Question

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)= 1/(2sin(x)cos(x)
(b)

lim

t ? pi/2?

a(t)= +infinity


(c)

lim

t ? 0+

a(t)= -infinity


(d)

lim

t ? pi/4

a(t)= 1


(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 [?/6,?/4] is ??

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

(i) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [?/6,b], as b approaches ?/6 from the right. The limiting value is -4/3

(j) Create a table of values to study the average rate of change of the area of the triangle on the time intervals [a,?/3], as a approaches ?/3 from the left. The limiting value is 4/3

Answer 1

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