Florida B.E.S.T. MA.912.C.4.5
The Standard
Analyze function graphs by using derivative graphs and the Fundamental Theorem of Calculus.
Florida B.E.S.T. Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use a graph of f' to determine where f rises, falls, and has local extrema. They interpret signed area under f' as the change in f and use an initial value to find later values. They also use changes in f' to describe concavity.
What Mastery Looks Like
- Given a graph of f' and one known value of f, students can calculate other values of f from signed areas. They correctly identify intervals of increase, decrease, concavity, and local extrema, with reasons based on the graph.
Common Misconceptions
- Students confuse the value of f' with the value of f. They add all areas as positive instead of using signed area. They may label every point where f' equals zero as an extremum, even without a sign change.
How to Assess It
- Exit ticket: Let f'(x)=2 on 0≤x≤2 and f'(x)=-1 on 2≤x≤5, with f(0)=4. Ask students to find f(5) and identify where f has a local maximum.
Lesson moves
Ways to Teach It
Place unit-square tiles above and below a piecewise derivative graph, then total signed area to find changes in the original function.
Ask students: How can the derivative be zero without the original function having a local maximum or minimum?
Run a card sort matching derivative graphs with possible original graphs, and require students to justify each match using signs and areas.
Use a velocity-time graph to calculate position changes and identify when a moving object reverses direction.
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Related Standards
- MA.912.C.3.3
Determine where a function is decreasing and increasing using its derivative.
- MA.912.C.3.6
Sketch graphs by using first and second derivatives. Compare the corresponding characteristics of the graphs of f, f' and f".
- MA.912.C.2
Develop an understanding for and determine derivatives.
- MA.912.C.2.6
Find derivatives of inverse functions.
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