The Derivative

[nycmathguy]

1. In what way is a limit related to the derivative?

2. Is the formula here called the difference quotient? If so, why give it that name?

3. Is the derivative the instantaneous rate of change?

4. Can you explain the figure below in simple words?

Here is how math.net explains the figure. I am trying to make sense of the geometric interpretation of a derivative.

 

[MathLover1]

1. The derivative of function f at x=c is the limit of the slope of the secant line from x=c to x=c+h as h approaches 0. Symbolically, this is the limit of [f(c)-f(c+h)]/h as h→0

2. (f(x+h)-f(x))/h is called the formula of difference quotient.

3.yes, the derivative, f (a) is the instantaneous rate of change of y = f(x) with respect to x when x = a. When the instantaneous rate of change is large at x1, the y-values on the curve are changing rapidly and the tangent has a large slope. The instantaneous rate of change is the change in the rate at a particular instant, and it is same as the change in the derivative value at a specific point.

4. I would explain it same way as above

 

[nycmathguy]

1. The derivative of function f at x=c is the limit of the slope of the secant line from x=c to x=c+h as h approaches 0. Symbolically, this is the limit of [f(c)-f(c+h)]/h as h→0

2. (f(x+h)-f(x))/h is called the formula of difference quotient.

3.yes, the derivative, f (a) is the instantaneous rate of change of y = f(x) with respect to x when x = a. When the instantaneous rate of change is large at x1, the y-values on the curve are changing rapidly and the tangent has a large slope. The instantaneous rate of change is the change in the rate at a particular instant, and it is same as the change in the derivative value at a specific point.

4. I would explain it same way as above

We will return to the derivative and rates of change ideas this weekend.