At a Glance - 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.
Example 1
For the function below, which is greater: f ' (0) or f ' (1)? |
Example 2
For the function shown below, which is greater: f ' (-1) or f ' (1)? |
Exercise 1
For the function f and value a, determine if f ' (a) is positive, negative, or zero.
Exercise 2
For the function f and value a, determine if f ' (a) is positive, negative, or zero.
Exercise 3
For the function f and value a, determine if f ' (a) is positive, negative, or zero.
Exercise 4
For the function f and value a, determine if f ' (a) is positive, negative, or zero.
Exercise 5
For the function f and value a, determine if f ' (a) is positive, negative, or zero.
Exercise 6
- For the function shown below, which is greater: f ' (x_{1}) or f ' (x_{2})?
Exercise 7
- For the function shown below, which is greater: f ' (x_{1}) or f ' (x_{2})?
Exercise 8
- For the function shown below, list in order from least to greatest: f ' (x_{1}), f ' (x_{2}), f ' (x_{3}).
Exercise 9
- For the function shown below, list in order from least to greatest: f ' (x_{1}), f ' (x_{2}), f ' (0).