AB Calculus
Inverses and Derivatives
Name: ___________________________________
Reminder: If (x, y) is a point on f(x), then (y, x) is a point on f −1 (x) !
1.
If f (x) =
a.
1
x − 3 and g(x) = x 3 , find
8
f −1 (g −1 (1))
Note: f ′(a,b) =
b.
g −1 ( f −1 (−3))
1
( f )′(b,a)
−1
Example: If f (1) = 2 and f ′(1) = 5 , then f −1 (2) = __________ and ( f −1 )′(2) = _____________.
2.
If h(x) = cos(x) + 3x , find
a.
h(0)
b.
(h −1 )′ (1)
3.
Let f and g be differentiable functions and let the values of f, g and the derivatives f ′ and g′ at
x = 1 and x = 2 be given by the table below. Determine the value of (g −1 )′(2) .
x
1
2
4.
f ′(x)
5
6
g(x)
2
π
g′(x)
4
7
Find the value of g ′(1) when g is the inverse of the function f (x) = 2sin x, −
A.
5.
f(x)
3
2
1
3
B.
−1
1
2
C.
D.
−
1
3
E.
Suppose g is the inverse function of a differentiable function f and G(x) =
f ′(3) =
A.
−
B.
4
C.
6
D.
−1
E.
1
2
1
. If f(3) = 7 and
g(x)
1
, find G ′(7) .
9
−5
π
π
≤x≤ .
2
2
−4