Why is the difference quotient important for calculus?
One of the cornerstones of calculus is the difference quotient. The difference quotient — along with limits — allows you to take the regular old slope formula that you used to compute the slope of lines in algebra class and use it for the calculus task of calculating the slope (or derivative) of a curve.
What does a difference quotient tell us about a function?
The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.
When would you use a difference quotient?
To find the slope of the secant line between two points in a function, we use difference quotients. Difference quotients are used with limits in the definition of the derivative and are important to understand, but can be difficult to compute and simplify.
What does it mean to evaluate the difference quotient?
Is the difference quotient always the same?
That’s it! The difference quotient is the same as the slope of the line through any two points, (x, f(x)) and (x + h, f(x + h)), on the function. … You also may remember that the slope of a line is the rate of change of y with respect to x.
How is the difference quotient related to the derivative?
The difference quotient formula is a part of the definition of the derivative of a function. By taking the limit as the variable h tends to 0 to the difference quotient of a function, we get the derivative of the function.