Question

(a) The sales of a book publication are expected to grow according to the function
S = 300000(1 − e
−0.06t), where t is the time, given in days.
(i) Show using differentiation that the sales never attains an exact maximum value.
(ii) What is the limiting value approached by the sales function?

Answer 1

SOLUTION

GIVEN

The sales of a book publication are expected to grow according to the function

$\sf{S = 300000(1 - {e}^{ - 0.06t} )}$

where t is the time, given in days.

TO DETERMINE

(i) Show using differentiation that the sales never attains an exact maximum value.

(ii) What is the limiting value approached by the sales function?

EVALUATION

Here it is given that the sales of a book publication are expected to grow according to the function

$\sf{S = 300000(1 - {e}^{ - 0.06t} )}$

(i) Differentiating both sides with respect to t we get

$\sf{S' = 300000 \times ( - {e}^{ - 0.06t} ) \times ( - 0.06)}$

$\sf{ \implies \: S' = 18000 \: {e}^{ - 0.06t}}$

Now for the maximum value of S we have

$\sf{ S' = 0}$

$\sf{ \implies \: 18000 \: {e}^{ - 0.06t} = 0}$

$\sf{ \implies \: {e}^{ - 0.06t} = 0}$

Which is absurd

More precisely S' > 0 for all real values of t

∴ The sales never attains an exact maximum value.

(ii) Here it is given that

$\sf{S = 300000(1 - {e}^{ - 0.06t} )}$

Now

$\displaystyle \sf{\lim_{t \to \infty } \: S }$

$= \displaystyle \sf{\lim_{t \to \infty } \: 300000(1 - {e}^{ - 0.06t} )}$

$= \displaystyle \sf{300000 \: \lim_{t \to \infty } \: (1 - {e}^{ - 0.06t} )}$

$= \displaystyle \sf{300000 \: (1 - \lim_{t \to \infty } \: {e}^{ - 0.06t} \: )}$

$= \displaystyle \sf{300000 \: (1 - 0 )}$

$= \displaystyle \sf{300000 \: }$

∴ The limiting value approached by the sales function = 300000

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