For the function y = f(x) , what does the quotient ∆y/∆x represent?

Question

a. the average rate of change for f(x) over the interval x1 # x # x2 b. the derivative of f at x1 c. the mean value of f(x) d. the instantaneous rate of change for f(x) at x1

Answer 1

It is the derivative of the function f(x). It is the slope of the curve at any point(Only a ratio between Change in Y to change in X). It is a General formula or notation irrespective of the value of X.

According to the value of X the value of dy/dx will vary. If it is a straight line dy/dx is constant. If it is a curve then it is a slope(rate of change) at that point(value of X).

Answer 2

(a) is the correct option. Let me also tell you what the other options mean. I think you can obviously figure out that (b) & (c) mean nothing. Now, if you take the limit of the quantity in the quotient as the change in the variable (here, x) approaches (or tends to) zero, you get (d), 'coz its the definition of (d).

Answer 3

d answer is a) avg change
when we giv delta it means avg because it is over a bigger interval while for derivative and instantaneous u need to giv limit of the function or use dx/dy
hope u understand it