Please give popular examples in life to illustrate the limit, what does the derivative mean! High Sc

Have you ever walked on a straight and long road, the left and right sides of the road are parallel, but at a glance, at the infinity of the road, the two lines intersect, you may think that this is just a look; Another example you can do on your own is to walk from your house to school, but only half the distance at a time: the first time 1 2, the second time 1 4, 1 8 , 1 16, so that it is halved indefinitely, and you will find that you will never get to school. There are many more examples like this, and they are very famous in the mathematical world, such as:

1) Remove the 1 3 in the middle of a line segment, and replace it with the two sides of a regular triangle (its length is the given line segment 1 3), and repeat it continuously, and a snowflake is formed, and the characteristic of this figure is that the area tends to be fixed, and the perimeter tends to infinity.

2) Nine equal parts of a cube, then dig out the middle one, and dig out the middle one of the remaining eight nine equal parts each, and repeat.

This figure is even more magical, with the volume tending to 0 and the area tending to infinity.

As for derivatives, they are directly understood as differentials. For example, if you divide a circle into n arcs, you will find that each arc is approximately equivalent to a straight line, and this discovery provides us with a way to find the area of the curved edge in a coordinate system.

I think just a few examples are easy enough to understand.

Say me!? You're so humorous

Limits, like when you run, hang a hundred dollars in front of your head. No matter how fast you run, you're only going to get close to it, but you can't touch it.

Geometrically, the derivative is to draw the distance you run as a curve over time, and each point on the curve is the value of the distance traveled (for example, at 1 minute and 0 seconds, the corresponding point on the curve is 15 meters, at 2 minutes and 1 second, it is 31 meters, etc.). At a certain point, for example, when you just run to 1 minute and 0 seconds, draw a tangent line through it, and the slope of this tangent line is the derivative of your running at 1 minute and 0 seconds, which is your instantaneous speed at this time.

Algebraically speaking, the derivative is: your instantaneous velocity at 1 minute and 0 seconds.

Imagine 1 x when x is large, for example, when 999999999999999999, 1 x is close to 0, so 0 is the limit of the function y=1 x when x approaches infinity! The derivative meaning is the rate of change, for example, the equation of free fall is h=1 2gt 2, and the rate of change of distance h at the time of its derivative is that the rate of change of distance to time at this time is the derivative of h, that is, the velocity is the derivative of h to t.

I didn't study well in the first year of high school upstairs.

If the nth power of a is b

So. log a(b)=n

That is to say, a derivative is an operation to find exponents knowing the base number and power.

Is a derivative the same as a derivative?

The limit is. Cycle = 1

Limit: Two related variables x and y, and y=f(x). When one of the variables x is infinitely close to (but not equal to) a fixed value that may be constant or infinite, the tendency of the other variable y, if this trend is also constant, then the limit exists and is this constant.

For example, if x and y are used to represent the time and distance of a uniform motion, then x and y are two related quantities, and this relation is y = f(x) = 3 x (let the speed of uniform motion be 3). Now we can say that when x ->0, the limit of y=f(x) is 0.

That is to say, when the amount of time of motion tends to 0, the distance of motion also tends to 0. When x ->2, the limit of y=f(x) is 6. That is to say, when the amount of time of motion tends to be 2, the distance of motion also tends to 6.

Derivative: To put it bluntly, it is the rate of change. This rate of change is, of course, also relative to two related variables.

It is the degree to which a change in one of them affects a change in the other quantity. For example, when x > 1, the derivative of y=x3 is greater than the derivative of y=x2 (this is not accurate, but it can be understood), and why, because when x changes a little, y=x 3 changes more than y = x 2, which means that y = x 3 is more affected by x than y = x 2. When x changes from 1 to 2 (an increase of 1), y = x 3 changes from 1 to 8 (an increase of 7), and y = x 2 changes from 1 to 4 (an increase of 3).

Looking at the image of these two functions, y=x3 is also steeper than y=x2.

If you look back, the definition of the derivative is actually as follows: the change in the dependent variable, the change in the independent variable, and then the limit.

This is a theoretical question. It's hard to give an example.

The expression of the derivative of the old demonstration of the extreme distribution table is f'(x0)=lim[x→x0][f(x)-f(x0)]/x-x0)。

Differential y=f(x), then dy=f'(x) dx limit form: 1) f'(x0)=lim(x→x0)[f(x)-f(x0)]/x-x0)2)f'(x)=lim( x 0)[f(x+ x)-f(x)] xd denotes the microvolt bridging fraction.

The limit of the derivative is the limit of the derivative expression. Mathematical analysis is based on the concept of limit, limit theory (including series) as the main tool to study the function of a discipline with the limit of the limit has the characteristics of uniqueness, boundedness, sign-preserving, inequality preservation, compatibility of band and real number operations, and relationship with subcolumns.

Take advantage of the limit. The thinking method gives the concepts of continuous functions, derivatives, definite integrals, divergence of series, partial derivatives of multivariate functions, divergence of generalized integrals, double integrals, and curvilinear integrals and surface integrals.

Derivative and limit finding are two completely different concepts. The limit is the premise of the derivative.

First of all, the derivative is generated from finding the tangent of the curve.

This problem arises, so the derivative can be used to find the slope of the tangent of the curve at any point.

Second, some infinitives can be solved using derivatives.

Limit (i.e., 0 0, infinity, infinity, etc.), this method is called "Lobida's law."

Take y=x as an example, when x tends to 1, y also tends to 1, which is the limit.

Deriving y=x from x yields y=2x, and the geometric meaning of this equation is that the slope of the tangent of the function at x point is 2x

That is, when x=1, y=2, it means that the slope of the tangent line of the function y=x at the point of x=1 is k=2

The reason why y=x derives y=2x after deriving x is to use the method of finding tangent lines to take two points on the image to form a straight line, and when the two points are constantly close to each other and finally become a point, the straight line is also the tangent of the image at the starting point. The method of deriving this process is the limit method. Therefore, derivation and limit finding are not the same in and of themselves.

You can see the answer downstairs @花苗贵树, which is very concise.

"The limit is just a number: the limit of x tending to x0 = f(x0). The derivative, on the other hand, is the instantaneous rate of change and is the slope of the function at that point x0. The derivative has one more part than the limit that expresses "process".

The idea of derivatives was first introduced by the French mathematician Fermat to study the problem of extreme values, but directly related to the concept of derivatives are the following two problems: first, the known laws of motion find velocity; Second, the tangent of the known curve is found. This was established by the British mathematician Newton and the German mathematician Leibniz in the course of studying mechanics and geometry, respectively.

The limit is the basis of the derivative, in a sense, the essence of the derivative is a limit, when the increment of the independent variable tends to zero, the limit of the ratio of the increment of the value of the function to the increment of the independent variable is the derivative. This limit reflects the trend of change of the function and describes the speed of change of the function.

One of the backgrounds of derivative research is to find the tangent of the curve, and the slope of the tangent of the curve at a certain point is the geometric meaning of the derivative, so to find the tangent slope of the function at a certain point is to find the derivative of the function at that point, and of course, to find the limit value of the slope of the secant.

It's easy to understand, first of all, you know that the derivative definition is lim(f(x)-f(x0)) (x-x0), and this equation is very important because it shows that a derivative can only exist if f(x)-f(x0) is an infinitesimal of the same order x-x0 (in this case the derivative is a non-zero constant) or a higher order infinity (in this case the derivative is 0). Conversely, if f(x)-f(x0) is a low-order infinitesimal of (x-x0), then the limit is determined by the derivative definition and must not exist.

After you understand this, continue to look at the following, the first one is the simplest, when a is not equal to 0, it means that f(x)-f(x0) and x-x0 are infinitesimal solutions, so the derivative exists and is equal to a, if a=0 means that f(x)-f(x0) is the higher order infinitesimal of x-x0, so the derivative is 0

Second, where k>1, shows that f(x)-f(x0) is the higher-order infinitesimal of x-x0, so the derivative must exist and must be 0

<> from the function b=f(a), we get the set of numbers a and b, in a, when dx is close to itself, the limit of the function at dx is called the differentiation of the function at dx, and the central idea of the differentiation is infinite division. Differentiation is the main part of the linearity of a function that focuses on a variable.

The derivative defines the rock formula:f'(x)=lim(h->0)[f(x+h)-f(h)]/h；lim(h→0)[f(0+h)-f(0-h)]/2h=2lim(h→0)[f(0-h+2h)-f(0-h)]/2h=lim(h->0)2f'(0-h) when f'(x) is continuous at x=0 to have lim(h->0)2f'(0-h)=2f'(0)。

Derivatives are local properties of functions. The derivative of a function at a point describes the rate of change of the function around that point. If the independent variables and the values of the function are real numbers, the derivative of the function at a certain point is the tangent slope of the curve represented by the function at this point.

The essence of derivatives is to perform a local linear approximation of a function through the concept of limits. For example, in kinematics, the derivative of the displacement of an object with respect to time is the instantaneous velocity of the object.

The above content reference: Foreland Encyclopedia - Derivatives.