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Proof of: if f(x) > g(x) for a < x < b then a∫b f(x) dx > g(x).
Because we get f(x) ≥ g(x) then we knows that f(x) - g(x) ≥ 0 on a ≤ x ≤ b and therefore by Property 8 proved as above we know that,
a∫b f(x) - g(x) dx > 0
We know as well from Property 4,
a∫b f(x) - g(x) dx = a∫b f(x) dx - a∫b g(x) dx
Therefore, we then get,
a∫b f(x) dx - a∫b g(x) dx > 0
a∫b f(x) dx > a∫b g(x) dx
Proof of: If m ≤ f(x) ≤ M for a ≤ x ≤ b then m (b - a)≤ a∫b f(x) dx ≤ M (b - a).
Provide m ≤ f(x) ≤ M we can utilize Property 9 on each inequality to write,
a∫b m dx < a∫b f(x) dx ≤ a∫b M dx
So by Property 7 on the left and right integral to find,
m(b -a) < a∫b f(x) dx ≤ M (b -a)
If A be the area of a right triangle and b one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is 2 Ab /√ b 4 +4A 2 . An
How many integers satisfy (sqrt n- sqrt 8836)^2 Solution) sqrt 8836 = 94 , let sqrt n=x the equation becomes... (x-94)^2 (x-94)^2 - 1 (x-95)(x-93) hence 93 8649 the number o
Chain Rule : We've seen many derivatives. However, they have all been functions similar to the following kinds of functions. R ( z ) = √z f (t ) = t 50
Q. How to calculate Mode? The mode of a data set is the value that is repeated most often in the data set. It has the highest frequency. There can be one, more than one, or n
A jeweler has bars of 18-carat gold and 12-carat gold. How much of every melted together to obtain a bar of 16-carat gold, weighing 120 gm ? It is given that pure gold is 24 carat.
#k1=f(Tn, Xn), k2=f (Tn + H.Y,Xn + H.Y.k1) Xn+1=Xn + H(a.k1+ b.k2) Find a relation between Y,a and b so that the method is second order consistent.
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1) Find the are length of r(t) = ( 1/2t^2, 1/3t^3, 1/3t^3) where t is between 1 and 3 (greater than or equal less than or equal) 2) Sketch the level curves of f(x,y) = x^2-2y^2
(-2x^2y4)(10xy^2)^3
1.If a+b=2b and ab+cd+ad=3bc,prove that a,b,c,d are in A.P 2.The nth term of an A.P is an+b.Find the sum of the series upto n terms.
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