Math 131 Form 222 Test 2 With Answers - La Sierra University Page 4
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47. (10 pts) An airplane flying at an altitude of 4 miles passes directly over a radar station ( = 4 in the diagram
below). When the airplane is 15 miles from the radar station ( = 15), the radar detects that the distance
is
changing at a rate of 220 miles per hour. What is the speed of the airplane. (Round answer to nearest mile
per hour).
2
2
2
Solution: Using the Pythagorean theorem we note
+ 4
=
, we are given
and
and we look to
find
. Then differentiating this implicitly with respect to , we find
2
2
[
+ 16] =
[
]
2
= 2
and so
=
2
2
When = 15, we find that
=
15
4
=
209, and we were given
= 220 miles per hour. Therefore,
15
=
220
228 miles per hour. That is the plane is travelling approximately 228 miles per hour.
209
8. (a) (3 pts) State the definition of derivative.
( + ∆ )
( )
Solution: The derivative of
at
is denoted by
( ) and defined by
( ) = lim
∆
∆
0
provided the limit exists.
2
(b) (7 pts) Let ( ) = 5
3 + 6. Use the definition of derivative to find
( ).
2
Solution: For ( ) = 5
3 + 6, we compute
( + ∆ )
( )
( ) =
lim
∆
∆
0
2
2
5( + ∆ )
3( + ∆ ) + 6
(5
3 + 6)
=
lim
∆
∆
0
2
2
2
5(
+ 2 ∆ + (∆ )
)
3
3∆ + 6
5
+ 3
6
=
lim
∆
∆
0
(∆ )(10 + 5∆
3)
=
lim
∆
∆
0
=
lim
10 + 5∆
3 = 10
3
∆
0
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