A differentiable function of time is given by F(t)=e^ 5t , (1+a t^ 2 )^ b

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A differentiable function of time is given by

F(t)=e^{5t}\,\bigl(1+a t^{2}\bigr)^{b},

where

a

and

b

are constants with

a>0

and

b\neq0

. (a) Compute

\dfrac{dF}{dt}

in terms of

t

a

, and

b

, leaving your answer factored. (b) Evaluate

\dfrac{dF}{dt}

t=0

and interpret the result in terms of how

F

initially changes with time. (c) Holding

t

fixed, determine

\dfrac{\partial F}{\partial a}

and describe how increasing

a

affects

F(t)

for a fixed instant

t>0

. Justify your reasoning.