# The Second Derivative

```BC 1
Intro to section 2.5
Name:
(1)
Consider the graph below of a population P as a function of time. The 10 may simply
represent 10, or it may represent 10 thousand or 10 million.
(a)
Describe the growth of the population over time. Be specific.
(b)
What may be said about P' ? How does P' change over time? Be specific, including
point A in your description. How does the graph of P change as P' changes?
BC 1
(2)
Name:
Consider the graph of ƒ on the right.
its entire domain?
(b) What happens to ƒ' as x increases?
(c) What does this imply about the derivative of ƒ' ?
(3)
Consider the graph of g on the right.
entire domain?
(b) What happens to g' as x increases?
(c) What does this imply about the derivative of g' ?
BC 1
Name:
Geometry of Derivatives
Suppose you are given the graph of f, the first derivative of a function f. Describe how you would
determine each of the following and explain clearly why your method is correct.
1.
Where the graph of f is decreasing.
2.
Where the graph of f has local maximums.
3.
Where the graph of f has points of inflection.
4.
Where the graph of f is concave down.
BC 1
Name:
Suppose you are given the graph of f, (f is the first derivative of f, which makes it the
second derivative of a function f). Using what you know about the relationship between a
function and its first derivative, describe how you would determine each of the following from
the graph of f and explain fully why your method is correct.
5.
Where the graph of f has points of inflection.
6.
Where the graph of f is concave down.
7.
Whether the graph of f has a local maximum or local minimum at x = a given f   a   0 .
BC 1
Name:
8. Give an example of a function that satisfies the given conditions. Be explicit about what the
value of a is in your example.
a. The function f has no local maxima or minima at x = a, yet f ( a )  0 .
b. The function f has no point of inflection at x = a, yet f (a)  0 .
```