A particle moves in the yz-plane along the curve represented by the vector-valued function Find the maximum value of
A)4
B)3
C)7
D)8

Question

A particle moves in the yz-plane along the curve represented by the vector-valued function  Find the maximum value of 
A)4 
B)3 
C)7 
D)8

Quizplus · Accepted Answer

To find the maximum value of the given function, we need to find the critical points. We can do this by finding where the derivative of the function is equal to zero or does not exist. The derivative of the function is: r'(t) = <2t, 3 - t^2, 3t> Setting r'(t) equal to zero gives us 2t = 0 and 3t = 0. This means t = 0 is a critical point. Now we need to check the endpoints of the curve. The curve starts at (0,3,0) and ends at (0,0,0). Plugging in t = 0 into the function gives us r(0) = <0, 3, 0>, which is the starting point of the curve. Therefore, the maximum value of the function is attained at the endpoint of the curve, which is (0,0,0). Plugging in t = 1 into the function gives us r(1) = <1,2,3>. The distance between the starting point and the endpoint is the magnitude of this vector: ||r(1) - r(0)|| = ||<1, -1, 3>|| = sqrt(11) So the maximum value of the function is sqrt(11), which is approximately 3.316. Therefore, the answer is choice C. To find the maximum value of the function, we need to find the critical points, where the derivative of the function is zero, or the derivative does not exist. The derivative of the function is 11ebbd4d_3f05_3820_8a0a_51ae4ea18646_TB8202_11d/dt = <0,2cos(t),-2sin(t)>. For the derivative to be zero, we need cos(t) = 0 and sin(t) = 0. This occurs at t = π/2 and t = 3π/2. To determine whether these are maximum or minimum points, we look at the second derivative. The second derivative is 11ebbd4d_3f05_3820_8a0a_51ae4ea18646_TB8202_11d^2/dt^2 = <0,-2sin(t),-2cos(t)>. At t = π/2, the second derivative is <0,-2,0>, which is negative, so we have a maximum value at this point. At t = 3π/2, the second derivative is also negative, so we have another maximum value at this point. Plugging in these values into the function, we get 7 for both cases, so the maximum value of the function is 7. Therefore, the answer is (C).

A Particle Moves in the Yz-Plane Along the Curve Represented