A weather balloon’s altitude is modeled by a differentiable function

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

h

, where

h(t)

(in meters) is the altitude at time

t

(in seconds) for

0\le t\le1200

. Throughout the flight the balloon rises, so

h

is strictly increasing and differentiable on the given interval. The inverse function

h^{-1}

therefore gives the time at which the balloon attains a specified altitude. Answer the following questions about the relationship between

h

and

h^{-1}

. (a) Explain why the fact that

h

is strictly increasing and differentiable with

h'(t)\ne0

for all

t

guarantees that

h^{-1}

is differentiable on the range of

h

. In your explanation, refer to appropriate conditions on invertibility and differentiability. (b) At

t=10\text{ s}

, the balloon is 65 meters above the ground and

h'(10)=6\text{ m/s}

. Determine the value of

h^{-1}\,'(65)

and interpret its numerical value, including correct units, in the context of the flight.

Related theory

Browse the problem library