Video: APCALC03AB-P1A-Q05-156190565246

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

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Video Transcript

Suppose that a company’s revenue of producing π‘₯ units of a certain item is given by the equation 𝑅 of π‘₯ equals 3000 plus 7.4π‘₯ plus 0.75π‘₯ squared. What is the instantaneous rate of change of 𝑅 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 𝑅 prime of π‘₯ or d𝑅 by dπ‘₯ as we’re looking for the rate of change of 𝑅 with respect to π‘₯ when π‘₯ is equal to 200. We can find the first derivative of our function 𝑅 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. It’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 𝑅 prime of π‘₯ is equal to 7.4 plus 1.5π‘₯.

Finally, we evaluate this derivative at the given moment. That’s when π‘₯ is equal to 200, giving 𝑅 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 𝑅 prime of 200 is equal to 7.4 plus 300; that’s 307.4. So by using differentiation to find the first derivative of our function 𝑅 of π‘₯ and then evaluating this first derivative 𝑅 prime of π‘₯ when π‘₯ is equal to 200, we see that the instantaneous rate of change of 𝑅 with respect to π‘₯ when π‘₯ equals 200 is 307.4.

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