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.
People who Shmooped this also Shmooped...
Advertisement
Advertisement
Advertisement