Calculus
Confusing Average and Instantaneous Rate of Change
Students use the slope over an interval as if it were the slope at one exact point.
Quick check
Why students make it
A secant line and a tangent line both measure change, so they can look like the same idea. The average rate of change uses two input values, while the instantaneous rate of change uses a derivative at one input value.
The rule done right
The average rate of change of on is . The instantaneous rate of change at is , the slope of the tangent line at that point.
Related resources
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Calculus, The Two Big Problems Infographic -
Calculus, The Definition of the Derivative Infographic -
Calculus, Tangent and Normal Lines Infographic -
Mean Value Theorem Infographic -
Calculus, The Derivative as a Function Infographic - Tangent Lines & The Derivative as a Limit Cheat Sheet
- Mean Value & Rolle's Theorems Cheat Sheet
- Derivatives & Differentiation Rules Cheat Sheet
- Linear Approximation & Differentials Cheat Sheet
- Calculus, Derivatives Worksheet
- Calculus, Related Rates Worksheet
- Calculus, Integrals Worksheet
- Mean Value Theorem & Rolle's Theorem Explorer Tool
- Derivative & Integral Visualizer Tool
- Function Graph Explorer Tool
- Limits & Continuity Explorer Tool
- Derivative Rules Step-by-Step Tool
- Derivatives & Tangent Lines Lab Lab
- Limits & Continuity Lab Lab
- FTC & Net Change Lab Lab