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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
Solve 3x plus 2y equals 0 and -9x plus y equals 42 using substitution method?
3x+2y = 0
-9x+y = 42 => y = 42+9x
Substitute the bottom rearranged equation into the top equation:
3x+2(42+9x) = 0
3x+82+18x = 0
21x = -84
Divide both sides by 21 to find the value of x:
x = -4
Substitute the value of x into the original equations to find the value of y:
So: x = -4 and y = 6
What is the domain of y equals square root of 1 minus x to the power of 2?
y=(√1)-x2
The domain is the set of numbers that "x" can be. In this equation "x" can be any real number.
The domain for this problem would be (-inf,inf)
*Inf= Infinity*
What is the vertex of y equals x2 plus 4x-3?
y = X^2 + 4X - 3
set polynomial = to 0
X^2 + 4X - 3 = 0
add 3 to each side
X^2 + 4X = 3
complete the square. take linear term (4) halve it, square it and add constant to other side, factor quadratic term
(X + 2)^2 = 7
subtract 7 from each side
(X + 2) - 7 = 0
Vertex = (-2,-7)
My TI-84 confirms this vertex with original equation
How do you integrate one over sqrt of x squared minus nine between limits five and ten?
5∫10[1/√(x2 - 9)] dx = 5∫10[dx/√(x2 - 32)]
Let x = 3 sec θ, where 0 < θ < π/2 or π < θ < 3π/2. Then, dx = 3 sec θ tan θ dθ and
√(x2 - 9) = √[(3 sec θ)2 - 9] = √[9(sec2 θ - 1)] = √(9 tan2 θ) = 3 tan θ
5∫10[dx/√(x2 - 32)]
= 5∫10[(√3 sec θ tan θ)/(3 tan θ)] dθ
= 5∫10 sec θ dθ
= ln |sec θ + tan θ|5|10 = ln sec 10 + tan 10 - sec 5 - tan 5 = -2.298
How do you use differentiation in general?
By in general I assume that you mean without using first principles.
For d/dx of any single term all you do is take the power down to the front and make a new power of the old power minus one:
e.g. d/dx(x2) = 2x(2-1) = 2x
d/dx(3x4) =4.3x(4-1) = 12x3
This also works for fractions and negative numbers:
e.g. d/dx(0.5x-1) = -1 times 0.5x(-1-1) = -0.5x-2.
How do you find x when 3 bracket 3x plus 1 bracket equals 4 bracket 2x plus3 bracket?
The answer is x = 9 and the workings and explanation are below for your information.
3(3x + 1) = 4(2x + 3)
First we need to multiply out the brackets. We will do this by multiplying each term inside the bracket with what is outside the bracket. So on the left side of the equation, for example, we will get:
9x (because this is 3x * 3) + 3 (because this is 3 * 1).
So when we multiply out both brackets we will get:
9x + 3 = 8x + 12
We now need to move all the "x's" to one side of the equation and all the constants (the numbers) to the other side. When we move anything from one side of an equation to the other side we have to change a positive to a negative and vice versa.
So we will get:
9x - 8x = -3 + 12
Now we can simplify both sides to:
x = 9.
Does this have a solution x2 plus x plus 1 equals 0?
No because the discriminant of this quadratic equation is less than zero.
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That just means that it has no real solutions.
The two solutions are:
x = - 1/2 + 1/2 i sqrt(3)
x = - 1/2 - 1/2 i sqrt(3)
What is the average arithmetic mean of 3x plus 1 and x-3?
1/2 (3x + 1 + x - 3) = 1/2 (4x - 2) = 2x - 1
What is the answer to the rational expression 2x plus 3 over 10x squared plus 21x plus 9?
(2x+3)/(10x2+21x+9)
Factor the quadratic expression in the denominator:
(2x+3)/(5x+3)(2x+3)
which becomes:
1/(5x+3) or (5x+3)-1
How do you solve the equation 6n plus 2 equals 22?
You are finding what n equals, so you need to turn the equation into n=_______.
- Subtract the 2 from both sides of the equal sign. --> 6n=20
- Divide 20 by 6 so the n will be by itself.-->n=20/6
20/6= 3.333, so n=3 1/3.
What happens to the graph of y equals x when the equation changes to y equals x 9?
The line y = x will shift up when you add a value to x and shift down when you subtract a value from x.
How do you convert x2 plus y2 equals z to standard form for y?
X2 + Y2 = Z
subtract X2 from each side
Y2 = - X2 + Z
take square root each side
Y = sqrt(- X2 + Z)
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What does x equal when 10x plus 8 equals 68?
x = 6
Why?
10x + 8 = 68
Subtract 8 from both sides: 10x = 60
Divide both sides by 10: x = 6
3x-1 equals 11 what is the value of x2 plus x?
3x-1=11 solves to x=4. Plug in 4 to get 42+4. The answer is 20.
Does anyone know the solution to the lame fly problem?
Using the Pythagorean theorum, the fly must walk in a diagonal line to the top corner of the room (this would be .608 feet, or 7.296 inches). Then he must walk from that top corner of the room in a diagonal line down the 30 foot wall (this will be 32.311 feet, or 387.7819 inches), then he must walk a diagonal line from the bottom corner of the room to the food (this is also .608 feet/7.296 inches).
Then we add all our measurements together to get 402.324 inches, or 33.527 feet.
