Question

Daily oil production by Pemex, Mexico's national oil company, can be approximated by $P(t)=-0.022 t^{2}+0.2 t+2.9$ million barrels $\quad(1 \leq t \leq 9)$ where $t$ is time in years since the start of $2000 .^{59}$ Find the derivative function $\frac{d P}{d t} .$ At what rate was oil production changing at the start of $2004(t=4) ?$ HINT [See Example 4.]

Name: Problem 48, Chapter 10: Finite Mathematics and Applied Calculus
Uploaded: 2021-09-26T10:43:17-08:00
Duration: 5 min 5 s
Channel: Charles Machakwa
Description: Daily oil production by Pemex, Mexico's national oil company, can be approximated by P(t)=-0.022 t^2+0.2 t+2.9 million barrels (1 ≤t ≤9) where t is time in years since the start of 2000 .^59 Find the derivative function (d P)/(d t) . At what rate was oil production changing at the start of 2004(t=4) ? HINT [See Example 4.]

   Daily oil production by Pemex, Mexico's national oil company, can be approximated by
$P(t)=-0.022 t^{2}+0.2 t+2.9$ million barrels $\quad(1 \leq t \leq 9)$
where $t$ is time in years since the start of $2000 .^{59}$ Find the derivative function $\frac{d P}{d t} .$ At what rate was oil production changing at the start of $2004(t=4) ?$ HINT [See Example 4.]

Finite Mathematics and Applied Calculus

Stefan Waner, Steven… 5th Edition

Chapter 10, Problem 48

Instant Answer

Solved by Expert Charles Machakwa

09/26/2021

Step 1

022 t^{2}+0.2 t+2.9$. We can use the power rule for differentiation, which states that the derivative of $t^n$ is $n t^{n-1}$. Show more…

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Daily oil production by Pemex, Mexico's national oil company, can be approximated by $P(t)=-0.022 t^{2}+0.2 t+2.9$ million barrels $\quad(1 \leq t \leq 9)$ where $t$ is time in years since the start of $2000 .^{59}$ Find the derivative function $\frac{d P}{d t} .$ At what rate was oil production changing at the start of $2004(t=4) ?$ HINT [See Example 4.]

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Key Concepts

Derivative

The derivative represents how a function changes with respect to changes in its input, offering a measurement of the rate at which one quantity varies as another changes.

Instantaneous Rate of Change

The instantaneous rate of change is the derivative evaluated at a specific point in time, providing the exact rate at which the function is changing at that moment.

Polynomial Differentiation

Differentiating a polynomial involves applying the power rule to each term: every term of the form t^n is differentiated to give n*t^(n-1), making the process straightforward due to the sum of its individual components.

Power Rule

The power rule is a fundamental differentiation rule stating that the derivative of t raised to the power n is n*t^(n-1). This rule is essential in efficiently obtaining the derivative of any polynomial term.

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