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A particle moving along a curve in the xy-plane has position (x(t), y(t)), at time t > 0, where
$$\frac{dx}{dt} = \left(\frac{t}{2} - 3\right)^{1/3} \quad \text{and} \quad \frac{dy}{dt} = 4te^{-t}.$$
At time t = 3, the particle is at the point (5, 4).
(a) Find the speed of the particle at time t = 3.
A particle moving along a curve in the xy-plane has position (x(t), y(t)), at time t > 0, where
$$\frac{dx}{dt} = \left(\frac{t}{2} - 3\right)^{1/3} \quad \text{and} \quad \frac{dy}{dt} = 4te^{-t}.$$
At time t = 3, the particle is at the point (5, 4).
(b) Write an equation for the line tangent to the path of the particle at time t = 3.
A particle moving along a curve in the xy-plane has position (x(t), y(t)), at time t > 0, where
$$\frac{dx}{dt} = \left(\frac{t}{2} - 3\right)^{1/3} \quad \text{and} \quad \frac{dy}{dt} = 4te^{-t}.$$
At time t = 3, the particle is at the point (5, 4).
(c) Is there a time t at which the particle is farthest to the right? If yes, explain why and give the value of t and the x-coordinate of the position of the particle at that time. If no, explain why not.
A particle moving along a curve in the xy-plane has position (x(t), y(t)), at time t > 3, where
$$\frac{dx}{dt} = \left(\frac{t}{2} - 3\right)^{1/3} \quad \text{and} \quad \frac{dy}{dt} = 4te^{-t}.$$
At time t = 3, the particle is at the point (5, 4).
Describe the behavior of the path of the particle as t increases without bound.
Describe the behavior of the path of the particle as t increases without bound.
NO CALCULATOR IS ALLOWED FOR THIS QUESTION.
The function g is defined on the closed interval [−4, 8]. The graph of g consists of two linear pieces and a semicircle, as shown in the figure above. Let f be the function defined by $f(x) = 3x + \int_5^x g(t)\,dt$.
(a) Find f(7) and f′(7).