Parametric
Given x(t) = 2t, y(t) = t^2-1. Find the rectangular equation by eliminating the parameter.
x = 2t
y = 1/4 x^2-1
x(t) = cos(t) and y(t) = sin(t) on the interval 0 less than or equal to t less than or equal to 2pi. Find the length of the curve on the given interval.
2pi
r(t) = <2t^2 +4t +1, 3t^3 -4t> then r'(t) =
r'(t) = <4t + 4, 9t^2 -4>
4r cos(Theta) = r^2
Convert into rectangular form
4 = (x-2)^2 +y^2
Find the area bounded by r = 5 sin(theta)
Given x(t) = t^1/2 and y(t) = 1/4 (t^2-4) for t greater than or equal to 0. Find dy/dx
t^3/2
x = 1-4t and y = 7t on the interval 0 less than or equal to t less than or equal to 2. Find the length of the curve on the given interval.
2 * the square root of 65
The path of a particle moving along a path in the xy-plane is given by the vector-valued function, f(t) = <t^2, sin(t)>. Find the slope of the path of the particle at t= 3pi/4
- square root of 2/ 3pi
What is the slope of the line tangent to the polar curve r = 1+2sin(Theta) at Theta = 0?
1/2
Find the area of one petal of the rose curve r = 2cos(3theta)
A = 2.356
A particle moves in the xy-plane so that its position for t greater than or equal to 0 is given by the parametric equations x(t) = 2kt^2 and y(t) = 3t, where k is a positive constant. When t = 2 the line tangent to the particle's path has a slope of 4. What is the value of k?
k = 3/32
Find the length of the curve r = 3+3sin(Theta) over the given interval 0 less than or equal to theta less than or equal to 2pi
24
The position of a particle moving in the xy-plane is given by x = t^2 +2t, y = 2t^2 -6t. What is the speed of the particle when t = 2?
2 * the square root of 10
r = 3theta at Theta = pi/2
Find the slope of the line tangent to the polar curve at the given Theta value.
-2/pi
Consider the polar curve defined by the function r(theta) = 2theta cos(theta), where 0 less than or equal to Theta less than or equal to 3pi/2.
Find the area of the region enclosed by the inner loop of the curve.
A = .5065
Consider the curve given by the parametric equations y = t^3 -12t and x = 1/2t^2-t. Find dy/dx in terms of t and then write an equation for the line tangent to the curve at the point where t = -1.
dy/dx = 3t^2 -12/t-1
y = 9/2(x-3/2)+11
Find the equation that gives the length of the path described by the parametric equations x = the square root of 5 and y = 3t-1 from 0 less than or equal to t less than or equal to 1
x(t) = 2 sin(t/2) and y(t) = 2cos(t/2) for time t > 0. Find the speed of the particle.
1
r = 5/3-cos(Theta) at Theta = 3pi/2
Find the slope of the line tangent to the polar curve at the given Theta value.
-1/3
Find but do not solve the integral that gives the total area of the region shared by both polar curves r = 2cos(theta) and r = 2sin(theta)
2 * the integral from 0 to pi/4 of cos^2(theta) -sin^2(theta) dTheta
Find the second derivative of the following parametric equations:
dy/dt = sin(t^2)
dx/dt = 4
d^2y/dx^2 = t/8 cos(t^2)
The length of the path described by the parametric equations x = cos^3(t) and y = sin^3(t), for 0 less than or equal to pi/2 is given by...
The integral from 0 to pi/2 of the square root of 9cos^4(t)sin^2(t) + 9sin^4(t)cos^2(t) dt
The instantaneous rate of change of the vector-valued function f(t) is given by f'(t) = <2+20t-4t^3, 6t^2+2t>. If f(1) = <5, -3>, what is f(-1)?
f(-1) = <1, -3>
Find the value(s) of Theta where the polar graph r = 2-2cos(Theta) has a horizontal tangent line on the interval 0 less than or equal to Theta less than or equal to 2pi.
Theta = 2pi/3 and 4pi/3
Find the area of the common interior of the polar curves r = 4sin(theta) and r=2
A = 2.913