h (a) Sketch the graph of r t = t i + t 2 j . Show that r t is a smooth vector -valued function but the change of parameter t = τ 3 produces a vector-valued function that is not smooth, yet has the same graph as r t . (b) Examine how the two vector-valued functions are traced, and see if you can explain what causes the problem.
h (a) Sketch the graph of r t = t i + t 2 j . Show that r t is a smooth vector -valued function but the change of parameter t = τ 3 produces a vector-valued function that is not smooth, yet has the same graph as r t . (b) Examine how the two vector-valued functions are traced, and see if you can explain what causes the problem.
(a) Sketch the graph of
r
t
=
t
i
+
t
2
j
.
Show that
r
t
is a smooth vector-valued function but
the change of parameter
t
=
τ
3
produces a vector-valued function that is not smooth, yet
has the same graph as
r
t
.
(b) Examine how the two vector-valued functions are traced, and see if you can explain what
causes the problem.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
a. Sketch the graph of r(t) = ti+t2j. Show that r(t) is a smooth vector-valued function but the change
of parameter t = 73 produces a vector-valued function that is not smooth, yet has the same graph as
r(t).
b. Examine how the two vector-valued functions are traced, and see if you can explain what causes the
problem.
What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy and Vx = -Uy) do not hold?
The acceleration of an object after t seconds is given by the vector-valued function
a(t)
.
If the object's initial position is ( – 9, – 3, 10), and the object's initial velocity is
find a vector-valued function 7 (t) representing the object's position at time t.
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