5.9 Connecting a Function, Its First Derivative, and its Second Derivative

Overview

Interpreting the graph of the function

• when a function is increasing, you know that the first derivative will be positive
• when a function is decreasing, the first derivative will be negative (the points in that range will be below 0)
• when a function is concave up, the second derivative is positive
• when a function is concave down, the second derivative is negative

• From (-∞,-2), the graph of g(x) is decreasing, therefore the graph of g'(x) over (-∞,-2) is negative
• From (-2,0.85), the graph of g(x) is increasing, therefore the graph of g'(x) over (-2,0.85) is positive
• From (0.85, 2.8), the graph of g(x) is decreasing, therefore the graph of g'(x) over (0.85, 2.8) is negative
• From (2.8,∞), the graph of g(x) is increasing, therefore the graph of g'(x) over (2.8,∞) is positive

• From (-∞,-0.5), the graph of g(x) is concave up, therefore the graph of g"(x) is positive, AND g'(x) is increasing
• From (-0.5, 1.5), the graph of g(x) is concave down, therefore the graph of g"(x) is negative, AND g'(x) is decreasing
• From (1.5, ∞), the graph of g(x) is concave up, therefore the graph of g"(x) is positive, AND g'(x) is increasing

• If f(x) has a relative minimum (the graph changes from decreasing to increasing), then the graph of f'(x) will change from negative to positive at that point
• If f(x) has a relative maximum (the graph changes from increasing to decreasing), then the graph of f'(x) will change from positive to negative at that point
• If f(x) has a point of inflection changing from concave up to concave down, then the graph of f'(x) will have a relative maximum AND f"(x) will change from positive to negative at that point
• If f(x) has a point of inflection changing from concave down to concave up, then the graph of f'(x) will have a relative minimum AND f"(x) will change from negative to positive at that point

• at x=-2, f(x) has a relative minimum, therefore the graph of f'(x) will change from negative to positive at x=-2
• at x=0.85, f(x) has a relative maximum, therefore the graph of f'(x) will change from positive to negative at x=0.85
• at x=2.8, f(x) has a relative minimum, therefore the graph of f'(x) will change from negative to positive at x=2.8
• at x=-0.5, f(x) has a point of inflection changing from concave up to concave down, therefore f'(x) will have a relative maximum and f"(x) will change from positive to negative at x=-0.5
• at x=1.5, g(x) has a point of inflection changing from concave down to concave up, therefore g'(x) will have a relative minimum and g"(x) will change from negative to positive at x=1.5

Interpreting the graph of the first derivative

• If f'(x) is positive, then the graph of f(x) is increasing

• If f'(x) is negative, then the graph of f(x) is decreasing

• If f'(x) is decreasing, then the graph of f(x) is concave down

• If f'(x) is increasing, then the graph of f(x) is concave up

• If f'(x) changes from positive to negative at x=k, then f(x) has a relative maximum at x=k

• If f'(x) changes from negative to positive at x=k, then f(x) has a relative minimum at x=k

• f'(x) is always positive, therefore f(x) is always increasing

• from (-∞,1), f'(x) is decreasing, therefore f(x) is concave down

• from (1,∞), f'(x) is increasing, therefore f(x) is concave up

• at x=1, f'(x) changes from decreasing to increasing, therefore f(x) has a point of inflection at x=1

Common mistakes

1. Misconception: The first derivative is always positive when the function is increasing > The first derivative can be positive even when the function is decreasing, as long as the rate of decrease is slowing down. Similarly, the first derivative can be negative even when the function is increasing, as long as the rate of increase is slowing down.
2. Misconception: The first derivative is always negative when the function is decreasing > The first derivative can be negative even when the function is increasing, as long as the rate of increase is accelerating. Similarly, the first derivative can be positive even when the function is decreasing, as long as the rate of decrease is accelerating.
3. Misconception: The first derivative is always zero at a local maximum or minimum > The first derivative is zero at a local maximum or minimum only if it is a flat maximum or minimum. If the maximum or minimum is "pointy," the first derivative will be non-zero at that point.
4. Misconception: The first derivative is always positive at a local minimum > The first derivative can be negative at a local minimum if the minimum is "pointy."
5. Misconception: The first derivative is always negative at a local maximum > The first derivative can be positive at a local maximum if the maximum is "pointy."
6. Misconception: The graph of the first derivative is always the slope of the original function > The graph of the first derivative gives the rate of change of the original function at each point, but it is not the same as the slope of the original function.
7. Misconception: The graph of the first derivative tells you the shape of the original function > The graph of the first derivative gives information about the rate of change of the original function, but it does not necessarily tell you the shape of the original function.