An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle in the first quadrant. (Connect the origin with the y-intercept and x-intercept of the tangent line.)

(a) The area of the right triangle is a(t)=  .

(b) lim t → pi/2⁻a(t)=

(c) lim t → 0⁺a(t)=

(d) lim t → pi/4 a(t)=  

(e) With our restriction on t, the smallest t so that a(t)=2 is  

(f) With our restriction on t, the largest t so that a(t)=2 is  

(g) The average rate of change of the area of the triangle on the time interval [π/6,π/4] is  .

(g) The average rate of change of the area of the triangle on the time interval [π/4,π/3] is  .

(h) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [π/6,b], as b approaches π/6 from the right. The limiting value is  .

(i) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [a,π/3], as a approaches π/3 from the left. The limiting value is  .

Transcribed Image Text:

P(t) P(t)

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