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Math 31 – Derivatives of Trigonometric Functions Review
1.
Evaluate the following limits:
a.
lim x cot x 
c.
omit
e.
g.
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b.
x 0
d.
 x
sin 2  
2 
lim
x 0
sin x
omit
lim
x 0
x 0
sin x

5x
1  cos x

x 0
sin x
lim
f.
h.
lim
1  cos 2 x

x 0 2 x  cos x  1
lim
tan x

tan 2 x
1
2.
Find
dy
for each of the following:
dx
a.
y  cos3  2 x  1
b.
y  sec  2 x  1
c.
 x
y  sin 2 x cos  
2
d.
y
e.
 1 
y  sin 

 2x 1 
f.
y  cot 2  x 2  2 
g.
y  tan 2  sin 3x 
h.
y
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3
x3
cos x
cos x
sec x  tan x
2
3.
Find an equation of the tangent for each of the following:
5
6
a.
y  2  cos 2 x at x 
b.
y  2sin x at x 
c.
y  csc x  2sin x at x 
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
3

6
3
4.
Determine the maximum and minimum points, the concavity and points of inflection for each of
the following:
a.
y   sin 2 x on the interval  0, 2 
b.
y  sin 2 x 
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x
on the interval  0,  
2
4
5.
Find the derivatives of the following using implicit differentiation:
a.
x 2 sin y 2  y cos x  6
b.
cos  x  y   cos  x  y   1
c.
x sin y  cos 2 y  cos y
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5
6.
A 6 m ladder is leaning against a wall and begins to slide. The foot of the ladder slides outward at
a rate of 0.2 m/s. At what rate is the angle between the ladder and the wall changing when the top
is 3 m from the ground?
7.
The hypotenuse of a right triangle has length of 10 cm. Determine its maximum possible area.
8.
A vehicle moves along a straight road at a speed of 4 m/s. A searchlight is located on the ground
20 m from the road and is focused on the vehicle. At what rate (in rad/sec) is the searchlight
rotating when the vehicle is 15 m from the point on the road closest to the searchlight?
9.
A lighthouse is situated 200 m from a straight shoreline. The light rotates clockwise at 2
revolutions per minute. At what speed is the beam of light moving along the shore when the angle

between the beam and the shore is ?
6
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6
Answer Key
1.
2.
3.
4.
a.
1
b.
1
5
c.
omit
d.
0
e.
0
f.
0
g.
omit
h.
a.
6cos2  2 x  1 sin  2 x  1
1
2
b.
6sec  2 x  1 tan  2 x  1  2 x  1
c.
1

 x
 x 
sin x  4cos x cos    sin x sin   
2
2
 2 

2
x2 sec x  x tan x  3 OR x2 sec2 x  3cos x  x sin x 
e.

f.
4 x cot  x 2  2  csc 2  x 2  2 
g.
6 tan  sin 3x  sec2  sin 3x  cos3x
h.
a.
cos x
6 3x  6 y  5 3  15  0
b.
3x  3 y  3 3    0
c.
6 3x  2 y   3  2  0
a.

  5

 3   7 
min  , 1 ,  ,  1 max 
,1 , 
,1
 
 3 
4
  4

 4   4  , inflection point  2 , 0  ,  , 0  ,  2 , 0 




a.
b.
 1 
cos 

 2x 1 
 2 x  1
2
2


 5


  3

min  , 0.06  , max 
, 0.28  , inflection point  , 0.11 ,  , 0.68 
 12

 12

4
  4

2
2 x sin y  y sin x
y 
cos x  2 x 2 y cos y 2
sin  x  y   sin  x  y 
y 
sin  x  y   sin  x  y 
 sin y
x cos y  2sin 2 y  sin y
0.067 rad/sec
25 cm 2
0.128 rad/sec
3200  m/min
c.
6.
7.
8.
9.
3
d.
b.
5.
3
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y 
7