T, N, B
How do you calculate the tangent unit vector for r(t)?
(r'(t))/(|r'(t)|
Suppose r(t) = < t, t2, t3 >
Find r'(t) x r''(t)
< 6t2, -6t, 2 >
Calculate the curvature of:
r(t)=<2t, 5cos(t), 5sin(t)>
K=5/29
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
What is the only mammal that can fly?
bat
If r(t) = < t2 - 4t, 1 + 5t, (1/3)t3 + (1/2)t2 >, find T(4)
(1/21) < 4, 5, 20 >
Find the parametric equations for the line tangent to the curve, r(t) = < 4cos(t), 4sin(t), 8cos(2t) > at the point
(2sqrt3, 2, 4)
x(t)=-2t+2sqrt(3)
y(t)=2sqrt(3)t+2
z(t)=-8sqrt(3)t+4
Find the point(s) on the graph of
y = 2(x - 2)3 - 4 at which the curvature is 0
(2, -4)
A particle moves with the position function, r(t)=<tlnt, t, e^-t> .
Find the velocity, speed and acceleration functions for the particle's motion.
v(t)=<1+lnt, 1, -e^-t>
a(t)=<1/t, 0, e^-t>
speed=|v(t)|=sqrt(2+2lnt+(lnt)^2+e^(-2t))
What is the slowest animal in the world?
3-toed sloth (yes, they are slower than snails)
(sea anemone would also be accepted)
Calculate N(t) if r(t)=<e^tcost,e^tsint,5>
N(t)=1/sqrt(2)<-sint-cost,-sint+cost,0>
OR N(t)=1/sqrt(2)<sint+cost,sint-cost,0>
Find r(t) if r(1) = < 1, 1, 0 > and
r'(t)=<5t^4, 6t^5, sqrt(t) >
r(t)=< t^5, t^6, 2/3t^(3/2)-2/3 >
Find the radius of curvature of
y = -3x3 + 2x2 - 4 when x = 1
26^(3/2)/14
A ball is thrown from ground level at an angle of elevation of 30o. If the initial velocity is 4m/s, at what time does the ball reach its maximum height? (Round your answer to 3 decimals)
t ≈ .204 seconds
How many hearts does an octopus have?
3
Calculate N(t) at the point (1, 2/3, 1)
if r(t)=<t^2, 2/3t^3, t>
<-1/3, 2/3, -2/3>
Write the vector function, r(t) that represents the curve of intersection of the two equations below
z=sqrt(x^2+y^2)
z=3+y
r(t)=<t, 1/6 (t^2-9), 1/6 (t^2+9)>
r(t)=<+-sqrt(6t+9),t,3+t>
r(t)=<+-sqrt(27-6t),t-3,t>
Determine the interval(s) on which the curve below is smooth
r(t)=<t^3,sqrt(t^2-1)>
(-∞, -1) U (1, ∞)
A baseball player at second base throws a ball 90 feet to the player at first base. The ball is released at a point 5 feet above the ground with an initial speed of 70 feet per second and at an angle of 15˚ with the horizontal. At what height does the player at first base catch the ball? (Answer must be correct to at least 2 decimal places.)
≈0.767 feet
Which animal has the largest eyes?
colossal squid
Find a fully simplified equation of the osculating plane of the curve
r(t)=<sin(2t), t, cos(2t)>
at the point (0, π, 1)
Hint: the osculating plane contains vectors T(t) and N(t)
x-2y+2π=0
Find the domain of the vector function
r(t)=<(t-1)/(t+1),sin(t), ln(49-t^2)>
(-7, -1) U (-1, 7)
Find the length of the curve
r(t)=<2t, cos(2t), sin(2t)>
0≤t≤1
2sqrt(2)
NO CALCULATORS!
A projectile is fired from ground level at an angle of 30˚ with the horizontal. The projectile must have a range (horizontal distance) of at least 200 feet before hitting the ground. Find the minimum initial velocity necessary for the projectile. (Note, your answer might have some roots in it, that's okay.)
v_0=(80sqrt(2))/root(4)(3)
Which mammal has the highest blood pressure?
giraffe