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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 do you simplify sec²x minus 1 divided by sec²x?
Use your identity to write the numerator in terms of tangent.
2x plus y equals 3 2y-6 equals -4x?
4x + 2y = 6 (first equation x 2)
4x + 2y = 6 (second equation rearranged)
No unique answer exists
What is 8x - 6 equals 10x plus 6?
Subtract 8x from each side: -6 = 2x + 6;
Subtract 6 from each side: -12 = 2x;
Divide each side by 2: x = -6
What does 7x squared minus 112 equals?
If x is a real number then, depending on the value of x, any value greater than or equal to -112.
Factored, that becomes 7(x +4)(x - 4)
It follows that x - 1 = 2x - 5 as each expression is equal to y.
Add 1 to each side: x = 2x - 4
Add 4 to each side: x + 4 = 2x
Subtract x from each side: 4 = x
y = x - 1 = 3
How do you solve x squared equal negative 28?
Take the square root of each side of the equation.
x = ± i sqrt(28)
What is the range of the function yx2?
The range of the function y=x^2 would be y is greater than or equal to 0 in this case. So pretty much just find the vertex of the function and what ever the y coordinate is set that as the lowest number for the range.
What is 3y plus 5 equals y minus 3?
3y+5=y-3
subtract the y from the other side then:
2y+5=-3
subtract the 5
2y=-8
then divide 2 by both sides and your final answer should be:
y= - 4
Assuming the equation is 3x - 6 + 3 = 15, here's how to solve it.
3x - 6 + 3 = 15 Combine like terms
3x - 3 = 15 Move Xs to one side and constants to another
3x = 18 Divide both sides by 3
x = 6
Check that x = 6 satifies the equation (it does)
What is the comparison test for series?
The comparison test states that if a series of positive numbers converges, and in another series, each of the corresponding terms is smaller, then it too must converge. Similarly, if a series of positive numbers diverges to infinity, and another series has each of its terms greater than the corresponding terms of the other, then it too diverges.
Why does the amplitude of a simple pendulum decrease?
Air resistance against the bob and string and friction in the pivot make the amplitude of a simple pendulum decrease.
What is answer x plus y equals 4?
This is a linear equation in two variables and the coordinates of each and every point on the line that it describes is a solution.
A single linear equation does not have an "answer".
It is a linear expression in two variables.
As an expression it cannot be solved. Furthermore, to solve equations in two variables you need at least two linear equations.
How does the graph of y equals 3x2-5 compare with the graph of y equals 3x2 plus 8?
The figures are exactly the same, but every point on the first graph
is exactly 13 below the corresponding point on the second one.
What is the relation of the forms y equals Psinx plus Qcosx and y equals Asin of x plus c?
A = Root (Q squared plus P squared)
C = 90 + tan inverse P/Q
... I think lol just worked it out just now =w=
How do you find antiderivatives?
Finding antiderivatives essentially involves "un-doing" the derivative. If the function you are antidifferentiating involves variables raised to a power, instead of multiplying by the power and decreasing the degree by one as you would do when taking the derivative, you add one to the degree of the power and multiply by the reciprocal of the new power .
For example, if the function you are antidifferentiating is x^2, you add one to the power (so now it is x^3) and multiply by the power's reciprocal (1/3). The antiderivative is (1/3)x^(3) plus an arbitrary constant.
If the function you are antidifferentiating involves sin and cos functions, use trig identities, partial fractions, u substitution, etc. Remember that you can always take the derivative of the expression you find as the antiderivative to verify its validity.
x3 + x = -2 => x3 + x + 2 = 0 => x3 + x2 - x2 - x + 2x - 2 = 0 => x2(x+1) - x(x+1) + 2(x+1) = 0 => (x+1)*(x2-x+2) = 0 Setting the first bracket equal to zero gives the only real solution, which is x = -1 The second bracket gives the complex roots, x = ½*[1 +or- i*sqrt(7)]
Sin x equals fourth fifths find cos x?
If sin(x) = four fifths, then cos(x) = three fifths.
Sin-1(0.8) = 53.13 degrees.
Cos(53.13) = 0.6.
There are many ways you could work this out. One of them is to use the relationship between the squares of the sine and cosine:
sin2(x) = 1 - cos2(x)
(4/5)2 = 1 - cos2(x)
16/25 = 1 - cos2(x)
-cos2(x) = - 1 + 16/25
cos2(x) = 1 - 16/25
cos2(x) = 9/25
cos(x) = ±3/5
You could also work it out using good ol' Pythagorean theorem:
Let:
a ≡ adjacent
o ≡ opposite
h ≡ hypotenuse
Then we know:
sine = o/h
cosine = a/h
h2 = a2 + o2
We already have our opposite and hypotenuse sides, as we're given those with the value of the sine. We need to work out the adjacent side then:
h2 = a2 + o2
∴ a2 = h2 - o2
∴ a = (h2 - o2)1/2
∴ a = (52 - 42)1/2
∴ a = (25 - 16)1/2
∴ a = 91/2
∴ a = ±3
And now that we have the length of the adjacent side, as well as the hypotenuse, we know the cosine of the angle as that is is the ratio between the two:
cosine = a/h
∴cosine = ±3/5
