Mathematics

Problem

Let

P(x)=m(x)\,n\bigl(k(x)\bigr)

, where

m

n

, and

k

are differentiable functions with

m(x)>0

and

n(u)>0

for all real numbers in their domains. In a manufacturing context,

x

represents the number of hours since production began,

m(x)

is the rate (units: widgets per hour) at which raw material enters an assembly process, and

n\bigl(k(x)\bigr)

is an efficiency factor that depends on an internal temperature

k(x)

(measured in degrees Celsius). (a) Describe qualitatively how an increase in

x

influences

P(x)

through each of the two factors. (b) Derive an explicit expression for

P'(x)

in terms of

m

m'

n

n'

, and

k'

. (c) Suppose

m(4)=120

m'(4)=6

k(4)=350

k'(4)=\tfrac{1}{2}

n(350)=0.83

, and

n'(350)=\!-0.002

. Interpret the numerical value and sign of

P'(4)

within the context of the manufacturing process, including appropriate units.