2012 Ch.3 Tables
Exploring Functions Through Tables
*A warm-up AP-type questions*
Objective: To examine tables of functions and determine how the function is behaving.
Example 1: For all x in the interval [3, 7], the function f has a positive first derivative and a negative second derivative.
Which of the following could be the table of values for f ?
Let’s consider the “finite differences” in the tables.
a)
b)
x
3
4
5
6
7
x
3
4
5
6
7
f(x)
19
16
13
10
7
c)
f(x)
19
23
28
34
41
d)
x
3
4
5
6
7
f(x)
19
24
28
31
33
x
3
4
5
6
7
f(x)
19
17
14
10
5
e)
x
3
4
5
6
7
f(x)
19
22
25
28
31
Example 2: Let f be a continuous function with selected values given in the table below.
x
f(x)
0
7
1
10
2
13
3
16
4
19
Now consider the “Finite differences” of the data. Is the function increasing at an increasing rate, increasing at a
decreasing rate, or increasing at a constant rate? What does that mean in terms of the first and second derivative?
Example 3: Let g be twice-differentiable function with selected values given in the table below
x
g(x)
0
6
1
11
2
18
3
27
4
38
5
51
Is the function increasing at an increasing rate, increasing at a decreasing rate, or increasing at a constant rate? What does
that mean in terms of the first and the second derivative?
Example 4: Let h be a twice-differentiable function with selected values given in the table below
x
h(x)
0
25
1
14
2
1
3
-14
4
-31
5
-50
Is the function decreasing at an increasing rate, decreasing at a decreasing rate, or decreasing at a constant rate? What does
that mean in terms of the first and the second derivative?
Example 5: Let d be twice-differentiable function with selected values given in the table below
x
d(x)
0
28
1
23
2
19
3
16
4
14
5
13
Is the function decreasing at an increasing rate, decreasing at a decreasing rate, or decreasing at a constant rate? What does
that mean in terms of the first and the second derivative?
Now let’s consider these AP questions
AP Question #1: The function is continuous on the closed interval [2, 4] and twice differentiable on the open interval (2,
4). If f (3)  2 and f ( x)  0 on the open interval (2, 4), which of the following could be a table of values for f?
a)
x
b)
f(x)
x
f(x)
2
2.5
2
2.5
3
5
3
5
4
6.5
4
7
b)
d)
x
f(x)
2
3
3
5
4
6.5
x
f(x)
2
3.5
3
5
4
7.5
x
f(x)
2
3
3
5
4
7
e)
AP Question #2: For all x in the closed interval [2, 5], the function f has a positive first derivative and a negative second
derivative. Which of the following could be a table of values for f?
a)
x
2
3
4
5
b)
f(x)
7
9
12
16
d)
x
2
3
4
5
c)
x
2
3
4
5
f(x)
7
11
14
16
e)
f(x)
16
14
11
7
x
2
3
4
5
f(x)
16
13
10
7
x
2
3
4
5
f(x)
16
12
9
7
The following are tables of values for some contestants in “The Biggest Loser” competition
Contestant #1
t [ weeks]
W(t) [weight in pounds
0
200
1
210
2
230
3
260
Contestant #2
t [ weeks]
W(t) [weight in pounds
0
300
1
290
2
270
3
240
Contestant #3
t [ weeks]
W(t) [weight in pounds
0
350
1
330
2
320
3
315
Contestant #4
t [ weeks]
W(t) [weight in pounds
0
350
1
330
2
310
3
290