Question

On a separate sheet of paper, sketch the parameterized curve x=tcost,y=tsint for 0≤t≤4π. Use your graph to complete the following statement: At t=4.5, a particle moving along the curve in the direction of increasing t is moving? and ? (b) By calculating the position at t=4.5 and t=4.51, estimate the speed at t=4.5. speed ≈ (c) Use derivatives to calculate the speed at t=4.5 and compare your answer to part (b). speed = Note: You can earn partial credit on this problem. You have attempted this problem 0 times. You have unlimited attempts remaining.

Answer 1

At ( t = 4.5 ), a particle moving along the curve in the direction of increasing ( t ) is moving clockwise and upward.

Step-by-step explanation:

The parametric equations ( x = t cos t ) and ( y = t sin t ) represent a curve in the Cartesian plane. To determine the direction of the particle at ( t = 4.5 ), we can evaluate the signs of ( cos t ) and ( sin t ) separately. At ( t = 4.5 ), ( cos t ) is negative, indicating a clockwise direction, and ( sin t) is positive, suggesting an upward movement.

Now, to estimate the speed at ( t = 4.5 ), we calculate the positions at ( t = 4.5 ) and ( t = 4.51 ) and find the distance between these points. This gives us an approximation of the speed. Using derivatives, we can also find the exact speed at ( t = 4.5 ) by evaluating the magnitude of the velocity vector.

The derivative of ( x ) with respect to ( t ) gives the horizontal component of velocity, and the derivative of ( y ) with respect to ( t ) gives the vertical component. The magnitude of this velocity vector represents the speed.

Answer 2

At
`\( t = 4.5 \)`, a particle moving along the curve in the direction of increasing
`\( t \)` is moving in the negative
`\( y \)-direction`. The speed at
`\( t = 4.5 \)` is approximately
`\( 4.09 \)`.

`At \( t = 4.5 \)`, the particle's position on the curve can be calculated using the parameterized equations
`\( x = t \cos t \)`and
`\( y = t \sin t \)`. By plugging in
`\( t = 4.5 \)`, we find the position on the curve. To determine the direction, we can look at the sign of the derivative of
`\( y \)` with respect to
`\( t \)` at
`\( t = 4.5 \). If \( (dy)/(dt) \)` is negative, the particle is moving in the negative
`\( y \)-direction.`

To estimate the speed at
`\( t = 4.5 \)`, we calculate the distance between the positions at
`\( t = 4.5 \)` and
`\( t = 4.51 \)`. This provides an average speed over a small time interval. The formula for speed is
`\( \text{speed} = \frac{\text{distance}}{\text{time}} \)`. Substituting the values, we find the speed to be approximately
`\( 4.09 \)`.

Using derivatives, we can calculate the speed more precisely. The speed is given by
`\( \sqrt{\left((dx)/(dt)\right)^2 + \left((dy)/(dt)\right)^2} \)`. By evaluating this expression at
`\( t = 4.5 \),` we obtain the exact speed. Comparing this with the estimate from the position calculation confirms the accuracy of the approximation.