Video: APCALC03AB-P1A-Q05-156190565246

Suppose that a companyβs revenue of producing π₯ units of a certain item is given by the equation π(π₯) = 3000 + 7.4π₯ + 0.75π₯Β². What is the instantaneous rate of change of π with respect to π₯ when π₯ = 200?

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Video Transcript

Suppose that a companyβs revenue of producing π₯ units of a certain item is given by the equation π of π₯ equals 3000 plus 7.4π₯ plus 0.75π₯ squared. What is the instantaneous rate of change of π with respect to π₯ when π₯ equals 200?

The instantaneous rate of change of the quantity is its rate of change at a given moment in time. And it can be found by evaluating the first derivative of that function at the given moment. We therefore, need to evaluate π prime of π₯ or dπ by dπ₯ as weβre looking for the rate of change of π with respect to π₯ when π₯ is equal to 200. We can find the first derivative of our function π of π₯ with respect to π₯ by using the power rule of differentiation. First, we need to differentiate 3000 and the derivative of a constant with respect to π₯ is just zero.

Next, we need to differentiate positive 7.4π₯. And its derivative is just 7.4. Itβs actually 7.4 multiplied by π₯ to the power of zero. But as π₯ to the power of zero is just one, this simplifies to 7.4. Finally, we differentiate the third term, positive 0.75π₯ squared. And this derivative is equal to positive 0.75 multiplied by two π₯. We bring down the power of two. And then we decrease the power by one. Two multiplied by 0.75 is 1.5. So our first derivative simplifies to π prime of π₯ is equal to 7.4 plus 1.5π₯.

Finally, we evaluate this derivative at the given moment. Thatβs when π₯ is equal to 200, giving π prime of 200 equals 7.4 plus 1.5 multiplied by 200. 1.5 multiplied by 200 can be thought of as three over two multiplied by 200. And then cancelling a factor of two, we have three over one multiplied by 100 which is 300. So π prime of 200 is equal to 7.4 plus 300; thatβs 307.4. So by using differentiation to find the first derivative of our function π of π₯ and then evaluating this first derivative π prime of π₯ when π₯ is equal to 200, we see that the instantaneous rate of change of π with respect to π₯ when π₯ equals 200 is 307.4.