A graphing calculator is recommended.
The displacement (in centimeters) of a particle s moving back and forth along a straight line is given by the equation 
s = 3 sin(𝜋t) + 3 cos(𝜋t),
 where t is measured in seconds. (Round your answers to two decimal places.)
(a)
Find the average velocity during each time period.
(i)
[1, 2]
  
Correct: Your answer is correct.
 cm/s
(ii)
[1, 1.1]
  
Correct: Your answer is correct.
 cm/s
(iii)
[1, 1.01]
  
Correct: Your answer is correct.
 cm/s
(iv)
[1, 1.001]
  
Incorrect: Your answer is incorrect.
 cm/s
(b)
Estimate the instantaneous velocity of the particle when 
t = 1.
  
Correct: Your answer is correct.
 cm/s

We are working with the displacement function of the particle given as \( s(t) = 3\sin(\pi t) + 3\cos(\pi t) \), and we need to find the average velocity over specific time intervals and estimate the instantaneous velocity at \( t = 1 \).

### (a) Average velocity:
The average velocity over a time interval \([t_1, t_2]\) is given by the formula:
\[
v_{\text{avg}} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]
We already know your answers for parts (i) through (iii) are correct, so we'll focus on part (iv), and then proceed to part (b).

#### (iv) Interval \([1, 1.001]\)
Let's calculate the average velocity for this interval:
- \( t_1 = 1 \)
- \( t_2 = 1.001 \)

First, we need to calculate \( s(1) \) and \( s(1.001) \).

\[
s(1) = 3\sin(\pi \cdot 1) + 3\cos(\pi \cdot 1) = 3\sin(\pi) + 3\cos(\pi) = 3(0) + 3(-1) = -3 \, \text{cm}
\]

Now calculate \( s(1.001) \):
\[
s(1.001) = 3\sin(\pi \cdot 1.001) + 3\cos(\pi \cdot 1.001)
\]
We need to approximate \(\sin(\pi \cdot 1.001)\) and \(\cos(\pi \cdot 1.001)\):
\[
\sin(\pi \cdot 1.001) \approx -0.00314, \quad \cos(\pi \cdot 1.001) \approx -0.999995
\]
\[
s(1.001) = 3(-0.00314) + 3(-0.999995) = -0.00942 - 2.999985 = -3.009405 \, \text{cm}
\]

Now, apply the average velocity formula:
\[
v_{\text{avg}} = \frac{s(1.001) - s(1)}{1.001 - 1} = \frac{-3.009405 - (-3)}{0.001} = \frac{-3.009405 + 3}{0.001} = \frac{-0.009405}{0.001} = -9.41 \, \text{cm/s}
\]

So, the correct average velocity for this interval is \( v_{\text{avg}} = -9.41 \, \text{cm/s} \).

### (b) Instantaneous velocity at \( t = 1 \)
The instantaneous velocity is the derivative of the displacement function \( s(t) \) with respect to time \( t \), i.e., \( v(t) = s'(t) \).

The displacement function is:
\[
s(t) = 3\sin(\pi t) + 3\cos(\pi t)
\]
The derivative \( v(t) \) is:
\[
v(t) = \frac{d}{dt} [3\sin(\pi t) + 3\cos(\pi t)] = 3\pi \cos(\pi t) - 3\pi \sin(\pi t)
\]
At \( t = 1 \):
\[
v(1) = 3\pi \cos(\pi \cdot 1) - 3\pi \sin(\pi \cdot 1) = 3\pi (-1) - 3\pi (0) = -3\pi
\]
Approximating \( \pi \approx 3.14159 \):


The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation s = 3 sin(𝜋t) + 3 cos(𝜋t), where t is measured in seconds. Find the average velocity during each time period and estimate the instantaneous velocity when t = 1.

Explore how to calculate the average and instantaneous velocities of a particle using the displacement equation s = 3 sin(𝜋t) + 3 cos(𝜋t). Learn about key trigonometric and calculus concepts to solve problems involving motion along a straight line. Includes detailed step-by-step solutions.

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