being hollow, the curves we refer to as conic sections are what results when you section the cones at various angles.
There are plenty of sites and books with pictures illustrating how to obtain the various curves through sectioning, so I won't bore you with more pictures here. And there are books and entire web sites devoted to the history of conics, the derivation and proofs of their formulas, and their various applications. I will not attempt to reproduce that information here.
This lesson, and the conic-specific lessons to which this page links, will instead concentrate on: finding curves, given points and other details; finding points and other details, given curves; and setting up and solving conics equations to solve typical word problems.
There are some basic terms that you should know for this topic:
center: the point (h, k) at the center of a circle, an ellipse, or an hyperbola.
vertex (VUR-teks): in the case of a parabola, the point (h, k) at the "end" of a parabola; in the case of an ellipse, an end of the major axis; in the case of an hyperbola, the turning point of a branch of an hyperbola; the plural form is "vertices" (VUR-tuh-seez).
focus (FOH-kuss): a point from which distances are measured in forming a conic; a point at which these distance-lines converge, or "focus"; the plural form is "foci" (FOH-siy).
directrix (dih-RECK-triks): a line from which distances are measured in forming a conic; the plural form is "directrices" (dih-RECK-trih-seez).
axis (AK-siss): a line perpendicular to the directrix passing through the vertex of a parabola; also called the "axis of symmetry"; the plural form is "axes" (ACK-seez).
major axis: a line segment perpendicular to the directrix of an ellipse and passing through the foci; the line segment terminates on the ellipse at either end; also called the "principal axis of symmetry"; the half of the major axis between the center and the vertex is the semi-major axis.
minor axis: a line segment perpendicular to and bisecting the major axis of an ellipse; the segment terminates on the ellipse at either end; the half of the minor axis between the center and the ellipse is the semi-minor axis.
locus (LOH-kuss): a set of points satisfying some condition or set of conditions; each of the conics is a locus of points that obeys some sort of rule or rules; the plural form is "loci" (LOH-siy).
One very basic question that comes up pretty frequently is "Given an equation, how do I know which sort of conic it is?" Just as each conic has a typical shape:
...so also each conic has a "typical" equation form, sometimes along the lines of the following:
parabola: Ax2 + Dx + Ey = 0 circle: x2 + y2 + Dx + Ey + F = 0 ellipse: Ax2 + Cy2 + Dx + Ey + F = 0 hyperbola: Ax2 – Cy2 + Dx + Ey + F = 0
These equations can be rearranged in various ways, and each conic has its own special form that you'll need to learn to recognize, but some characteristics of the equations above remain unchanged for each type of conic. If you keep these consistent characteristics in mind, then you can run through a quick check-list to determine what sort of conic is represented by a given quadratic equation.
Given a general-form conic equation in the form Ax2 + Cy2 + Dx + Ey + F = 0, or after rearranging to put the equation in this form (that is, after moving all the terms to one side of the "equals" sign), this is the sequence of tests you should keep in mind:
Are both variables squared?
No: It's a parabola. Yes: Go to the next test....
Do the squared terms have opposite signs?
Yes: It's an hyperbola. No: Go to the next test....
Are the squared terms multiplied by the same number?
Yes: It's a circle. No: It's an ellipse.
Classify the following equations according to the type of conic each represents:
A) Both variables are squared, and both squared terms are multiplied by the same number, so this is a circle. B) Only one of the variables is squared, so this is a parabola. C) Both variables are squared and have the same sign, but they aren't multiplied by the same number, so this is an ellipse. D) Both variables are squared, and the squared terms have opposite signs, so this is an hyperbola.
If they give you an equation with variables on either side of the "equals" sign, rearrange the terms (on paper or in your head) to get the squared stuff together on one side. Then compare with the flow-chart above to find the type of equation you're looking at.
You may have noticed, in the table of "typical" shapes (above), that the graphs either paralleled the x-axis or the y-axis, and you may have wondered whether conics can ever be "slanted", such as:
Yes, conic graphs can be "slanty", as shown above. But the equations for the "slanty" conics get so much more messy that you can't deal with them until after trigonometry. If you wondered why the coefficients in the "general conic" equations, such as Ax2 + Cy2 + Dx + Ey + F = 0, skipped the letter B, it's because the B is the coefficient of the "xy" term that you can't handle until after you have some trigonometry under your belt. You'll probably never have to deal with the "slanty" conics until calculus, when you may have to do "rotation of axes". Don't be in a rush. It's not a pretty topic.
By nature of the logarithm, most log graphs tend to have the same shape, looking similar to a square-root graph:
y = sqrt(x)
y = log2(x)
--> The graph of the square root starts at the point (0, 0) and then goes off to the right. On the other hand, the graph of the log passes through (1, 0), going off to the right but also sliding down the positive side of the y-axis. Remembering that logs are the inverses of exponentials, this shape for the log graph makes perfect sense: the graph of the log, being the inverse of the exponential, would just be the "flip" of the graph of the exponential:
y = 2x
y = log2(x)
comparison of the two graphs, showing the inversion line in red
In order to graph this "by hand", I need first to remember that logs are not defined for negative x or for x = 0. Because of this restriction on the domain (the input values) of the log, I won't even bother trying to find y-values for, say, x = –3 or x = 0. Instead, I'll start with x = 1, and work from there, using the definition of the log.
Since 20 = 1, then log2(1) = 0, and (1, 0) is on the graph.
Since 21 = 2, then log2(2) = 1, and (2, 1) is on the graph.
Since 3 is not a power of 2, then log2(3) will be some messy value. So I won't bother with graphing x = 3.
Since 22 = 4, then log2(4) = 2, and (4, 2) is on the graph.
Since 5, 6, and 7 aren't powers of 2 either, I'll skip them and move up tox = 8.
Since 23 = 8, then log2(8) = 3, so (8, 3) is on the graph.
The next power of 2 is 16: since 24 = 16, then log2(16) = 4, and (16, 4) is on the graph.
The next power of 2, x = 32, is too big for my taste; I don't feel like drawing my graph that wide, so I'll quit at x = 16.
The above gives me the point (1, 0) and some points to the right, but what do I do for x-values between 0 and 1? For this interval, I need to think in terms of negative powers and reciprocals. Just as the left-hand "half" of the exponential function had few graphable points (the rest of them being too close to the x-axis), so also the bottom "half" of the log function has few graphable points, the rest of them being too close to the y-axis. But I can find a few:
Since 2–1 = 1/2 = 0.5, then log2(0.5) = –1, and (0.5, –1) is on the graph.
Since 2–2 = 1/4 = 0.25, then log2(0.25) = –2, and (0.25, –2) is on the graph.
Since 2–3 = 1/8 = 0.125, thenlog2(0.125) = –3, and (0.125, –3) is on the graph.
The next power of 2 (as x moves in this direction) is1/16 = 2–4, but the x-value for the point (0.0625, –4) seems too small to bother with, so I'll quit with the points I've already found.
Listing these points gives me my T-chart:
Drawing my dots and then sketching in the line (remembering not to go to the left of the y-axis!), I get this graph:
(a) and f
(x) respectively. The restriction in both the cases is that 'the limit must exist' differentiability of a function. 
