Mastering Calculus: The Difference Quotient For Accurate Derivatives And Tangent Lines

Difference quotient for f at x=a

The graph of f is concave downward on the interval a

The difference quotient for a function f at x=a is a mathematical expression that represents the slope of the secant line between two points on the graph of the function. More specifically, it is defined as:

[f(a+h) – f(a)] / h

where h is a small change in the value of x (typically approaching zero), and (a,h) and (a+h, f(a+h)) are two points on the graph of the function.

Geometrically, this expression represents the slope of the line that passes through the points (a,f(a)) and (a+h, f(a+h)) as h approaches zero. This is equivalent to the instantaneous rate of change of f at x=a, which is also known as the derivative of f at x=a. The difference quotient is an important concept in calculus and is commonly used to calculate derivatives and slopes of tangent lines.

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