Calculus
Confusing Concavity with Increasing and Decreasing
Students see a graph is concave up and say the function is increasing.
Quick check
Why students make it
Concave up feels like the graph is going upward, so students mix up the shape of the graph with the direction the function values move. They forget that a graph can be decreasing while its slope is becoming less negative.
The rule done right
Increasing and decreasing are decided by the sign of . Concavity is decided by the sign of , so means slopes are increasing, not necessarily that is increasing.
Related resources
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Concavity and Inflection Points Infographic -
First Derivative Test Infographic -
Curve Sketching with Derivatives Infographic -
Increasing and Decreasing Functions Infographic -
Monotonicity and the Derivative Infographic - Concavity & Inflection Points Cheat Sheet
- Curve Sketching Reference Cheat Sheet
- Derivatives & Differentiation Rules Cheat Sheet
- Mean Value & Rolle's Theorems Cheat Sheet
- Tangent Lines & The Derivative as a Limit Cheat Sheet
- Calculus, Derivatives Worksheet
- Calculus, Optimization Worksheet
- Calculus, Related Rates Worksheet
- Curve Sketching Explorer Tool
- Function Graph Explorer Tool
- Derivative & Integral Visualizer Tool
- Mean Value Theorem & Rolle's Theorem Explorer Tool
- Derivative Rules Step-by-Step Tool
- Derivatives & Tangent Lines Lab Lab
- Optimization & Related Rates Lab Lab
- Limits & Continuity Lab Lab