Worksheet # 1: Applications of Limits Problem 1. Sensitivity of

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Worksheet # 1: Applications of Limits
Problem 1.
Sensitivity of Measurements: Suppose that f is a function
of x. If x = x0 + ∆x, then we define ∆f = f (x0 + ∆h) − f (x0)
and ∆f /∆x measures how much the value of f changes when x
is changed.
With GPS, radio signals give us h up to a certain measurement
error (See Fig. 2 and Fig. 3). The question is how accurately
can we measure L. To decide, we find ∆L
∆h . In other words, these
variables are related to each other. We want to find how a change
in one variable affects the other variable.
The planet Quirk is flat. GPS satellites hover over Quirk at an
altitude of s = 20, 000 km (see Fig. 3 above). See how accurately
you can estimate the distance L from the point directly below the
satellite to a point on the planet surface knowing the distance h
from the satellite to the point on the surface in two cases. (The
letter h is for hypotenuse.)
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We see that
p
p
2
2
L(h) = h − s = h2 − (20000)2
a. Use a calculator or a spread sheet to compute ∆L/∆h for
h = h0 ± ∆h = 25, 000 ± ∆h, and ∆h = 1, 10−1, 10−2. Write
an estimate of L in the form
|L(h0 + ∆h) − L(h0)| = |∆L| ≤ C|∆h|
choosing the simplest integer C that works for all six cases
(the first case is filled in for you, double check it).
∆h
-1
1
-0.1
0.1
-0.01
0.01
∆L
-1.667
∆L/∆h 1.667
b. Do the same for h = 20, 001 ± ∆h, and ∆h = 1, 10−1, 10−2.
(the first case is filled in for you, double check it).
∆h
-1
1
-0.1
0.1
-0.01
0.01
∆L
-200.002
∆L/∆h 200.002
c. Compute the limit:
p
p
2
2
(h0 + ∆h) − s − h20 − s2
∆L
0
L (h0) = lim
= lim
∆h→0 ∆h
∆h→0
∆h
d. Now go back to part a. above with h0 = 25, 000. How does
L0(h0) compare with your choice of C in part a.?
e. For h0 = 20, 001 how does L0(h0) compare to your choice of
C found in part b. above?
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Problem 2. Lots of Trig Limits Consider the unit circle:
A C
θ
O
D
Determine the following limits:
area of sector AOB
θ→0 area of ∆ AOD
(a) lim
area of sector AOB
θ→0 area of ∆ COB
(b) lim
area of sector AOB
θ→0 area of ∆ AOB
(c) lim
length of segment AB
θ→0 length of segment CB
(d) lim
length of segment AB
θ→0
length of arc AB
(e) lim
length of segment AB
θ→0 length of segment AD
(f) lim
B