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
(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
1 answer:
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