Let and be positive

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

Problem

Let

P(t)

and

Q(t)

be positive, differentiable functions of time

t

. Consider their ratio

R(t)=\dfrac{P(t)}{Q(t)}.

Suppose that at a particular instant

t=t_0

you know the numerical values of

P(t_0)

Q(t_0)

\dfrac{dP}{dt}(t_0)

, and

\dfrac{dQ}{dt}(t_0)

. (a) Write

\dfrac{dR}{dt}

explicitly in terms of

P

Q

\dfrac{dP}{dt}

, and

\dfrac{dQ}{dt}

. No simplification is necessary. (b) Explain, without calculating any numerical value, how the signs of

\dfrac{dP}{dt}

and

\dfrac{dQ}{dt}

together determine whether

R

is increasing or decreasing at

t=t_0

. Your explanation should reference the formula from part (a). (c) At

t=t_0

, suppose

P=8

Q=4

, and

\dfrac{dP}{dt}=2

. Determine the inequality that

\dfrac{dQ}{dt}

must satisfy so that

R

is decreasing at

t=t_0

, and justify your reasoning.