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[College Advanced numbers] how to distinguish continuous points, removable breakpoints, infinite breakpoints and oscillatory breakpoints?
In high numbers, a breakpoint is usually either the first type or the second type
Just compare the left and right limits of the function at the discontinuity
If left limit = right limit, it is a removable breakpoint; if not equal, it is a jumping breakpoint; If at least one of the left and right limits is infinite (does not exist), it is an infinite discontinuity. As for the oscillation discontinuity, there are only sine function, cosine function and some periodic functions (elementary functions)
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How to easily understand breakpoints. A higher number
Undefined point
Find the function z = SiNx + siny + sin (x + y) (0
When the partial derivative of Z to x = cosx + cos (x + y) = 0, cosx = - cos (x + y) = cos (pi-x-y), so x = pi-x-y. similarly, when the partial derivative of Z to y = 0, y = pi-x-y. so x = y = pi / 3. At this time, z = 3, radical 3 / 2
When x = 0, z = 2siny, maximum 2, minimum 0. It is the same when y = 0. When x = pi / 2, z = 1 + root 2Sin (y + pi / 4) maximum 1 + root 2, minimum 2
To sum up, the maximum value of Z is 3 radical 3 / 2 and the minimum value is 0
The minimum value of the function y = (square of X + 5) divided by (x2 + 4) = 4 under the root sign is [(x ^ 2 + 4) + 1] / root sign (x ^ 2 + 4) = (x ^ 2 + 4) / root sign (x ^ 2 + 4) + 1 / root sign (x ^ 2 + 4). How did this step come out
（m+n) / p = m / p + n / p
[(x ^ 2 + 4) + 1] / root sign (x ^ 2 + 4)
=(x ^ 2 + 4) / root sign (x ^ 2 + 4) + 1 / root sign (x ^ 2 + 4)
Find the minimum value of function y = x square under the root sign + 9 + x square under the root sign - 10x + 29 Where x square + 9 and x square - 10x + 29 are under the root sign! It's the sum of two roots! The correct answer seems to be five times the root 2,
5 times root 2
First, let's look at this problem: make a rectangular coordinate system, determine two points a (0, - 3) and B (5,2), a moving point P moves on the x-axis, and find the sum of the distances from point P to two points ab. obviously, the expression listed in this problem is your problem stem. So what is the minimum value? Obviously, when p is the intersection of the connecting line between two points AB and the x-axis, the minimum value is: under the root sign (5 square + 5 square) = 5 times root 2
Find the minimum value of function y = root sign (x square - 10x + 29) + root sign (x square + 9)
This is a typical example of solving problems with the combination of number and shape. The solution is as follows (Note: √ represents the root, x ^ 2 represents the square of x) y = √ (x ^ 2-10x + 29) + √ (x ^ 2 + 9), that is, y = √ [(X-5) ^ 2 + 2 ^ 2] + √ (x ^ 2 + 3 ^ 2). In the above formula, y is regarded as point (x, 0) to point a (5,2) and point B (0,3) in the plane rectangular coordinate system
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