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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 limit as x approaches 0 of cos squared x by x?
The limit of cos2(x)/x as x approaches 0 does not exist.
As x approaches 0 from the left, the limit is negative infinity.
As x approaches 0 from the right, the limit is positive infinity.
These two values would have to be equal for a limit to exist.
What type of linear system is 7x plus 2y equals 16 -21x-6y equals 24?
Do you mean 7x+2y = 16 and 21x-6y = 24?
If so then it is simultaneous equation with the solution being: x = 12/7 and y = 2
This brings you back to 16.
Whenever you times a number by something and then divide it by the same number, the answer will always go back to where it started.
Use completing the squares solve 2x2-3x-1 equals 0?
y = 2x2 - 3x - 1 = 0 After graphing this equation on my calculator, I found that there are in fact two answers, but neither of them are integers. This means that it is impossible to factor this quadratic equation in the form (x + a)(x + b) with a and b being integers.
What is the quadratic for x2-13x plus 40?
Assume this quadratic equation equals zero. Then :-
x2 - 13x + 40 = 0, this can be factored (x - 8)(x - 5) = 0
This holds true when either x - 8 = 0 or x - 5 = 0
When x - 8 = 0 then x = 8 : When x - 5 = 0 then x = 5
How do you factor 4x minus t squared?
(4x - t2)
= [(2√x)2 - t2] (This is now the difference of squares)
= (2√x - t)(2√x + t)
Let M be the subset of 2x2 matrices A such that det(A)=0 and tr(A'A)=4 (I shall use ' to denote transpose).
Recall that one of the three definitions of a k-dimensional manifold is:
M c R^n is a k-dimensional manifold if for any p in M there is a neighborhood W c R^n of p and a smooth function F:W-->R^n-k so that F^-1(0) = M intersection W and rank(DF(x))=n-k for every x in M intersection W.
In short, what we need to do is find a function F so that the inverse image of the zero vector under F gives M and the rank of the derivative of F is equal to the dimension of the codomain of F.
Let an arbitrary 2x2 matrix be written as:
[a c]
[b d]
Then the two constraints that define M are
1) ad-bc=0
2) a^2+b^2+c^2+d^2=norm(a,b,c,d)^2=4
Define F:R^4-->R^2 by F(a, b, c, d)=(ad-bc, norm(a,b,c,d)^2-4). Then clearly F^-1(0,0)=M. Furthermore, F is smooth on M because the multiplication and addition of smooth functions (a, b, c, d) is also smooth. (Note that we have taken the neighborhood W to be some superset of M. This guaranteed to exist because if we interpret the set of 2x2 matrices as R^4, every point in M has norm 2, so any ball centered at the origin with length greater than 2 will contain M).
All that remains to be done is to check that rank(DF(x))=2 for every x in M. Observe that [DF]= [d -c -b a]
[2a 2b 2c 2d]
We shall now argue by contradiction. Suppose rank(DF) did not equal 2 for every x in M. Then we know that the two rows are linearly dependent i.e.
h[d -c -b a] + k[2a 2b 2c 2d] = 0 and h and k are not both 0.
Suppose h is 0. Then we have 2ka = 2kb = 2kc = 2kd = 0, and since k cannot also be 0, this implies that a=b=c=d=0, therefore norm(a,b,c,d)^2=0. But, (a,b,c,d) must be in M, so this is a contradiction. Hence h cannot be 0.
Now suppose h is nonzero. Then we can divide it out and there exists a, b, c, d and a constant k so that
[d -c -b a] + k[2a 2b 2c 2d] = 0 i.e. we have:
d + 2ka = -c + 2kb = -b + 2kc = a + 2kd = 0. From this we can substitute to obtain:
a(1 - 4k^2) = b(4k^2 - 1) = c(4k^2 - 1) = d(1 - 4k^2) = 0 and hence
a^2(1 - 4k^2)^2 = b^2(4k^2 - 1)^2 = c^2(4k^2 - 1)^2 = d^2(1 - 4k^2)^2 = 0.
Note that (1 - 4k^2)^2 = (4k^2 - 1)^2. Now, adding the four above expressions together we get:
(a^2 + b^2 + c^2 + d^2)(1 - 4k^2)^2 = norm(a,b,c,d)^2(1 - 4k^2)^2 = 0. But, since we require (a,b,c,d) to be in M, this reduces to 4(1 - 4k^2)^2 = 0. This implies that
1 - 4k^2 = 0, and hence k = +/-(1/2).
Now, if k=+1/2, then we have a + d = b - c = 0, therefore d = -a and b = c. Since we have det(A) = 0 as one of our constraints on M, this implies that ad - bc =
-(a^2) - (b^2) = -(a^2 + b^2) = 0, which implies a = b = 0, by the property of norms. But, if a = b = 0, then (a,b,c,d) = (0,0,0,0) and hence norm(a,b,c,d)^2 = 0, which is a contradiction.
We can argue analogously for the case where k = -1/2. Hence, assuming that rank(DF) is not 2 for some (a,b,c,d) in M leads to a contradiction, so we conclude that rank(DF)=2 for all x in M.
Finally, from this result, we conclude that since F:R^4-->R^2=R^(4-2), M must be a 2-dimensional manifold.
Factor the trinomial below Enter each factor as a polynomial in descending orderx2 plus 18x plus 45?
x2 + 18x + 45 = (x + 15)(x + 3).
Solve x2 plus 18x plus 4 equals 0?
This quadratic equation which will have two solutions can be solved by completing the square or by using the quadratic equation formula.
Completing the square:
x2+18x+4 = 0
(x+9)2+4 = 0
(x+9)2+4-81 = 0
(x+9)2 = 77
x+9 = + or - the square root of 77
x = -9 + or - the square root of 77
If you're not too sure about the procedure of completing the square your maths tutor should be familiar with it.
How do you prove one - tan square x divided by one plus tan square xequal to cos two x?
(1 - tan2x)/(1 + tan2x) = (1 - sin2x/cos2x)/(1 + sin2x/cos2x)
= (cos2x - sin2x)/(cos2x + sin2x)
= (cos2x - sin2x)/1 = (cos2x - sin2x) = cos(2x)
