Simple rate of change and derivative problem 1. Given the function y = x ^ 3-2, when x = 2, △ Y / △ x =? 2. Find the average change rate of y = x ^ 2-2x + 1 near x = - 2

Simple rate of change and derivative problem 1. Given the function y = x ^ 3-2, when x = 2, △ Y / △ x =? 2. Find the average change rate of y = x ^ 2-2x + 1 near x = - 2

The rate of change of a point, that is, LIM (△ x tends to 0) △ Y / △ x, is the derivative of the point
If you have learned how to derive, it is very easy to do it directly. Otherwise, you have to use the limit to do it
The first derivative y '= 3x ^ 2
When x = 2, y '= 12
So △ Y / △ x = 12
Second, the same y '= 2x-2
When x = - 2, y '= - 6
△y/△x=-6

Derivative and rate of change
The radius of a circle is r, and the radius increases after uniform expansion. Note: it seems that the problem is not the area difference divided by the original area

To solve this problem, we investigate the knowledge of integral,
The area formula of circle is s = f (x) = π X & # 178;, (x is radius, X ≥ 0)
The rate of change of the area of the circle is set to t
∫f(x)=πx³/3
t={∫f(R+r)-∫f(R)}/{∫f(R)-f(0)}=((r+R)³-R³)/R³=(1+r/R)³-1

Rate of change and derivative
When the object moves in a horizontal straight line, s (T) = 5T - 2T, find the velocity when t = 0

S(t) = 5t - 2t²
v(t)=S'(t)=5-4t
v(0)=5-4*0=5