Prep Session Topic: Interpreting Graphs (f, f’, and f”)

What the AP exam expects:

Analysis of graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

Concept of the derivative• Derivative presented graphically, numerically, and analytically• Derivative interpreted as an instantaneous rate of change• Derivative defined as the limit of the difference quotient• Relationship between differentiability and continuity

Derivative as a function• Corresponding characteristics of graphs of ƒ and ƒ’• Relationship between the increasing and decreasing behavior of ƒ and the sign of ƒ’

Second derivatives• Corresponding characteristics of the graphs of ƒ, ƒ’, and ƒ “• Relationship between the concavity of ƒ and the sign of ƒ ‘• Points of inflection as places where concavity changes

Applications of derivatives• Analysis of curves, including the notions of monotonicity and concavity• Optimization, both absolute (global) and relative (local) extrema

From AP College Board

What this means:

You will be expected to take a graph of a function’s derivative and describe properties of the original function. These properties include increasing and decreasing behavior, local and absolute extrema, concavity and points of inflection, differentiability and continuity. (Eventually, you will also be expected to find the value of an integral). You will be expected to justify at every step for all of the above concepts.

The overriding concepts for the interpretation of graphs are all precalculus concepts. In order to fully understand the calculus, you

must understand the underlying precalculus. As a calculus student, you will apply these concepts to derivative graphs. A. B.

A. = ___________

Interval(s) where

Interval(s) where

Interval(s) where is increasing

Interval(s) where is decreasingLocal MaximaLocal MinimaInterval(s) where is concave up

Interval(s) where is concave downPoint(s) of inflection

B. = ___________

Interval(s) where

Interval(s) where

Interval(s) where is increasing

Interval(s) where is decreasingLocal MaximaLocal MinimaInterval(s) where is concave up

Interval(s) where is concave downPoint(s) of inflection

Question 1: How would a precalculus student justify the extrema on A?

Question 2: How would a precalculus student justify the point of inflection on B?

is increasing ↔ is positive

is decreasing ↔ is negative

is concave up ↔ is positive

is concave down ↔ is negative

= 0 ↔ MIGHT have a local extrema at x

= 0 ↔ MIGHT have a point of inflection at x

is increasing ↔ is positive

is decreasing ↔ is negative

= 0 ↔ MIGHT have a local extrema at

x

= 0 ↔ MIGHT have a point of inflection at x

A. B.

Question 3: How would a calculus student justify the extrema on A?

Question 4: How would a calculus student justify the point of inflection on B?

The graph below represents , the derivative of a function .

Question 5: Where is > 0? What does this mean about ?

Question 6: Where is < 0? What does this mean about ?

Question 7: Where is >0? What does this mean about ?

Question 8: Where is < 0? What does this mean about ?

BIG QUESTION 9: Are there any extrema on ? Why?

BIG QUESTION 10: Are there any points of inflection ? Why?

MC 1

MC 2

MC 3

The graph of a twice differentiable function is shown above. Which of the following is true?

MC 4

MC 5

FR 2003 AB4 (a)-(c) only

FR 2008 AB4 (a),(c), and (d) only

FR 2006 MODIFIED AB 3 (a) and (b) only

The graph of the six line segments shown below is the derivative of the function g; that is, the graph shown is .

(a) Find g’(4) and g”(4).

(b) Does g have a relative maximum, a relative minimum, or neither at . Justify your answer.

Solutions 2003 AB4

Solutions 2006 AB3

Solutions 2008 AB4

Note: AP Free Response Questions and Solutions can be found at AP Central.