Here’s the Graph of the Derivative ….
Tell me About the Function
Lin McMullin
NCTM Annual Meeting
Philadelphia, Pennsylvania
April 27, 2012
AP Calculus
Free-response Questions
Standard
Synthesis
• Area – Volume
• Rate / Accumulation
• Differential Equation
• Particle Motion
• Functions
• Table Stem
• Power Series (BC)
• Graph Stem
• Polar, Parametric &
Vector (BC)
• Family/Generic
Functions
2011 AB 4
AP Calculus
Graph-stem Questions
2011 AB 4 / BC 4 The graph of the derivative Stem: Graph
(a) f defined by integral
-
-
5.4
5.3
4.4
5.4
-
4.4
5.4
(b) max/min
4.3, 4.7
(3.6)
4.3
Justify
-
-
-
4.4
3.4
4.3
Give reason
-
-
-
(d) Average ROC
2.1
(P1)
2.4
4.3
3.2
4.2
FTC
Evaluate from graph g, g' (and g'')
(c) POI
MVT
x 2
AP Calculus
Graph-stem Questions
1996 AB 1
x 2
AP Calculus
Graph-stem Questions
1996 AB 1
Increasing
Decreasing
Absolute
Maximum
Local Max
POI
Local Min
Absolute
POI Minimum
POI
Down
Increasing
Up
Down
Up
AP Calculus
Graph-stem Questions
Year &
Number
Mean
%9
% Zero
2003 AB 4
2.68
n/a
20.4
2004 AB 5
2.63
1.0
28.0
2006 AB 3
3.24
4.0
22.0
2008 AB 4
2.60
2.3
29.7
2009 AB 1
4.67
4.9
9.6
2009 AB 6
2.07
0.2
23.8
2010 AB 5
1.75
0.3
38.9
2011 AB 5
2.44
0.4
29.5
x 2
AP Calculus
Graph-stem Questions
+

0
–


–
0



0


+
f '(x)

x 2
AP Calculus
Graph-stem Questions
Method 1 Think Tangent Line
• The derivative is the slope of the tangent line
and we see the graph mimics the tangent line,
so we turn that around and use these ideas ….
• Winplot Demo
x 2
AP Calculus
Graph-stem Questions
Feature
Conclusion
y' > 0
y is increasing
y' < 0
y is decreasing
y' changes - to +
y has a local minimum
y' changes + to -
y has a local maximum
y' increasing
y is concave up
y' decreasing
y is concave down
y' extreme values
y has points of inflection
x 2
AP Calculus
Justifications
Conclusion
Justification
y is increasing
y' > 0
y is decreasing
y' < 0
y has a local minimum
y' changes - to +
y has a local maximum
y' changes + to -
y is concave up
y' increasing
y is concave down
y' decreasing
y has points of inflection
y' extreme values
x 2
AP Calculus
Graph-stem Questions
2009 AB 1
x 2
AP Calculus
Graph-stem Questions
2006 AB 3
x 2
AP Calculus
Graph-stem Questions
Method 2 Accumulation
• The given might include the graph of f  t  and
ask about
g  x   g  c    f  t  dt
x
c
• Here we use the Fundamental Theorem of
Calculus to see that g   x   f  x  with the initial
condition  c, g  c  
x 2
Graphing without Derivatives

g  x   7   f  t  dt
x

0
y  f t 

(The graph of f consists of a

semicircle and two line segments)
     








Tell me all the usual stuff about the graph of g  x  and
explain your reasoning without mentioning the derivative or
any concepts related to the derivative.
x 2
Graphing without Derivatives
Winplot Demo 2
Antiderivative. Initial Condition = (0,0)
x 2
Graphing without Derivatives
g  x   7   f  t  dt
x
0
g  x   g  6    f  t  dt , such that g  0   7
x
6


y  f t 


     








x 2
Graphing without Derivatives



2

      1




1
3
1 

x 2
Graphing without Derivatives
g  x   g  6    f  t  dt , such that g  0   7
x
6



2
 
 

  1




1
3
1 

x
g(x)
-6
?
-4
?+
-2
? + 2
0
? + 2 –1 = 7
? = 8 – 2
2
3
4
x 2
Graphing without Derivatives
g  x   g  6    f  t  dt , such that g  0   7
x
6
x
g(x)
-6
8 – 2
-4
8–
-2
8
0
7
2
7–3=4

3
4–1=3

4
3+1=4



2
 
 

  1


1
3
1 

x 2
Graphing without Derivatives
x
g(x)
Location
Feature
Concavity
–6
8 – 2
(-6, 8 – 2)
Absolute minimum
–4
8–
–6≤x≤–2
Increasing
–2
8
(– 2, 8)
0
7
–2≤x≤3
2
4
(3,3)
3
3
3≤ x≤4
4
4
(4, 4)
Absolute Maximum
Decreasing
Local Minimum
Increasing
Endpoint Maximum



2
 
 

  1




1
3
1 

x 2
Graphing without Derivatives

 

  1

Concavity
(– 6, 8 – 2)
Absolute minimum
–
–6≤x≤ –4
Increasing
–4≤x≤ –2

 
Feature
(– 4, 8 – 

2
Location
3
1 

–
Increasing
Down
Absolute Maximum
Down
–2≤x≤0
Decreasing
Down
0≤x≤2
Decreasing
Down
(– 2, 8)
1
Point of Inflection
Up



(2, 4)
2≤x≤3
(3,3)
3≤ x≤4
(4, 4)
Point of Inflection
–
Decreasing
Up
Local Minimum
Up
Increasing
Up
Endpoint Maximum
–
x 2
Graphing without Derivatives


<=== Antiderivative. Initial Condition = (0,7)












Function






h  x   g   x   x  0
g x  x
AP Calculus
Graph-stem Questions
7  x  2  h  x   0
2  x  5  h  x   0
yx
Multiple-choice
2008 Part 1 A:
2008 Part 1B:
9 – Absolute max/min
76 – Increasing
10 – Riemann sum
77 – Limits
11 – Graph: Indentify
graph of derivative
84 – Relative Max/min
17 – FTC, POI
86 – Velocity table: identify
position graph
21 – Increasing (Motion)
27 – Slope Field
2008: 10 MC + 1 FR = 26.7%
2003: 9 MC + 1 FR = 24%
Suggestions
• Find and assign all you can of questions with a
graph from you text, from past exams ….
• Use released AP questions
• Cumulative tests and assignments
• Go further with each questions; find other
features that can be determined from the
graphs you have
• Practice justifications; have students explain
why the graphs have these features
• Use at the end of the year to draw together
disparate topics
Information
• E-mail: lnmcmullin@aol.com
• The PowerPoint slides, Handout and the
Winplot files, are here:
www.LinMcMullin.net
Click on AP Calculus
• They are also at the conference website
Here’s the Graph of the Derivative ….
Tell me About the Function
Lin McMullin