Functions and are differentiable and strictly monotonic on their respective domains

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Functions

f

and

g

are differentiable and strictly monotonic on their respective domains. The following values are known: •

f(1)=3

and

f'(1)=5

•

g(3)=2

and

g'(3)=-4

•

g^{-1}(2)=3

(equivalently,

g(3)=2

as stated above) Define the composite function

Q(x)=f^{-1}\Bigl(g^{-1}\bigl(\sqrt{x}\bigr)\Bigr)

for

x>0

. (a) Using differentiable function notation, write an explicit expression for

Q'(x)

that makes clear all layers of composition. (b) Determine

Q'(4)

. (c) Suppose

x

increases from

4

4.04

. Use a tangent–line approximation based on

Q'(4)

to estimate the corresponding change in

Q(x)

. State any assumptions you make.