Profit Karla Harby Communications, a small company of...

Profit Karla Harby Communications, a small company of science writers, found that its rate of profit (in thousands of dollars) after $t$ years of operation is given by $$P^{\prime}(t)=(3 t+3)\left(t^{2}+2 t+2\right)^{1 / 3}$$ (a) Find the total profit in the first three years. (b) Find the profit in the fourth year of operation. (c) What is happening to the annual profit over the long run?

Profit Karla Harby Communications, a small company of science writers, found that its rate of profit (in thousands of dollars) after $t$ years of operation is given by
$$P^{\prime}(t)=(3 t+3)\left(t^{2}+2 t+2\right)^{1 / 3}$$
(a) Find the total profit in the first three years.
(b) Find the profit in the fourth year of operation.
(c) What is happening to the annual profit over the long run?

Key Concepts

Rate of Change

Understanding the rate of change is fundamental in linking how a variable, such as profit, evolves over time. In this problem, analyzing the rate function allows one to determine immediate trends and, after integration, deduce the overall profit accumulation and its long-term behavior.

Integration and the Fundamental Theorem of Calculus

This concept involves using integration to recover a function from its derivative. By integrating the rate of profit over a specific time interval, one obtains the total accumulated profit over that period. The Fundamental Theorem of Calculus links differentiation and integration in a way that enables one to compute the net change of a function via its derivative.

Definite Integration

Definite integration is utilized to calculate the cumulative change (or total profit, in this context) over a set interval by evaluating the integral with corresponding limits. It formalizes the idea of accumulation by summing up infinitely many infinitesimally small changes given by the rate of profit.

Asymptotic Analysis

This concept is used to study the behavior of functions as the variable approaches very large values. In the context of profit analysis, asymptotic analysis helps determine the trend of annual profit over a long period, such as whether it will continue growing, level off, or eventually decline.

Transcript

00:01 For problem 55, we are given this function of p prime of t.

00:07 And for part a, we're asked to find the total profit.

00:10 So to do that, we are going to use u substitution and integrate.

00:16 So we will let this equal u.

00:18 So u equal whatever's inside the parentheses.

00:30 And that means du equals 2t plus 2.

00:35 I'm just integrating on both sides d t and i'm going to divide both sides by two so it would be one half d u and i divide this side by two which gives me t plus one d t and now since the question asks us in the first three years i have to integrate between zero and three so i do the integral between zero and three and 3 t plus 3 is basically t plus 1 times 3 so i can write a times 3 here and then we have the t plus 1 dt inside here so that would be 1 half to u and then you add the u times u to the 1 3rd power so you can move the 1 half to the front which gives us 3 halves and then the integral between 0 and 0 and 3 3.

02:02 3 u to the 1 third power d u and now since we're doing u substitution these values these bounds would not be correct because those bounds were for t now we have to plug in plug in the bounds so the lower bound is 0 so we plug in 0 into this equation 0 plus 2 times 0 plus 0 and that would give us the upper bound which is 2 and then the lower bound or that would give us the lower bound and now for the upper bound we plug in 3 and this gives us 17 so we change the bounds now so this would be 17 and this would be 2 and now we just do the integral so the integral of you to the 1 3rd power is u to the 4 thirds power times 3 over 4 and don't forget to bring the 3 halves in here too and this would be integrated from 2 to 17 so this gives us 9 over 8 and then if you plug in the bounds you get 17 to the 4 thirds minus 2 to the 4 thirds and this gives us about 46 .34 so in thousands this would be $46 ,341.

04:30 And this is the answer for part a.

04:34 Now for part b, we're asked to do the same thing, but then the profit is to be found in the fourth year of operation.

04:43 So that means that our bounds would change.

04:46 It would just be from 3 to 4...

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