Based on the following table of values for the function w(t), answer the questions below.
t and w(t) are a table with the following values:
(100,2.6), (110,7.3), (120,12.7), (130,19), (140,26.1), (150,34.1)
a) Which of the following best describes the derivative of w(t)?
A. changes sign
B. always positive
C. always negative
b) Which of the following best describes the second derivative of w(t)?
A. always negative
B. changes sign
C. always positive
Sketch a graph of this function. Then the answers will be obvious.
Damon,
is it always positive and changes sign?
slope is always positive
the slope keeps increasing so second derivative is positive
To answer these questions, we need to analyze the given table of values for the function w(t).
a) To determine the behavior of the derivative of w(t), we need to look for patterns in how w(t) changes as t increases. The derivative measures the rate of change of the function.
Looking at the table, we can observe that as t increases, the values of w(t) also increase. This suggests that the function w(t) is generally increasing.
However, to confirm this, we need to calculate the difference between consecutive values of w(t) and see how it changes.
For example, the difference between the first two values of w(t) is 7.3 - 2.6 = 4.7. Similarly, the difference between the next two values is 12.7 - 7.3 = 5.4.
If we continue to calculate the differences for the remaining values, we get:
(130, 19) - (120, 12.7) = 6.3
(140, 26.1) - (130, 19) = 7.1
(150, 34.1) - (140, 26.1) = 8
By analyzing these differences, we can see that they are all positive. This indicates that the function w(t) is increasing, and therefore, the derivative of w(t) is always positive.
Therefore, the answer to question a) is B. always positive.
b) To determine the behavior of the second derivative of w(t), we need to examine how the first derivative (rate of change of w(t)) changes.
Using the values we calculated for the first differences (7.3 - 2.6, 12.7 - 7.3, etc.), we can observe that they are increasing.
For example:
5.4 - 4.7 = 0.7
6.3 - 5.4 = 0.9
7.1 - 6.3 = 0.8
8 - 7.1 = 0.9
As we can see from these differences, they are all positive and increasing. This suggests that the rate of change of w(t) is increasing as t increases, indicating that the second derivative of w(t) is always positive.
Therefore, the answer to question b) is C. always positive.