A hiking trail is roughly in the shape of a circle with radius 2000 ft . Suppose that a rectangular coordinate system is set up with the origin at the center of the circle. Donna and Eddie start walking from point A 2000 , 0 at 220 ft/min . Eddie goes off-trail and walks directly toward the center of the circle and Donna walks along the trail counterclockwise. a. Find the coordinates of each person after 5 min . Round the coordinates to the nearest foot b. How far apart are Eddie and Donna after 5 min ? Round to the nearest foot. c. If Eddie can no longer walk because he sprains his ankle after foolishly leaving the trail, at what bearing would Donna walk to find him? Round to the nearest tenth of a degree.
A hiking trail is roughly in the shape of a circle with radius 2000 ft . Suppose that a rectangular coordinate system is set up with the origin at the center of the circle. Donna and Eddie start walking from point A 2000 , 0 at 220 ft/min . Eddie goes off-trail and walks directly toward the center of the circle and Donna walks along the trail counterclockwise. a. Find the coordinates of each person after 5 min . Round the coordinates to the nearest foot b. How far apart are Eddie and Donna after 5 min ? Round to the nearest foot. c. If Eddie can no longer walk because he sprains his ankle after foolishly leaving the trail, at what bearing would Donna walk to find him? Round to the nearest tenth of a degree.
Solution Summary: The author calculates the coordinates of Donna and Eddie after each person walks for 5min from point A.
A hiking trail is roughly in the shape of a circle with radius
2000
ft
. Suppose that a rectangular coordinate system is set up with the origin at the center of the circle. Donna and Eddie start walking from point
A
2000
,
0
at
220
ft/min
. Eddie goes off-trail and walks directly toward the center of the circle and Donna walks along the trail counterclockwise.
a. Find the coordinates of each person after
5
min
. Round the coordinates to the nearest foot
b. How far apart are Eddie and Donna after
5
min
? Round to the nearest foot.
c. If Eddie can no longer walk because he sprains his ankle after foolishly leaving the trail, at what bearing would Donna walk to find him? Round to the nearest tenth of a degree.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties
to find lim→-4
1
[2h (x) — h(x) + 7 f(x)] :
-
h(x)+7f(x)
3
O DNE
17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t).
(a) How much of the slope field can
you sketch from this information?
[Hint: Note that the differential
equation depends only on t.]
(b) What can you say about the solu-
tion with y(0) = 2? (For example,
can you sketch the graph of this so-
lution?)
y(0) = 1
y
AN
(b) Find the (instantaneous) rate of change of y at x = 5.
In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of
change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the
following limit.
lim
h→0
-
f(x + h) − f(x)
h
The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule
defining f.
f(x + h) = (x + h)² - 5(x+ h)
=
2xh+h2_
x² + 2xh + h² 5✔
-
5
)x - 5h
Step 4
-
The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x).
-
f(x + h) f(x) =
= (x²
x² + 2xh + h² -
])-
=
2x
+ h² - 5h
])x-5h) - (x² - 5x)
=
]) (2x + h - 5)
Macbook Pro
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