Tangent Lines and Derivatives
The following phrases all mean the same thing:
- the slope of f at a
- the derivative of f at a
- f ' (a)
- the slope of the tangent line to f at a
- the instantaneous rate of change of f at a
Since f ' (a) (the derivative of f at a) is equal to the slope of the tangent line to f at a, we can determine the sign of f ' (a) by looking at the tangent line to f at a.
Sample Problem
Consider this function:
The tangent line to f at a is sloping upwards. This means the slope of the tangent line to f at a is positive. Since the slope of the tangent line to f at a equals f ' (a), we know f ' (a) is also positive.
We can also tell the sign of f ' (a) by looking at the function f and not thinking about tangent lines.
- If f is increasing at a then f ' (a) is positive.
- If f is decreasing at a then f ' (a) is negative.
- If f "flattens out" at a then f ' (a) is zero.
We can also use tangent lines to compare values of f ' at different points.
Remember these pictures. We'll use them again when we discuss concavity and second derivatives.