From 1 to 1000, there are several multiples of 2 and 3 at the same time There should be algorithms

From 1 to 1000, there are several multiples of 2 and 3 at the same time There should be algorithms

The least common multiple of 2 and 3 is 6
6*166=9961000
So from 1 to 1000, there are 166 multiples of 2 and 3

1 find the derivative of function y = x square outside x = x0!
2. If f (x) = ax + 4f '= 2, find the value of A. 3. Find the curve y = 2x + 1 at P (- 1,

1.
y'=2x
When x = x0, y '= 2x0
two
f'(x)=a
F '= 2, so a = 2
three
Y = 2x ^ 2 + 1
When x = - 1, y '= 4x = - 4, which is the slope, the formula of point slant formula Y-3 = K (x - (- 1)) = - 4 (x + 1)
That is y + 4x + 1 = 0

How to prove: when y is a continuous periodic function, its derivative is also a periodic function, and the periods are equal. (including specific steps)

Let f (x) be a continuous differentiable periodic function with period T
f'(x+T)=lim(t->0) [f(x+T+t)-f(x+T)]/t
=lim(t->0) [f(x+t)-f(x)]/t
=f'(x)
So f '(x) is also a periodic function with period T

Is the derivative of a differentiable periodic function a periodic function?

Yes!
The derivative of a differentiable periodic function is a periodic function!

Is the derivative of a periodic function also a periodic function
If f (x) is a differentiable function with period T, then f '(x) must be a periodic function with period T?

Yes
Periodic function means that f (x) = f (x + T). For X in the domain of definition, t is its period
Then f '(x) = LIM ((f (x + Δ x) - f (x)) / Δ x)
=lim((f(x+t+Δx)-f(x+t))/Δx)=f'(x+t)
So f '(x) is also a periodic function with period T

If the integrand is a periodic function, is the original function a periodic function?
RT, the proof is given

Not necessarily, counter examples can be given
For example, y = sin (x) + 1 is a function with a period of 2 * PI,
But its ∫ YDX = - cos (x) + X is not a periodic function
That's it!

If the derivative function is a periodic function, then the original function is also a periodic function! Is the proposition correct? (please give a counter example or a proof process)

F (x) = SiNx + X is not a periodic function, but f '(x) = cosx + 1 is a periodic function
The original proposition does not hold

A continuous function is a periodic function, so its original function must be a periodic function, right?

Not necessarily. For example, 1 + cos [x], its original function x + sin [x] is obviously not a periodic function

Is the first derivative of a function differentiable everywhere continuous? Why?

Not necessarily. Let me give you a counterexample
f(x)=x²sin(1/x) x≠0
0 x=0
The function is differentiable everywhere in real number, but the derivative function is discontinuous at x = 0
You can try to do it yourself. If you need me to do it for you,

If the derivative of a function exists at some point, is the derivative continuous at this point
Although I know this sentence should be wrong, I can't find a counterexample. Can you help me find a counterexample

If the derivative of a function exists at a certain point, its derivative may not be continuous at this point
　f(x) = (x^2)sin(1/x),x ≠ 0,
　　= 0,x = 0
It is differentiable everywhere on R, but its derivative function is discontinuous at x = 0