AP Calculus AB and BC
Increasing vs Concave Up
Increasing means the first derivative is positive, so the function is rising. Concave up means the second derivative is positive, so the slope itself is rising. The two are completely independent: a function can be increasing while concave down.
Increasing
Use when: You are describing whether the quantity is going up or down.
Concave up
Use when: You are describing whether the rate of change is going up or down.
Side by side
| Increasing | Concave up | |
|---|---|---|
| Derivative involved | ||
| Describes | The function | The slope |
| Plain language | Rising | Rising faster and faster |
| Can happen together | Yes | Yes |
All four combinations occur. A quantity growing but slowing down is increasing and concave down; a quantity falling ever faster is decreasing and concave down. Interpretation questions on free response are usually testing exactly this pairing.
In context the second derivative is often the more interesting one. A company whose revenue is increasing but concave down is still growing, just less impressively each quarter.
Frequently asked questions
If a function is concave up, must it be increasing?
No. A parabola opening upward is concave up everywhere but decreases to the left of its vertex.
In the CED: Unit 5: Analytical Applications