Download Lesson Plan #6

1
Lesson Plan #32
Date: Wednesday November 23rd , 2011
Class: AP Calculus
Topic: Related rates involving trigonometric functions.
Aim: How do we solve related rates questions involving
trigonometric functions?
Objectives:
1) Students will be able to solve related rates questions
involving trigonometric functions
HW# 32:
1) A rocket travels vertically from a launch pad 10 km
away from an observer with a telescope. At a certain
moment the angle between the telescope and the

ground is 3 and it is changing at a rate of 0.5 rad/min. What is the rocket's velocity (in km/min) at that moment?
Do Now:
1) Given A 
1
dA
ab sin  , find
dt
2
Answer
2) For Do Now 1), evaluate
dA
da
db
d
 2,
 7,
 0 and   30o
when a  5, b  12,
dt
dt
dt
dt
Answer
3) In words, describe a problem that you are solving with Do Now examples 1 and 2.
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
When we are finding rates of change of two or more variables that are changing with respect to time, these rates are called Related
Rates. Today, let’s see how we find Related Rates involving trig functions.
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1) A hot-air balloon, rising straight up from a level field, is tracked by a range finder 500 ft from the lift-off point. At the
moment the range finder’s angle of elevation is

, the angle is increasing at a rate of 0.14 radians/minute.
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How fast is
the balloon rising?
y
Range Finder

500 feet
Ans.
2) Find the rate of change in the angle of elevation of the camera shown in the figure 10 seconds after liftoff, if the height of
rocket is given by the position function s  50t
2
s

2000 ft.
Ans.
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Sample Test Questions:
1) If f ( x)  sec x  csc x, then f ' ( x) 
f ( x)  sec 2 x  csc 2 x
E) sec x tan x  csc x cot x
A) 0
B)
D) sec x tan x  csc x cot x
2) If
B) 0
C) 1
E) 2 
D) 2
f ( x)  sin 2 (3  x), then f ' (0) =
A) -2cos3
B) -2sin3cos3
5) If f ( x)  x cos
A)
csc x  sec x
f ( x)  cos 2 x, then f " ( ) =
A) -2
3) If
C)

2
B)
D) 2sin3cos3
E) 6sin3cos3
1
2
, then f '   equals
x
 
2
D) 
C) -1

6) If sin( xy)  y , then
A) sec(xy)
C) 6cos3

2
E) 1
dy
equals
dx
B) ycos(xy)-1
C)
1  y cos( xy)
x cos( xy)
D)
y cos( xy)
1  x cos( xy)
E) cos(xy)
Review for Exam:
1) A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. Find
the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters
above the ground
3m/sec
Ans.
30 m
2) Find the point on the curve y 
Observer
\
x that is a minimum distance from the point 4,0
Ans.