The following phrases all mean the same thing:
- 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.
Practice:
For the function below, which is greater: f'(0) or f'(1)? | |
Draw the tangent lines to f at 0 and 1. Which is greater? The tangent line to f at 0 is steeper than the tangent line to f at 1, and so has a greater slope.
This means f'(0) > f'(1).
| |
For the function shown below, which is greater: f'(-1) or f'(1)? | |
We draw the tangent lines to f at the two points in question: Since both slopes are negative, the one that is "less negative" will be closer to zero, and so will be greater. The tangent line to f at 1 has the "less negative" slope, so f'(1) > f'(-1). | |
For the function f and value a, determine if f'(a) is positive, negative, or zero.
Answer
- f'(a) is zero because the tangent line to f at a is horizontal (has a slope of 0).
For the function f and value a, determine if f'(a) is positive, negative, or zero.
Answer
- f'(a) is negative because the tangent lint to f at a slopes downwards:
For the function f and value a, determine if f'(a) is positive, negative, or zero.
Answer
- f'(a) is negative because the tangent lint to f at a slopes downwards:
For the function f and value a, determine if f'(a) is positive, negative, or zero.
Answer
- f'(a) is zero because the tangent line to f at a is horizontal (has a slope of 0).
For the function f and value a, determine if f'(a) is positive, negative, or zero.
Answer
- f'(a) is positive because the tangent line to f at a slopes upwards:
- For the function shown below, which is greater: f'(x_{1}) or f'(x_{2})?
Answer
- Draw the tangent lines to f at x_{1} and x_{2}:
The slope of the tangent line to f at x_{1}, also known as f'(x_{1}), looks like about 0. The slope of the tangent line to f at x_{2}, also known as f'(x_{2}), looks decidedly more negative. We conclude that f'(x_{1}) is greater.
- For the function shown below, which is greater: f'(x_{1}) or f'(x_{2})?
Answer
- Draw the tangent lines to f at x_{1} and _{x2}:
The slope of the tangent line to f at x_{1}, also known as f'(x_{1}), looks positive but shallow. The slope of the tangent line to f at x_{2}, also known as f'(x_{2}), looks positive and much steeper. We conclude that f'(x_{2})>f'(x_{1}).
- For the function shown below, list in order from least to greatest: f'(x_{1}), f'(x_{2}), f'(x_{3}).
Answer
- Draw the tangent lines to f at x_{1}, x_{2}, x_{3}:
The slope of the tangent line to f at x_{1}, also known as f'(x_{1}), is positive and fairly steep.The slope of the tangent line to f at x_{2}, also known as f'(x_{2}), looks like it's about zero.The slope of the tangent line to f at x_{3}, also known as f'(x_{3}), is negative. Thereforef'(x_{3})<f'(x_{2})<f'(x_{1}).
- For the function shown below, list in order from least to greatest: f'(x_{1}), f'(x_{2}), f'(0).
Answer
- Draw the tangent lines to f at x_{1}, x_{2}, 0:
The slope of the tangent line to f at x_{1}, also known as f'(x_{1}), is positive and fairly steep.The slope of the tangent line to f at x_{2}, also known as f'(x_{2}), is negative and fairly steep.The slope of the tangent line to f at 0, also known as f'(0), is zero. Thereforef'(x_{2})<f'(0)<f'(x_{1}).