The relationship between derivative and differential, derivative can calculate tangent and slope, so what does differential calculate? indicating contrast

The relationship between derivative and differential, derivative can calculate tangent and slope, so what does differential calculate? indicating contrast

The differential of the independent variable (x) is △ x → 0, which is denoted as DX
 
The differential of function (y) is called Dy, which is equal to the derivative of function multiplied by the differential of independent variable
That is dy = y & # 39; DX
 
We know that (△ Y / △ x) &; △ x = △ y, that is, the change of average rate of change (△ Y / △ x) multiplied by X is equal to the change of Y
When △ x → 0, the average rate of change (△ Y / △ x) becomes the instantaneous rate of change, i.e. Y & # 39;, then the above formula can be written as
Y & # 39; DX = Dy, Dy means a very small change of Y
 
Geometric meaning: derivative, also known as tangent slope. Tangent is approximated by secant
As shown in the figure, take two points on the curve, take the connecting line (secant) of the two points as the hypotenuse to make a right triangle. The slope of the secant is the ratio of the opposite side to the adjacent side (the ratio of the two right sides), that is, the average rate of change. The adjacent side can represent △ x, and the opposite side is △ y,
When △ x → 0 (the adjacent side infinitely shrinks) becomes DX, the opposite side (also infinitely shortens) is Dy, and the secant becomes a tangent, and its slope is what you call derivative
 
 
One thing to add: Dy and DX are infinitesimals & nbsp;

If differential (i.e. derivative) represents the slope of the original function or the rate of change of the original function, then what does integral represent?

In terms of geometric meaning, the first-order integral represents the area of the integrand image enclosed by the coordinate axis in the two-dimensional plane, and the second-order integral represents the volume of the integrand image enclosed by the coordinate axis in the three-dimensional space,

What kind of binary function is partial derivative but not differentiable?
Although there is such a function, what causes it to be differentiable but not differentiable

The existence of partial derivatives is a necessary and insufficient condition for differentiability,
A sufficient and unnecessary condition for the continuity of partial derivatives to be differentiable,
The function that can be partial derivative but not differentiable is the case that the partial derivative exists in the neighborhood, but the partial derivative is discontinuous at the point of discussion
[the above statement can not be taken as a whole, because it may be differentiable, but the partial derivatives are discontinuous]
To judge whether a partial derivative is differentiable or not, we have to grasp the definition of differentiability
△z-dz=o（ρ）

Given that the image of a quadratic function passes (3, - 2) (2, - 3) and the axis of symmetry is a straight line x = 1, find the relationship of the quadratic function

x=1
Then y = a (x-1) ² + K
Substituting
-2=4a+k
-3=a+k
So a = 1 / 3, k = - 10 / 3
So y = x & # 179 / / 3-2x / 3-3

The numbers in the figure represent the area of two rectangles and a right triangle, and the area of the other triangle is______ ．

Because Ao × od = 16, OC × OE = 18, so Ao × OD × OC × OE = 16 × 18, and OD × OE = 6 × 2 = 12, so OA × OC = 16 × 18 △ 12 = 24, so the area of the other triangle is 24 △ 2 = 12, answer: the area of the other triangle is 12. So the answer is: 12

What quadrant is the image of the first order function y = (nx-z) - 4x + 5 in

We have known that for the standard formula y = KX + B, K is the slope, and B is the intercept on the y-axis. So when n-4 = 0, the image is a straight line parallel to the x-axis. If 5-Z > O (intersecting with the y-axis), the image is in 1,2

Z = ln (x-2y) for the second order partial derivative
Isn't the second partial derivative four?

Of X, - 1 / (x-2y) ^ 2
Y, - 4 / (x-2y) ^ 2

Disjunctions of appearance

Good looks

In the same rectangular coordinate system, draw the graph of function y = 2x, y = 2x + 3, y = 2x-3
x -2 -1 0 1 2
y=2x -4 -2 0 2 4
y=2x+3 -1 1 3 5 7
y=2x-3 -7 -5 -3 -1 1
① Look at these three images. The shape of these three function images is () and the inclination is ()
② The image of the function y = 2x passes through the origin, and the function y = 2x + 3 intersects the Y axis at the point (), that is, it can be seen as the result of translating () unit lengths of the line y = 2x to (); similarly, the function y = 2x-3 intersects the Y axis at the point (), that is, it can be seen as the result of translating () unit lengths of the line y = 2x to ()
③ The graph of a linear function y = KX + B is a (). When b > 0, it is obtained by y = KX translating () unit length to (); when B < 0, it is obtained by y = KX translating () unit length to ()
④ (1) in the same rectangular coordinate system, the image of y = - 2x + 3 is obtained by translating the straight line y = - 2x to () units, and the image of y = - 2x-5 is obtained by translating the straight line y = - 2x to () units
(2) Translate the line y = - x + 1 downward by 2 units to get the line ();
(3) The line y = 1 / 2x-2 can be obtained by translating the line y = 1 / 2x + 3 to () units

① (2) the image of the function y = 2x passes through the origin, and the function y = 2x + 3 intersects with the Y axis at the point (0,3), that is, it can be seen as the result of the translation (3) unit lengths of the straight line y = 2x upward. Similarly, the function y = 2x-3 intersects with the Y axis

A 30cm long wire is divided into two parts, each part is bent into an equilateral triangle, and their area and minimum value are

Let one part of the wire be x and the other 30-x
The area of the two parts is calculated as
Three times x ^ 2 under 36 percent root,
3 times (30-x) ^ 2 under 36% root
The sum of the two parts is equivalent to finding out
The point with the smallest value of x ^ 2 + (30-x) ^ 2
15, obviously,
The area of the two parts can be calculated as
2 / 25 times under the root 3