[Knowledge Point] the derivative of number theory

1. Preface

I started the annual combinatorial math class again. This time last year I can say is not understand a word, now a year has passed, although a high has passed, and then eggs Ah! Three hours, learning. Permutation combinations, function limits, derivative, differential, definite integral and indefinite integral, I'm going to ask the math group how long it took to learn ... So we can only have some supplements after class.

2. Function Change Rate

Any function, there is its own rate of change. The rate of change is not a change, so do not take units. and observe the Y.

When x=0->1, the function y value increases the y=f[1]-f[0]=4, then the rate of change is: y/x=4/1=4;

When x=1->2, the function y value increases the y=f[2]-f[1]=1, then the rate of change is: y/x=1/1=1.

It can be seen that the change rate of function decreases with the increase of X value.

3. Function limit

Contact physics A compulsory knowledge, can not help thinking: when we calculate the speed of a moving object, obviously can only find a certain time period of the average speed, the book is written on the concept is that when t enough hours is instantaneous speed. How small is small enough? Now let's do an experiment. There is an object whose displacement formula is: h[i]=-4.9t^2-13.1t. Assume that the instantaneous velocity of the t=2 is required. We first consider the situation near t=2, take a moment any time the t, T is the amount of change, now divided into t<0 and t>0 two case discussion:

When T<0, v= (h[2]-h[2+ T])/(n (t)) = (4.9 t^2+13.1 t)/-t=-4.9 t-13.1;

When t=-0.01, v=-13.051; when t=-0.001, v=-13.0951;

When t=-0.0001, v=-13.09951; when t=-0.00001, v=-13.099951;

When t=-0.000001, v=-13.0999951; when t=-0.0000001, v=-13.09999951;

When T<0, v= (h[2+ t]-h[2])/((t)-2) = ( -4.9 t^2-13.1 t)/t=-4.9 t-13.1;

When t=0.01, v=-13.149; when t=0.001, v=-13.1049;

When t=0.0001, v=-13.10049; when t=0.00001, v=-13.100049;

When t=0.000001, v=-13.1000049; when t=0.0000001, v=-13.10000049.

We found that when T approached 0 o'clock, V approached a definite value of 13.1. Therefore, we can determine that when t=2, the instantaneous velocity of an object is -13.1m/s.

In general, the standard notation is:

Said, "when t=2, t approaches 0 o'clock, the average speed v approaches the determined value-13.1".

4. The concept of derivative

Generally, the instantaneous rate of change in the function y=f (x) at x=x0 is:

We call it the derivative of function y=f (x) at X=x0, which is recorded as F ' (x0) or y ' |x=x0, i.e.:

5, the geometric meaning of the derivative

First determine the two times the origin of the function line Y=f (x), take any point on the line is P, as its tangent, the other line to take any point of PN, Connection ppn. Mobile PN,:

We found that when the point PN approached P, the secant PPN approached the tangent pt. First, it is easy to know that the slope of the secant PPN is:

As PPN Wireless approaches PT, the KN will also approach the PT K value. Set X=x (Pn)-X (P), the derivative of function f (x) at point P is the slope K of the tangent pt, i.e.:

6. Calculation of derivative

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[Knowledge Point] the derivative of number theory