▶️ Answer/Explanation
Solution
Correct Answer: C
1. Calculate average velocity:
\[ \text{Average velocity} = \frac{p(16)-p(1)}{16-1} = \frac{(\sqrt{16}-2)-(\sqrt{1}-2)}{15} = \frac{(4-2)-(1-2)}{15} = \frac{3}{15} = \frac{1}{5} \]
2. Find instantaneous velocity:
\[ p'(t) = \frac{d}{dt}(\sqrt{t} – 2) = \frac{1}{2\sqrt{t}} \]
3. Set equal and solve:
\[ \frac{1}{2\sqrt{t}} = \frac{1}{5} \implies 2\sqrt{t} = 5 \implies \sqrt{t} = \frac{5}{2} \implies t = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]
Why other options are incorrect:
– A) 1: Incorrect, this is the starting time
– B) \( \frac{121}{25} \): Doesn’t satisfy the equation
– D) 25: Exceeds the maximum possible value from the equation
The correct time when instantaneous velocity equals average velocity is \( \boxed{\frac{25}{4}} \).
▶️ Answer/Explanation
Solution
Correct Answer: B
1. Calculate \( f(1) \) and \( f(5) \):
\( f(1) = 3(1)^{2} – (1)^{3} = 3 – 1 = 2 \)
\( f(5) = 3(5)^{2} – (5)^{3} = 75 – 125 = -50 \)
2. Compute average rate of change:
\( \text{Average rate} = \frac{f(5)-f(1)}{5-1} = \frac{-50-2}{4} = \frac{-52}{4} = -13 \)
▶️ Answer/Explanation
Solution
Correct Answer: A
1. Calculate function values at endpoints:
\( y(0) = \cos(0) = 1 \)
\( y\left(\frac{\pi}{2}\right) = \cos(\pi) = -1 \)
2. Compute average rate of change:
\( \text{Average rate} = \frac{y\left(\frac{\pi}{2}\right) – y(0)}{\frac{\pi}{2} – 0} = \frac{-1 – 1}{\frac{\pi}{2}} = \frac{-2}{\frac{\pi}{2}} = -\frac{4}{\pi} \)
▶️ Answer/Explanation
Solution
Correct Answer: C
1. Calculate function values at endpoints:
\( f(0) = (0)^{4} – 5(0) = 0 \)
\( f(3) = (3)^{4} – 5(3) = 81 – 15 = 66 \)
2. Compute average rate of change:
\( \text{Average rate} = \frac{f(3) – f(0)}{3 – 0} = \frac{66 – 0}{3} = 22 \)