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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
How would you graph 2x plus 3y equals 6?
Find x when y = 0: 3y = 6 so y = 2. Mark the point (0,2) Find y when x = 0: 2x = 6 so x = 3 Mark the point (3, 0) Draw a line joining these two points and extend in both directions.
How do you do the area of bounded y2 equals 9xand2x equals 9y?
The basic steps you must follow are:
- Simplify the two equations to be functions of y.
- Find the intersections of the two functions.
- Integrate the difference of the two functions on between the two intersection points.
Here are the steps for this specific problem:
Simplifying the equations:
Simplify y2=9x to be a function of y.
y=±√(9x)
y=±3√(x)
(Keep the ± for now.)
Simplify 2x=9y to be a function of y
(2/9)x=y
y=2/9x
Finding the intersection of the two functions:
±3√(x)=y=2/9x
±3√(x)=2/9x
±27/2√(x)=x
x=0
±27/2=√(x)
729/4=x
x=729/4
(Now, since neither of the solutions is negative, the ± may be removed from the first function.)
Integrating the difference of the two functions:
Let A=area bounded by the two curves.
Let y1=3√(x), the first function.
Let y2=2/9x, the second function.
A=∫(y1-y2)dx
(y2 is subtracted from y1 because it is the lower function on that interval. Graph the two functions; it makes sense.)
A=∫(3√(x)-2/9x)dx on [0, 729/4]
A=∫(3x1/2-2/9x)dx on [0, 729/4]
A=2x3/2-1/9x2+C on [0, 729/4]
Now, find the particular solution on the interval [0, 729/4]
A=[2(729/4)3/2-1/9(729/4)2+C]-[2(0)3/2-1/9(0)2+C]
A=19683/4-59049/16+C-C
A=19683/4-59049/16=19683/16=1,230.1875
The area of the region bounded by y2=9x and 2x=9y is 1,230.1875 units2.
How can you write a function that represents all possible antiderivatives of a given function?
Given a function f(x) find any anti-derivative, F(x). The set of all possible derivatives is obtained by adding a term not involving x which can take any value.
So F(x) + C is a general derivative, where C can take any value.
Yes. According to the Commutative Property of Multiplication, 3x is = to x3 but, of course, the second term breaks the rule on "Numerical Coefficient first before Literal Coefficient"
What is the x intercept of 3y equals 18?
3y =18
y = 6
That is a line intercepting y axis at 6 but parallel to he x axis - there is no x intercept
- it never crosses the x axis
What is 2x-7 plus 4x equals 35?
2x - 7 + 4x + 7 = 35 + 7 {add 7 to both sides, the sevens on the left side cancel}
2x + 4x = 42
6x = 42 {combine the 'x' terms}
x = 7 {divide both sides by 6}. Check in original: 2(7) - 7 + 4(7) = 14 - 7 + 28 = 35.
They're asking you to find a value for x, which in this case is 4, since 3 times 4 equals 12.
2x minus 3y equals 11 5x plus 2y equals 18?
Presumably this is a simultaneuos equation question that can be solved using the elimination method:
2(2x-3y = 11)
3(5x+2y = 18)
Multiply all terms in the top equation by 2 and in the bottom by 3:
4x-6y = 22
15x+6y =54
Add both equations together:
19x = 76
Divide both sides by 19 to find the value of x:
x = 4
Substitute the value of x into the original equations to find the value of y:
Therefore: x = 4 and y = -1
1000000000*2.6 = 2600000000 or, more simply, 2.6*109 or 2.6 billion.
What is the y value of y equals -x2 plus 2x?
That's going to depend completely on the value of 'x'.
There are an infinite number of things it could be.
