If f (x, y, z) = z-x ^ 2-2y ^ 2, then the directional derivative of F at (1,1,3) along the direction of the upper normal of the surface z = x ^ 2 + 2Y ^ 2 at this point is__ .

Z = x ^ 2 + 2Y ^ 2g (x, y, z) = z-x ^ 2-2y ^ 2gx (x, y, z) = - 2x Gy (x, y, z) = - 4Y GZ (x, y, z) = 1 substituting x = 1, y = 1n = (- 2, - 4,1) EF / El = EF / ex * (- 2) / radical 21 + EF / ey * (- 4) / radical 21 + EF / EZ * 1 / radical 21 = (4x + 16y + 1) / radical 21 substituting x = 1, y = 1 directional derivative = radical 21

Let m be the set of all functions f (x) satisfying the following properties,
It is known that M is a set of all functions f (x) satisfying the following properties. For function f (x), there exists a constant k such that for any two independent variables x1, X2 in the domain of function f (x), all | f (x1) minus f (x2) | less than or equal to K | X1 minus x2 | holds. (1) the known function g (x) = ax ^ 2 + BX + C belongs to M. write out the conditions that real numbers a, B, C must satisfy. (2) for the element H (x) = root sign (x + 1), X is greater than or equal to 0, Find the minimum value of the constant K satisfying the condition (3) use the conclusion that | Sina | is less than or equal to | a | to prove that the function p (x) = asin (AX) belongs to m, where a is a real constant

(1) When a = 0, G (x) = BX + C makes / g (x1) - G (x2) / ≤ K / x1-x2 / i.e. / bx1-bx2 / ≤ K / x1-x2 / ‖ solution / B / ≤ K, C ∈ r when a ≠ 0, G (x) = ax & # 178; + BX + C makes / g (x1) - G (x2) / = / ax1 & # 178; + bx1-ax2 & # 178; - bx2 / ≤ K / x1-x2 / solution / a (x1 + x2) + B / ≤ 1

Let e ^ y + xy = 1 determine the implicit function y = y (x), then y '(0) =?
Does Y '(0) mean x = 0 or y = 0?

Let x = 0 get f '(0) e ^ y + xy = 1, then e ^ y = 1-xy, both sides of X derive from X, e ^ y * y' = (XY) '= y + XY', then y '= Y / (e ^ Y-X); when x = 0, substitute e ^ y + xy = 1 to get y = 0, then y' (0) = Y / (e ^ Y-X) ｜ (x

Let y = y (x) be the implicit function determined by the equation E ^ y + xy = e, and find y ^ n (0)

Let x = 0 be substituted into the equation E ^ y + xy = e to get e ^ (Y (0)) + 0 × y (0) = e, which is reduced to e ^ (Y (0)) = E
So y (0) = 1
So y ^ n (0) = 1

51.2 * 8.1 + 11 * 9.25 + 537 * 0.19 use the simplest method to do it in the shortest time, not calculator

Original formula = 51.2 * 8.1 + 11 * 9.25 + 53.7 * 1.9 = 51.2 * 8.1 + 11 * 9.25 + 51.2 * 1.9 + 2.5 * 1.9 = 51.2 (8.1 + 1.9) + 11 * 9.25 + 2.5 * 1.9 = 512 + 1.1 * 92.5 + 2.5 * 1.9 = 512 + 1.1 * (90 + 2.5) + 2.5 * 1.9 = 512 + 99 + 2.5 (1.9 + 1.1) = 618.5

What are the essential differences in the University's understanding of the concept and definition of function?

The concept of functions understood in junior high school is the same as that understood in senior high school. They are all mapped from a subset of real number set to a subset of real number set through a certain function, but the mapped functions are different. Junior high school students learn nothing more than linear function and quadratic function, and they learn nothing more than seeking the maximum value. Senior high school students will have a lot of exponential function, logarithmic function and so on, and the times will be higher, Even functions can be regarded as infinite polynomials (expanded into power series), when they study, they will use the method of derivation to study monotonicity, slightly add some differentiability and continuity, and study a little diversity
The function learned in university is quite deep. The domain of definition is complex field, and the range of value is complex field. This subject is called complex variable function; the domain of definition is probability space, and the range of value is (separable) Banach space (real number field or complex number field is separable Banach space, and the probability theory of senior mathematics majors will study abstract function, while the probability theory of junior mathematics majors will study real number field), This subject is called probability theory; the domain of definition is function space, and the domain of value is complex number field. This subject is called functional analysis. The concept is still similar to junior high school and senior high school, but because the research set can be expanded from number field to very abstract space, such as Lebesgue function space, LP space, Lie group, and other measurable space, plus many new tools, such as differential tool, series theory, etc, Another example is to define the function class through quotient space, and treat the function which is almost equal everywhere as a function (just like the odd number as 1, even number as 0, and carry out congruence operation). Many and many extensions will give birth to many interesting and difficult subjects. But from the concept point of view, it only generalizes the domain of definition and range of value; from the skill point of view, it will use a lot of tools, There are analytic and algebraic
Therefore, you can see that set theory is the foundation of mathematics, not function. I remember teacher Orsay said when he gave us training that, to tell you the truth, set is the foundation of mathematics. I didn't believe it at that time. Later, I learned all the time and found the importance of set. Mathematical research has also been studying the structure of set and the promotion of set
If there is a reward, give it more. I have no wealth. I have a question to ask. For my sake of typing so many words, give it more

0.1 times 2 / 7 is equal to?

One out of 35

The number of subsets of finite set
The set of n elements has () subsets
The set of n elements has () proper subsets
The set of n elements has () nonempty subsets
The set of n elements has () nonempty proper subsets

1: 2 ^ n
2: 2 ^ n-1
3: 2 ^ n-1
4: 2 ^ n-2
It's just time for us to review here in senior three

Equation: 5 times 3 / 7 + x = (56 + x) times 7 / 15 process

5×3/7+x=(56+x)×5/7
15/7+x=40+5/7x
x-5/7x=40-15/7
2/7x=265/7
x=132.5

X divided by two-thirds equals seven out of nine

X = 14 / 27, right

If f (x, y, z) = z-x ^ 2-2y ^ 2, then the directional derivative of F at (1,1,3) along the direction of the upper normal of the surface z = x ^ 2 + 2Y ^ 2 at this point is__ .