▶️ Answer/Explanation
Solution
Correct Answer: D
Step 1: Understand that speed is the derivative of position (x'(t))
For a linear function x(t), the derivative is constant.
Step 2: Analyze the options:
• A) Gives position at t=10, not speed
• B) Gives average position over 10 seconds
• C) Approximates speed using symmetric difference
• D) Gives the exact derivative (speed) for linear functions
Step 3: Since x(t) is linear, its graph is a straight line
The slope of y = x(t) gives the constant speed at all times.
▶️ Answer/Explanation
Solution
Calculate average rates of change to approximate velocity:
\( t = 0 \) to \( t = 20 \): \( \frac{105 – 0}{20} = 5.25 \, \text{ft/s} \)
\( t = 20 \) to \( t = 40 \): \( \frac{300 – 105}{20} = 9.75 \, \text{ft/s} \)
\( t = 40 \) to \( t = 60 \): \( \frac{900 – 300}{20} = 30 \, \text{ft/s} \)
\( t = 60 \) to \( t = 80 \): \( \frac{2400 – 900}{20} = 75 \, \text{ft/s} \)
The velocity increases over time, with the largest rate of change at later intervals. Among the options, \( t = 80 \) has the highest preceding rate (75 ft/s) and continues to increase.
Answer: D) \( t = 80 \)
▶️ Answer/Explanation
Solution
Speed is \( |x'(t)| \). At \( t = 4 \), the graph shows \( x(4) = 20 \), and the particle is stationary (slope = 0), so \( x'(4) = 0 \). Thus, speed = 0.
Answer: A) 0