Vector-Valued Functions and Space Curves
Calculus on Vector-Valued Functions
Arc Length and Curvature
Motion in Space
100
What is the name of a function in the form:
vecr(t)=f(t)hati+g(t)hatj
a plane curve
100
How do you calculate the tangent unit vector for r(t)?
(vecr'(t))/(||vecr'(t)||
100
The derivative to the tangent unit vector with respect to the arc-length parameter is...
Curvature!
100
if r'(t) represents an object's position at time t, what is the meaning of: r'''(t)
the object's acceleration at time t
200
Give the defining characteristic for a function's binormal unit vector.
a unit vector that is orthogonal to both the unit tangent vector and the unit normal vector
200
Sketch the plane curve given by:
vecr(t)=<cos(t), 2sin^2(t)>
200
Find the curvature and the radius of curvature at the given point.
y=1/2x^2+2 at x=sqrt(3)
k=1/8
r=8
200
A ball is thrown from ground level. It has an angle of elevation of 30o. If the initial velocity was 4m/s what time does the ball reach its maximum height?
t=.20408 seconds
300
Determine the domain of r and the interval of t where r(t) is continuous if:
vecr(t)=<ln(t), 2t-1, t>
t in (0,infty)
300
Find the indefinite integral:
int (thatj+t^2hatk) times(hati+thatj+thatk)dt
(t^3/3-t^4/4)hati+t^3/3hatj-t^2/2hatk
300
Calculate the curvature of:
vecr(t)=<2t, 5cos(t), 5sin(t)>
K=5/29
300
A projectile is fired from a canon at ground level with an angle of elevation of 20o. The projectile has a range of 80 meters. Find the minimum initial velocity. (gravity is 9.8m/s)
"Range "= x=(v_0)^2/9.8sin(2theta)=80
v_0=sqrt(((80)(9.8))/sin(40^("o"))) approx 34.9m/s