Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 1 of 8
Name_________________________________________ Date________________________ Period______
Worksheet 4.8—Inverse & Inverse Trig Functions
Show all work. No calculator unless otherwise stated.
1. Find the derivative with respect to the appropriate variable. Simplify your expression
(a)
( )
12
secyx
−
=
(b)
2
1arccosys s s= − +
(c)
( )
1
arcsin 2
y
x
=
(d)
1
cot 1yt
−
= −
(e)
1
csc
2
x
y
−
=
(f)
( )
sin arc co syt=
(g)
arctanym m=
(h)
(i)
arcsin 3x
y
x
=
Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 2 of 8
(j)
2
1
4 4 arcsin
22
x
yxx
!"
!"
= − +
$%
$%
&'
&'
(k)
11
sin cosytt
−−
=+
2. If a particle’s position is given by
( )
( )
12
tanxt t
−
=
, find the particle’s velocity at
1t =
.
3. (Calculator Permitted) Find the equation for the tangent line (in slope-intercept form) for the tangent to
the graph of y at the indicated point.
(a)
1
secyx
−
=
at
2x =
(b)
1
sin
4
x
y
−
"#
=
$%
&'
at
3x =
Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 3 of 8
4. A particle moves along the x-axis so that its position at any time
0t ≥
is given by
( )
arctanxt t=
.
(a) Prove that the particle is always moving to the right by analyzing the velocity function.
(b) Prove that the particle’s velocity is always decreasing by analyzing the acceleration function.
(c) What is the limiting position of the particle as t approaches infinity.
5. If
( )
53
21fx x x x=+ +−
and
( )
( )
( )
( )
fgx x gfx==
find (
(a)
( )
1f
(b)
( )
3g
!
.
Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 4 of 8
6. If
( )
cos 3hx x x=+
, find a)
( )
0h
and b)
( )
( )
1
1h
−
"
.
7. Find the equation of the tangent line to the graph of
2
arctan 1xx yy+=−
at
,1
4
π
"#
−
%&
'(
.
8. If
( )
1
3
8
fx x= −
and
( )
3
gx x=
, find
(a)
( )
( )
11
1fg
−−
o
(b)
( )
( )
11
4gg
−−
−o
(c)
( )
( )
11
3gf
−−
−o
Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 5 of 8
_____ 9. Find the value of
( )
1f
when
( )
11
5sin 6 tanfx x x
−−
=+
.
(A)
3
π
(B)
2
π
(C)
4
π
(D)
7
2
π
(E)
5
2
π
_____ 10. Simplify the expression
( )
( )
1
sin tanfx x
−
=
by writing it in algebraic form.
(A)
( )
2
1
1
fx
x
=
+
(B)
( )
2
1fx x=+
(C)
( )
2
1
x
fx
x
=
−
(D)
( )
2
1
x
fx
x
=
+
(E)
( )
2
1
1
fx
x
=
−
(F)
( )
2
1fx x= −
_____ 11. Determine
( )
fx
!
when
( )
1
2
tan
6
x
fx
x
−
"#
=
$%
$%
−
&'
.
(A)
( )
2
6
6
fx
x
!
=
+
(B)
( )
2
6
x
fx
x
!
=
+
(C)
( )
2
6
x
fx
x
!
=
−
(D)
( )
2
6
6
fx
x
!
=
−
(E)
( )
2
1
6
fx
x
!
=
−
Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 6 of 8
_____ 12. Find the derivative of f when
( )
2
5arcsin 25
5
x
fx x=+−
(A)
( )
5
5
x
fx
x
−
"
=
+
(B)
( )
2
25
x
fx
x
!
=
−
(C)
( )
2
5
25
fx
x
!
=
−
(D)
( )
1
5
fx
x
!
=
+
(E)
( )
1
5
fx
x
!
=
−
(F)
( )
5
5
x
fx
x
+
!
=
−
_____ 13. Find the derivative of f when
( )
( )
2
1
3 sinfx x
−
=
.
(A)
( )
1
2
6sin
1
x
fx
x
−
"
=
−
(B)
( )
1
2
3sin
1
x
fx
x
−
"
=
−
(C)
( )
1
2
6sin
1
x
fx
x
−
"
=
+
(D)
( )
1
2
6 cos
1
x
fx
x
−
"
=
−
(E)
( )
1
2
3cos
1
x
fx
x
−
"
=
+
(F)
( )
1
2
3cos
1
x
fx
x
−
"
=
−
Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 7 of 8
_____ 14. Determine if
1
1
lim sin
62
x
x
x
−
→∞
+
$%
&'
+
()
exists, and if it does, find its value.
(A)
6
π
(B)
2
π
(C) 0 (D)
3
π
(E)
4
π
(F) DNE
_____ 15. Let f be a twice-differentiable function and let g be its inverse. Consider the following
equations:
I.
( )
( )
gfx x=
,
( )
( )
fgx x=
II.
( )
( )
( )
( )
( )
( )
( )
2
0fgx gx fgxgx
!! ! ! !!
+=
III.
( )
( )
( )
1
gx
fgx
!
=
!
Which one do f, g satisfy?
(A) none of them (B) I and II only (C) I only (D) III only (E) II only (F) II and III only
(G) I and III only (H) all of them
_____ 16. Find the value of
( )
1g
!
when g is the inverse of the function
( )
2sinfx x=
,
22
x
ππ
−≤≤
.
(A)
1
3
(B)
1−
(C)
1
2
(D)
1
3
−
(E) 1 (F)
1
2
−
Calculus Maximus WS 4.8: Inverse & Inverse Trig Functions
Page 8 of 8
_____ 17. Suppose g is the inverse function of a differentiable function f and
( )
( )
1
Gx
gx
=
. If
( )
37f =
and
( )
1
3
9
f
!
=
, find
( )
7G
!
.
(A)
5−
(B) 4 (C) 6 (D)
1−
(E)
4−
_____ 18. Find
dy
dx
when
( )
tan 2 2xy x− =
(A)
2
2
8
14
dy x
dx
x
=
+
(B)
2
2
8
14
dy x
dx
x
= −
+
(C)
2
2
4
2
dy y
dx
x
= −
+
(D)
2
2
4
2
dy y
dx
x
=
+
(E)
2
2
8
14
dy x
dx
y
= −
+
(F)
2
2
8
2
dy y
dx
x
=
+