Calculus Answers

Calculate the velocity and acceleration vectors, and speed for
r(t)=⟨cos(4t),cos(t),sin(t)⟩
when t=7π/4.

Find r(t) and v(t) given acceleration
a(t)=⟨t,9⟩,
initial velocity
v(0)=⟨−1,3⟩,
and initial position
r(0)=⟨0,0⟩.

Evaluate the integral:

∫(t to 0) (3si+(3s^2)j+17k)ds

Use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces x^2+y^2=64 and z=3x^4.

r(t)=<_,_,_>

Match the surfaces with the appropriate descriptions.

1. z=x^2
2. x^2+y^2=5
3. x^2+2y^2+3z^2=1
4. z=2x^2+3y^2
5. z=2x+3y
6. z=y^2−2x^2
7. z=4

A. elliptic paraboloid
B. hyperbolic paraboloid
C. nonhorizontal plane
D. horizontal plane
E. parabolic cylinder
F. ellipsoid
G. circular cylinder

Find the derivative of the vector function.
a)r(t) =<e^(-t),t-t^3,ln t>
b)r(t) =(1/(1 +t)i+(t/(1 +t))j+((t^2)/(1 +t))k

Find the derivative of the vector function.
a)r(t) =<e^(-t),t-t^3,ln t>
b)r(t) =(1/(1 +t)i+(t/(1 +t))j+((t^2)/(1 +t))k

1)Find the limit.
a:)lim t->0 (e^(-3t)i+(t^2/sin^2 t)j|+ cos 2tk)
b:)lim t->1((t^2-t)/(t-1)i+sqrt(t+ 8)j+(sin pi t)/(ln t) k)

2) . Find a vector equation and parametric equations for the line segment that joins P(-1,2,2) and Q(-3,5,1).

1)Find the limit.
a:)lim t->0 (e^(-3t)i+(t^2/sin^2 t)j|+ cos 2tk)
b:)lim t->1((t^2-t)/(t-1)i+sqrt(t+ 8)j+(sin pi t)/(ln t) k)

2) . Find a vector equation and parametric equations for the line segment that joins P(-1,2,2) and Q(-3,5,1).

A Norman window with a perimeter of 20 units has the shape of a rectangle surmounted by a
semi-circle. If its base is x units long and the height of the rectangle is y units long, write a mathematical model for its area in terms of x.