In Chapter 3, for any piecewise function, if the left and right derivatives exist at the piecewise point, can we say that the function is continuous at the tapping point? Both left and right numbers exist, but they are not necessarily equal

In Chapter 3, for any piecewise function, if the left and right derivatives exist at the piecewise point, can we say that the function is continuous at the tapping point? Both left and right numbers exist, but they are not necessarily equal

No
For example, y = | x | is continuous at the segment point x = 0;
The sign function y = SGN (x) is discontinuous at the point x = 0
The above functions have left and right derivatives at the piecewise point

Let z = f (x, y) be determined by the equation Z ^ 3-3xyz = a ^ 3,

By using the derivative rule of implicit function, X is derived from both ends
3z^2*∂z/∂x-(3yz+3xy∂z/∂x)=0
Namely
∂z/∂x=yz/(z^2-xy)

What is the second order partial derivative of the implicit function z = Z (x, y) determined by the equation x + y + Z = e ^ Z?

The answer is shown in the picture

Simple calculation of 1 / 7 △ 7 + 7 △ 1 / 7, 15 △ 8 / 9 + 15 × 7 / 8

1 / 7 △ 7 + 7 △ 1 / 7 = 1 / 49 + 49 = 49 and 1 / 49
15÷8/9+15×7/8=15×9/8+15×7/8=15×（9/8+7/8）=15×2=30

Xiaoming and Xiaogang bought 25 exercise books. If Xiaoming gave Xiaogang three, half of Xiaoming's exercise books would be equal to one third of Xiaogang's exercise books,
How many exercise books did Xiao Ming and Xiao Gang buy?

Suppose Xiaogang bought x, Xiaoming bought (25 - x)
(25-X-3)/2=(X+3)/3
X=12
So Xiaogang bought 12 and Xiaoming bought 13

English translation
The Flow Diagram is keyed or numbered and illustrates the sequence of production and inspection from receiving through shipping.

The flow chart is marked and numbered to illustrate the sequence of production and inspection from receiving to shipping

16 degrees, 21 minutes 36 seconds 18 degrees 33 minutes + 22 degrees 55 minutes 90 degrees - 37 minutes 11 seconds

In degrees, minutes and seconds, 11.16 degrees = 11 ° 9.6 '= 11 ° 9'54'
Use degree to express 18 degrees, 21 minutes and 36 seconds = 18 ° 21.6 '= 18.36 degree
18 degrees 33 points + 22 degrees 55 points = 40 ° 88 '= 41 ° 28‘
90 degrees - 37 minutes 11 seconds = 52 ° 49‘

In a cylindrical glass container with a bottom radius of 10 cm and a water depth of 8 cm, an iron block with a length and width of 8 cm and a height of 15 cm should be placed in the container. (1) if the iron block is placed horizontally in the water, how many cm does the water surface rise? (2) if the iron block is placed vertically in the water, how many cm does the water surface rise?

（1） When the iron block is placed horizontally (water can submerge the iron block): (8cm) &# 178; × 15cm △ 10 &# 178; cm &# 178; × 3.14) = 960cm &# 179; △ 314cm &# 178; = 3.057cm answer: the water surface rises about 3.1cm. (2) when the iron block is placed vertically (assuming that water submerges the iron block n cm), the equation is as follows: (8cm) &# 17

4 8 / 9 16 / 27 () 36 / 125 216 / 49 according to the rule, what should be filled in the brackets? I want the reasoning process!
64/25
Tell me more about it. I can't understand it. Can you tell me how it came from one by one!

64/25
4 can be regarded as 4 / 1
The numerator of odd term and denominator of even term satisfy n ^ 2
The denominator of odd term and the numerator of even term satisfy n ^ 3
So the numerator of () should be 4 ^ 3 = 64 and the denominator 5 ^ 2 = 25

Examples of positive and negative proportion and disproportion
30 for each

Fixed working hours, total amount of work and working efficiency. (positive proportion) fixed total amount of work, working hours and working efficiency. (negative proportion) surface area and volume of cylinder. (out of proportion)