Find the functions that give the speed of the two objects in Example 2, for t ≥ 0 (Corresponding to the graphs in Figure 14.15 ). Example 2 Comparing trajectories Consider the trajectories described by the position functions r ( t ) − 〈 t , t 2 − 4 , t 3 4 − 8 〉 , for t ≥ 0, and R ( t ) − 〈 t 2 , t 4 − 4 , t 6 4 − 8 〉 , for t ≥ 0, Where t is measured in the same time units for both functions. a. Graph and compare the trajectories using a graphing utility. b. Find the velocity vectors associated with the position functions. Figure 14.15
Find the functions that give the speed of the two objects in Example 2, for t ≥ 0 (Corresponding to the graphs in Figure 14.15 ). Example 2 Comparing trajectories Consider the trajectories described by the position functions r ( t ) − 〈 t , t 2 − 4 , t 3 4 − 8 〉 , for t ≥ 0, and R ( t ) − 〈 t 2 , t 4 − 4 , t 6 4 − 8 〉 , for t ≥ 0, Where t is measured in the same time units for both functions. a. Graph and compare the trajectories using a graphing utility. b. Find the velocity vectors associated with the position functions. Figure 14.15
Solution Summary: The author explains the scalar function which can give the speed of the 2 objects.
Find the functions that give the speed of the two objects in Example 2, for t ≥ 0 (Corresponding to the graphs in Figure 14.15).
Example 2 Comparing trajectories
Consider the trajectories described by the position functions
r
(
t
)
−
〈
t
,
t
2
−
4
,
t
3
4
−
8
〉
, for t ≥ 0, and
R
(
t
)
−
〈
t
2
,
t
4
−
4
,
t
6
4
−
8
〉
, for t ≥ 0,
Where t is measured in the same time units for both functions.
a. Graph and compare the trajectories using a graphing utility.
b. Find the velocity vectors associated with the position functions.
Figure 14.15
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.
Let f be a function whose graph consists of 5 line segments and a semicircle as shown in the figure below.
Let g(x) = √ƒƒ(t) dt .
0
3
2
-2
2
4
5
6
7
8
9
10
11
12
13
14
15
1. g(0) =
2. g(2) =
3. g(4) =
4. g(6) =
5. g'(3) =
6. g'(13)=
The expression 3 | (3+1/+1)
of the following integrals?
A
Ов
E
+
+
+ +
18
3+1+1
3++1
3++1
(A) √2×14 dx
x+1
(C) 1½-½√ √ ² ( 14 ) d x
(B) √31dx
(D) So 3+x
-dx
is a Riemann sum approximation of which
5
(E) 1½√√3dx
2x+1
2. Suppose the population of Wakanda t years after 2000 is given by the equation
f(t) = 45000(1.006). If this trend continues, in what year will the population reach 50,000
people? Show all your work, round your answer to two decimal places, and include units. (4
points)
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