Parametric Derivatives
For x=4*sin(t) and y=2*cos(t), determine the value of the slope at t=pi/4.
What is -1/2?
For x=t+et and y=1-et, determine the value of the area from t=-3 to t=0 with respect to the x-axis.
What is 5/2+1/(2e6)?
Determine the value of the area inside the closed curve r2=2*sin(3*θ).
What is 4?
For x=(2*t+3)3/2/3 and y=t+t2, determine the value of the arc length from t=0 to t=3.
What is 21/2?
For x=-sqrt(t+1) and y=sqrt(3*t), determine the value of the slope at t=3.
What is -2?
For x=3*sin(t) and y=cos(t)+2, determine the value of the area from t=pi/2 to t=pi with respect to the y-axis.
What is 3*pi/4?
Set up an expression to determine the area inside the closed curve r=5*sin(4*θ).
What is the integral from θ=0 to θ=pi/4 of 4*(5*sin(4*θ))2?
For x=8*cos(t)+8*t*sin(t) and y=8*sin(t)-8*t*cos(t), determine the value of the arc length from t=0 to t=pi.
What is 4*pi2?
For x=sin(2*pi*t) and y=cos(2*pi*t), determine the Cartesian equation of the tangent line at t=-1/6.
What is y-1/2=sqrt(3)*(x+sqrt(3)/2)?
For x=t2 and y=t6-2t4, determine the value of the area above the curve and below the x-axis.
What is 4/3?
Determine the value of the area inside the closed curve r=3+2*cos(θ).
What is 11*pi?
Determine the value of the arc length of r=sin2(θ/2) from θ=0 to θ=pi.
What is 2?
For x=1/(t+1) and y=t/(t-1), determine the Cartesian equation of the tangent line at t=2.
What is y-2=9*(x-1/3)?
For x=sin(t) and y=cos(2*t), determine the value of the area below the curve and above the x-axis.
What is 2*sqrt(2)/3?
Set up an expression that determines the area shared by the curves r=1 and r=2*sin(θ).
What is the integral from θ=0 to θ=pi/6 of (2*sin(θ))2 plus the integral from θ=pi/6 to θ=pi/2 of (1)2?
Set up an expression that determines the arc length of all the small petals in one orbit of r=1-2*sin(3*θ).
What is the 3 times the integral from θ=pi/18 to θ=5*pi/18 of sqrt((1-2*sin(3*θ))2+(-6*cos(3*θ))2)?
For x=cos(t) and y=1+sin(t), determine whether the curve is concave up or concave down at t=pi/2.
What is concave down because the value of the second derivative is -1?
For x=2*t-t3 and y=ln(t), determine the value of the area below the curve, above the x-axis, and to the right of the y-axis.
What is (4*sqrt(2)-5)/3?
Set up an expression that determines the area outside the curve r=2+2*cos(θ) and inside the curve r=2-2*sin(θ).
What is the integral from θ=3*pi/4 to θ=7*pi/4 of ((2-2*sin(θ))2-(2+2*cos(θ))2)/2?
Set up an expression that determines the arc length of one orbit of r=1+cos(3*θ/2).
What is the the integral from θ=0 to θ=4*pi of sqrt((1+cos(3*θ/2))2+(-3*sin(3*θ/2)/2)2)?