AP® Calculus BC · Free response

A particle in the plane: tangent line, direction, acceleration and distance

write it first, then score yourself point by point

Parametric, vector or polar functions · Part A: graphing calculator allowed · 9 points

Start writingAll 29 questions

The question

Name: ______________________   Date: ____________   A particle in the plane: tangent line, direction, acceleration and distance (9 points)

Table

A particle with given velocity components

A particle moves in the xyxy-plane for 0≤t≤50 \le t \le 5. Its velocity components are given below, and at time t=2t = 2 the particle is at the point (3,−1)(3, -1). A graphing calculator is required.
Velocity components of the particle and its position at t=2t = 2
QuantityValue
dxdt\frac{dx}{dt}t+1\sqrt{t + 1}
dydt\frac{dy}{dt}3cos⁡(1.2t)−13\cos(1.2t) - 1
(x(2),y(2))(x(2), y(2))(3,−1)(3, -1)

  1. (a) Find the slope of the path of the particle at time t=2t = 2. Write an equation for the line tangent to the path at the point where t=2t = 2. [2 points]

  2. (b) For 0<t<50 < t < 5, on what open interval is the yy-coordinate of the particle decreasing? Give a reason for your answer. [2 points]

  3. (c) Find the acceleration vector of the particle at time t=2t = 2. [2 points]

  4. (d) Find the total distance traveled by the particle over the time interval 0≤t≤50 \le t \le 5. Find the xx-coordinate of the particle's position at time t=5t = 5. [3 points]