A very way to understand this very logically.
First know what is multiplication?
It is recurring addition.
So if we multiply 3 by (-2), that means:-
(-2)+(-2)+(-2)= -6
Now see the examples below:
4 X (-4) = -16
3 X (-4) = -12
2 X (-4) = -8
1 X (-4) = -4
0 X (-4) = 0
Have you noticed a trend. Notice carefully that as the numbers which are being multiplied by (-4) are decreasing 1 by 1 the product is increasing by (+4).
So for the numbers 4,3,2,1,0 the products are:-
-16
-16+4= -12
-12+4= -8
-8+4= -4
-4+4=0
So what will be in the next step when we do:
(-1) X (-4)=?
The series will continue 0+4 an will result to a positive 4
and it will go like this:
(-1) X (-4)= 4
(-2) X (-4)= 8
(-3) X (-4)= 12
+4 is added in each step. This was the best explanation to this according to me which was told to me by my teacher which he made himself.
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Your answer will depend on the parameters of the instructions. If you're looking for the first derivative, simply use the product rule by changing the denominator to a negative exponent and bringing it up (take the negative square root of the quantity x-2 to the top). Then, follow the rules of calculus and algebra. Wow, that's a mess. Let's see... you get "the quantity x cubed plus 6x squared plus 3x plus 1 times the quantity -1(x-2) raised to the negative second plus the quantity x-2 raised to the negative first times the quantity 3x squared plus 12x plus 3." This is because of the Product Rule. Simplifying (by factoring out (x-2) raised to the negative second and combining like terms) gives us "(x-2) raised to the negative second times the quantity 2x cubed minus 24x minus 7." This can also be written as "2x cubed minus 24x minus 7 all over the quantity x-2 squared." f'(x)= 2x^3-24x-7 (x-2)^2
The first constraint implies x*y=363, while you are finding the minimum of x+3y. You can do this with multivariable calculus approaches, but I will assume the simplest method is desired and use single-variable calculus methods. Since xy=363, y=363/x Substitute this into x+3y, and you get: f(x)=x+3(363/x)=x+1089/x Since you have this function as a function of a single variable now, you can minimize it like any other single variable function. f'(x)=1-1089x-2 by setting f'(x)=0, the solutions x=33 and x=-33 are found. x=-33 is an extraneous solution, since you are required to have positive numbers. So, you must investigate x=33 and confirm it is a minimum. If x=33 is a minimum of f(x), then values less than x=33 will yield a negative value in f'(x), since the function f(x) must slope downward up until it reaches a minimum. Conversely, values greater than x=33 will yield positive values in f'(x). 1<33 f'(1)=-1088 -1088<0, so f'(1) is negative 40>33 f'(40)=511/1600 511/1600>0, so f'(40) is positive. In a final summating statement: Since the function f(x)=1-1089/x2 has a critical point at x=33 and since values less than x=33 result in negative values of f'(x) and values greater than x=33 result in positive values of f'(x), x=33 is a local minimum of f(x). To get your final final answer: x=33 y=363/x implies y=363/33=11 x=33,y=11
There are many pairs of numbers that add of to 64. One is (1,63), and the product is 63. Another is (2,62), and the product is 124. There seems to be something missing from the question, perhaps some constraint on the product. Please restate the question.
A product in maths is found by multiplying numbers together. Not sure about the use of the word special?
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