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).
Example 2
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:
Example 3
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:
Example 4
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 ata is horizontal (has a slope of 0).
Example 5
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:
Example 6
For the function shown below, which is greater: f ' (x1) or f ' (x2)?
Answer
Draw the tangent lines to f at x1andx2:
The slope of the tangent line to f at x1, also known as f ' (x1), looks to be about 0. The slope of the tangent line to f at x2, also known as f ' (x2), looks decidedly more negative; f ' (x1) is definitely greater.
Example 7
For the function shown below, which is greater: f ' (x1) or f ' (x2)?
Answer
Draw the tangent lines to f at x1 and x2:
The slope of the tangent line to f at x1, also known as f ' (x1), looks positive but shallow. The slope of the tangent line to f at x2, also known as f'(x2), looks positive and much steeper. It's safe to say f ' (x2) > f ' (x1).
Example 8
For the function shown below, list in order from least to greatest: f ' (x1), f ' (x2), f ' (x3).
Answer
Draw the tangent lines to f at x1, x2, x3:
The slope of the tangent line to f at x1, also known as f ' (x1), is positive and fairly steep.The slope of the tangent line to f at x2, also known as f ' (x2), looks like it's about zero.The slope of the tangent line to f at x3, also known as f ' (x3), is negative. Thereforef ' (x3) < f ' (x2) < f ' (x1).
Example 9
For the function shown below, list in order from least to greatest: f ' (x1), f ' (x2), f ' (0).
Answer
We'll start by drawing the tangent lines to f at x1, x2, and 0:
The slope of the tangent line to f at x1, also known as f ' (x1), is positive and fairly steep.The slope of the tangent line to f at x2, also known as f ' (x2), is negative and fairly steep.The slope of the tangent line to f at 0, also known as f ' (0), is zero. Thereforef ' (x2) < f ' (0) < f ' (x1).
