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Calculus
The branch of mathematics that deals with the study of continuously changing quantities, with the use of limits and the differentiation and integration of functions of one or more variables, is called Calculus. Calculus analyzes aspects of change in processes or systems that can be modeled by functions. The English physicist, Isaac Newton, and the German mathematician, G. W. Leibniz, working independently, developed calculus during the 17th century.
25,068 Questions
What is the difference between gradient and vector notation?
In the name of God;
It must be mentioned that a vector has two important characteristics; 1- direction and 2- quantity. in other word for identification a vector these two characteristics must be defined. for example when we speak about displacement of a body it must has direction and quantity.
but about gradient, it has a general mean: difference.
Also a specified mean may be defined for it: "any increase or decrease in a vector or scalar field". it is a vector field.
What is the integral of 2x plus seven divided by x square plus 2x plus 5?
Hopefully I did this one correctly, if anyone sees an error please correct it. This is the problem:
∫(2x+7)/(x2+2x+5)
I rewrote the integral as:
2∫x/(x2+2x+5) + 7∫1/(x2+2x+5)
Both of these parts of the integral is in a form that should be listed in most integral tables in a calculus text book or on-line. From these tables the integral is the following:
2*[(1/2)ln|x2+2x+5| - (1/2)tan-1((2x+2)/4)] + 7*[(1/2)tan-1((2x+2)/4)]
Combining like terms gives the following:
ln|x2+2x+5| + (5/2)*tan-1((2x+2)/4)
Is x squared minus x minus twelve a prime?
Just like numbers, polynomials can be prime in some way, although this is usually referred to as being irreducible. Your polynomial is not in fact irreducible since it can be decomposed into (x-4)(x+3). This also means that plugging in any integer value for x gives a factorisation for the number x^2-x-12, which is (x-4)(x+3), so the value of x^2-x-12 for any integer x can only be prime if either x-4 = 1,-1 or x+3 = 1,-1 so for x=5,3,-2,-4. Now for x=5 we get 25-5-12=8, which is not prime. For x=3 we get 9-3-12=-6, which is not prime either. For x=-2 we have 4+2-12=6, which again is not prime. And for x=-4 we have 16+4-12=8 again, which of course is still not prime. So not only is your polynomial reducible, also any value it takes when plugging in an integer x will be composite. An alternative way of establishing this fact would have been to note that this polynomial only takes even values for any integer x and that the value 2 is not in fact attained for any integer x.
Negative 1 fifth x plus 4 equals 29?
-1/5 x + 4 = 29
Subtract 4 from each side:
-1/5 x = 25
Multiply each side by -5:
x = -125
6x plus 9 equals -19-8x what is the value of x?
This equation that you are asking of is a complex equation that inverse operations are needed in to solve it. First, use the inverse operation and add 8x to 6x. The 8x on the right side of the equation will cancel out. The 6x and the 8x will equal to 14x: 8x+6x+9=-19-8x+8x which now equals 14x+9= -19
Secondly, eliminate the constant terms by subtracting 9 on both sides of the equation: 14x+9-9= -19-9
Now the problem is 14x= -28
Third step - isolate the variable and divide 14 by -28. X should now equal -2.
14x/14= -28/14 which should equal x= -2
You can check and see that this is accurate by multiplying -2 to each coefficient that had x as their variable: 6(-2)+9=-19-8(-2)
-12+9= -19+16 which equals -3=-3
0 is the identity under addition for integers, rationals, reals, complex numbers. This means that 0+x = x+0 = x for all x.
The above answer is correct, but with some mild "un0carefulness" as we call 0 the identity BEAUSE 0 + x = x + 0 = x for all x in those sets.
As a matter of fact, we want something in our sets with an operation to be the identity, we just decided to call it 0.
In other words, we DEFINE 0 as a element in a group, ring, field, or other number systems where 0 + x = x + 0 = x for all x.
By the way, if you don't like it, you can call it something else. Like @ + x = x + @ = x for all x. You just have to define it. Then you can use it on your papers :P After all, 0 without a definition is just a symbol.
An invented term that is too vague to answer when queried in this manner.
What are the 3 possibilities of a system of equation?
1. It can have a unique solution. On a graph this would be single point of intersection.
2. It can no no solutions such as two parallel lines on a graph.
3. It can have an infinite number of solutions such as two equations that represent the same line on a graph.
You can make up examples to see this very easily.
For number 3, take any linear equation such as y=x+1
Now multiply both sides of the equation by 10 10y=10x+10. The solution to the system of those two equations is all real numbers, an infinite solution.
Now for number 2, take any line and just find a parallel line. Easy to do by simply making sure it has the same sloe but a different y intercept.
For the last one, take two lines that intersect. This will most often be the case if you randomly pick two linear equation. Say y =x+1 and y=3x+13. Different slope and different y intercept.
What is the range of y equals -2x plus 3?
The Range is all the Real numbers you can think of.
The Domain is also all Reals.
What are the steps to this question 4x plus 11-2x plus 13 equals 32?
Given .........................................4x+11-2x+13=32
Combine like terms ....................2x+24=32
Subtract 24 from both sides ......2x=8
Divide both sides by 2 ...............x=4
