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.
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.
Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect.
Chapter 12 Solutions
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Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
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