Based on the following table of values for the function w(t), answer the questions below.

t= 100 110 120 130 140 150

w(t)= 2.6 7.3 12.7 19 26.1 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

t and w(t) are a table with corresponding values.

(100,2.6), (110,7.3), (120,12.7), (130,19), (140,26.1), (150,34.1)

Sketch the graph first.

You will notice it's increasing and concave up. This should help with answering.

(if not, the solution is always positive for both)

To find the answers to these questions, we need to understand the concepts of derivatives and second derivatives.

The derivative of a function tells us how the function is changing at any given point. If the derivative is positive, it means the function is increasing. If the derivative is negative, it means the function is decreasing. If the derivative changes sign, it means the function is first increasing and then decreasing, or vice versa.

The second derivative of a function tells us how the derivative of the function is changing. If the second derivative is positive, it means the function is concave up (like a "U" shape). If the second derivative is negative, it means the function is concave down (like an "n" shape). If the second derivative changes sign, it means the function changes concavity.

Let's apply these concepts to the given table of values for the function w(t).

a) To determine the sign of the derivative of w(t), we can look at how the values of w(t) are changing.

From t=100 to t=110, the values of w(t) increase from 2.6 to 7.3, indicating that the function is increasing.

From t=110 to t=120, the values of w(t) increase from 7.3 to 12.7, indicating that the function is still increasing.

From t=120 to t=130, the values of w(t) increase from 12.7 to 19, indicating that the function is still increasing.

From t=130 to t=140, the values of w(t) increase from 19 to 26.1, indicating that the function is still increasing.

From t=140 to t=150, the values of w(t) increase from 26.1 to 34.1, indicating that the function is still increasing.

Since w(t) is always increasing, the derivative of w(t) is always positive. Therefore, the answer is B. always positive.

b) To determine the sign of the second derivative of w(t), we need to analyze how the derivative of w(t) is changing.

From the observations made in part a), we see that the function is continuously increasing from t=100 to t=150. This suggests that the derivative of w(t) is positive over the entire interval.

Since the derivative is positive, the concavity of the function must be positive. Therefore, the second derivative of w(t) is always positive. The answer is C. always positive.

In summary:
a) The derivative of w(t) is always positive (B. always positive).
b) The second derivative of w(t) is always positive (C. always positive).