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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
Find the point of intersection of y equals x2-6x plus 11 and the lines y equals x and y equals 2x?
y = x^2 - 6x + 11
x + y = 2x
x + y = 2x
x - x + y = 2x - x
y = x
Substitute x for y in the first equation:
y = x^2 - 6x + 11
x = x^2 - 6x + 11
x - x = x^2 - 6x - x + 1
0 = x^2 - 7x + 11
x = [-(-7) ± √[(-7)^2 - 4(1)(11)]]/(2)(1)
x = (7 ± √5)/2
x = (7 + √5)/2 or x = (7 - √5)/2
x ≈ 4.618 or x ≈ 2.382
Since y = x, then y ≈ 4.618 or y ≈ 2.382.
Thus, we can say that the solution of the system are (2.382, 2.382) and (4.618, 4.618).
Why it is necessary for a function to be one-one for existing its inverse?
If it is not 1-1 then you will not know where the inverse value is to be mapped.
For example,
f(x) = x2 , being an even function, is not 1-to-1.
Thus, for example, (-3)2 =9 = 32
Suppose g(x) were the inverse of f(x). What would g(9) be? -3 or 3? There is no way of telling.
The only way around it is to define the domain of x to be non-negative numbers and then f would be 1-to-1 so that g was a well-defined function whose range was the non-negative numbers.
Incidentally, the domain of f could just as well be defined as the non-positive numbers.
Which method would you use to solve this equation5x plus 3y equals 10?
There are an infinite number of solutions to a single equation in two variables.
So, in order to save myself time, the first thing I would do is go out and look for
another equation to put along with the first one.
Yes, because you can rewrite it as:
y = -x/10
Which is a line.
When you graph the above equation, the graph passes the vertical line test - meaning that the graph never intersects with any vertical line more than once.
What property is x plus 3 equals 5 then 2 plus 3 equals 5?
Well, if x+3=5 and 2+3=5, I'm guessing x should equal 2
What is the answer to -2x plus y equals 5?
The answer is any pair of numbers, just as long as the 'y'-number is
5 more than double the 'x'-number. There are an infinite number of them.
You miight think that if there are that many, then it's pretty easy to stumble across
a pair of numbers that's an answer. Unfortunately, there's a bigger infinity of pairs
of numbers that are not answers.
How many gallons of water are contained in a volume of 22 acre-feet?
22 acre-feet equates to 7,169,000 gallons of water.
When do you say that the graph is a function?
A function takes one input and assigns to it exactly one output, so a graph qualifies as a function if it passes the vertical line test (run a vertical line across the entire plane; the function should only cross your line once no matter where you are testing).
What is the answer to 3x2-6x-7 equals 0 in quadratic form?
I would not even attempt to factor this, so the quadratic formula.
X = - b (+/-) sqrt(b2 - 4ac)/2a
a = 3
b = - 6
c = - 7
X = - (- 6 ) (+/-) sqrt[(- 6)2 - 4(3)(- 7)]/2(3)
X = 6 (+/-) sqrt(36 + 84)/6
X = 6 (+/-) sqrt(120)/6
X = 6 (+/-) sqrt( 22 * 30)/6
X = [6 (+/-) 2sqrt(30)]/6
----------------------------------------exact answer less clean up
How do you solve the equation x plus 6 equals x plus 8?
X + 6 = X + 8
This is a nice example of an equation that has no solution.
If you subtract 'X' from both sides of the "equation", it says: " 6 = 8 " .
This should immediately lead us to suspect that something here is not kosher, (that is to say, unorthodox).
What is x if x plus 10 percent of x equals 9705.22?
10 percent = 0.1
'x' plus 10 percent of 'x' = 1.1x
1.1x = 9,705.22
x = 9,705.22/1.1 = 8,822.927 (rounded)
There cannot be a polygon with 6 diagonals.
A polygon with n vertices has n*(n-3)/2 diagonals.
If this is 6, then n*(n-3)/2 = 6
therefore n*(n-3) = 12
and so n^2 - 3n - 12 = 0
The only solutions to this equation are n = 1.5-sqrt(57)/2 or 1.5+sqrt(57)/2.
However, n, being the NUMBER of vertices cannot be an irrational number and so there is no solution.
3x - y equals 3 y equals 2x plus 2?
3x-y = 3y
3x = 4y
y = 0.75x
3x-0.75x = 2x+2
2.25x = 2x+2
0.25x = 2
x = 8
y = 6
