How big is my heart???(Find the area of the enclosed region)

2 2f (x) x 1 x g(x) x 1 x

WARM UP - Calculator active

Warm Up1) NO Calculator If x = t3 – t and y = (3t + 1)1/2, then

(a) Determine dy/dx at t = 1

(b) Write an equation of the line tangent to the curve at t = 1.

2) Calculator An object moving along a curve has position (x(t), y(t)) with dx/dt = cos(t2) and dy/dt = sin(t3). At time t = 0, the object is at position (4,7).

(a) Where is the particle when t = 2?

(b) Determine the speed of the particle at time t = 2.

Arc Length and AP Practice

Let’s Review…The length of a continuous function f(x) on the interval [a,b] is equal to

b2

a1 (f ' (x)) dx

For parametric equations…The length of a continuous curve x(t), y(t) on the interval a < t < b is equal to

b2 2

a(x' (t)) (y'(t)) dt

Determine the length of the curve x = t2 + 1, y = 4t3 + 3

on the interval -1 < t < 0

Consider the curve given by the parametric equations x = 2cos t and y = 2sin t. Determine the length of the curve for t = 0 to t = 2.

Calculator ActiveA particle is moving along a curve so that its position at time t is (x(t), y(t)), where x(t) = t2 – 4t + 8 and y(t) is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known that dy/dt = tet-3 – 1.

(a) Find the speed of the particle at time t = 3 seconds.

(b) Find the total distance traveled by the particle for 0 < t < 4.

Calculator ActiveA particle is moving along a curve so that its position at time t is (x(t), y(t)), where x(t) = t2 – 4t + 8 and y(t) is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known that dy/dt = tet-3 – 1.(c) Find the time t, 0 < t < 4, when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer.

Calculator ActiveA particle is moving along a curve so that its position at time t is (x(t), y(t)), where x(t) = t2 – 4t + 8 and y(t) is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known that dy/dt = tet-3 – 1.(d) There is a point with x-coordinate 5 through which the particle passes

twice. Find each of the following.(i) The two values of t when that occurs(ii) The slopes of the lines tangent to the particle’s path at that

point.(iii) The y-coordinate of that point, given y(2) = 3 + 1/e