(1 point) For the given position vectors r(t)r(t) compute the
unit tangent vector T(t)T(t) for the given value of tt .
A) Let r(t)=〈cos5t,sin5t〉
Then T(π4)〈
B) Let r(t)=〈t^2,t^3〉
Then T(4)=〈
C) Let r(t)=e^(5t)i+e^(−4t)j+tk
Then T(−5)=
4. Let r(?) = �?, 4 3 ? 3/2, ?2 �. (a) Find T, N, and B at the
point corresponding to ? = 1. (b) Find the equation of the
osculating plane at the point corresponding to ? = 1. (c) Find the
equation of the normal plane at the point corresponding to ? =
1
Consider the following vector function. r(t) =<3t, 1/2 t2,
t2> (a) Find the unit tangent and unit normal vectors T(t) and
N(t).
(b). Find the curvature k(t).
Find T(t), N(t), aT, and aN at the given time t for the space
curve r(t). [Hint: Find a(t), T(t), aT, and aN. Solve for N in the
equation a(t)=aTT+aNN. (If an answer is undefined, enter
UNDEFINED.)
Function Time
r(t)=9ti-tj+(t^2)k t=-1
T(-1)=
N(-1)=
aT=
aN=
3) Solve the initial value problems
c) R′ + (R/t) = (2/(1+t2 )) , R(1) = ln 8.
e) ) cos θv′ + v = 3 , v(π/2) = 1.
5) Express the general solution of the equation x ′ = 2tx + 1 in
terms of the erf function.
7) Solve x ′′ + x ′ = 3t by substituting y = x ′
9) Find the general solution to the differential equation x ′ =
ax + b,...
If T1 and T2 are independent exponential
random variables, find the density function of R=T(2) -
T(1).
This is for the difference of the order statistics not of the
variables, i.e. we are not looking for
T2 - T1. It is implied that they are both
from the same distribution. I know that
fT(t) = λe-λt
fT(1)T(2)(t1,t2) = 2
fT(t1)fT(t2) =
2λ2 e-λt1
e-λt2 , 0 < t1 <
t2 and I need to find fR(r).
From Mathematical Statistics and...
At what point do the curves r1(t) = t, 3 − t, 35 + t2 and r2(s)
= 7 − s, s − 4, s2 intersect? (x, y, z) = Find their angle of
intersection, θ, correct to the nearest degree. θ = °
All vectors are in R^ n. Prove the following statements.
a) v·v=||v||2
b) If ||u||2 + ||v||2 = ||u + v||2, then u and v are
orthogonal.
c) (Schwarz inequality) |v · w| ≤ ||v||||w||.
Given the vectors u1 = (2, −1, 3) and u2 = (1, 2, 2) find a
third vector u3 in R3 such that
(a) {u1, u2, u3} spans R3
(b) {u1, u2, u3} does not span R3