Conceptual
Problems:
Relating f, f’, and f”
Problem A
Problem B
Conceptual Problems
Inability to see derivative as a function, only a value
 Derivative is object but not as an operation

◦ Derivative vs. Differentiation vs. “Finding the derivative”

Derivative at a point predicting local behavior
◦ Symbolically
◦ Graphically
◦ From a table
Even after understanding the relationship between f and f”,
making the connection between f’ and f”
 Notation & Meaning of Notation

◦ Writes f’(x)=3 when they mean f’(4)=3

Translating from a forward proficiency f -> f’-> f” to “given
multiple representations, how can we relate them?”
Skill vs. Understanding

If Problem A was given in terms of the
polynomials, far fewer students would have
difficulty identifying which is the derivative
and second derivative. An important part of
understanding the concept of derivative is
multiple representations. (graphs, tables, etc.)

If the skill of finding the derivative at a given
point is over-stressed, the understanding of
the derivative as a function may not
materialize.
Language

General use of “it”; “it is concave up because it is
positive” vs. “the graph of f is concave up on the
given interval because its second derivative is
positive on that interval”
◦ The issue is primarily learning to attach the correct
characteristic to a specific noun, and then giving the
proper implication

Early in the Calc curriculum, it needs to be made
clear what the proper communication of an
answer looks like
◦ Students need to be instructed on the difference
between meaning and “meaning by association”
 f” is concave up because it is positive
Ability to Transport Knowledge

Associating the rules of the common specific
examples to general rules for the given operation
(improper generalization)
◦ Taking the derivative eliminates one “hump” of a graph;
then trying to apply this rule to a sine graph

Even if there is a strong understanding of the rules,
applying that rule to a alternate situation can still
prove difficult.
◦ After achieving an understanding of Problem A with the
prompts; f, f’, f”, not being able to solve the same
problem with the prompts; position, velocity, and
acceleration.
Practice






Given a function; find the first and second derivatives in symbolic form
OR
Given a function; graph the function along with the graphs of the first and
second derivatives
Given the graph of a function, graph the first and second derivatives
Problem A
Given the graph of the first or second derivative and enough additional
information; find an approximate graph of the original function
OR
Given the first or second derivative in symbolic form along with enough
additional information; find the symbolic form of the original function
Problem B
Do a problem similar to Problem B from information in a table