Question Video: Finding the Rate of Change of a Polynomial Function in a Real-World Context | Nagwa Question Video: Finding the Rate of Change of a Polynomial Function in a Real-World Context | Nagwa

# Question Video: Finding the Rate of Change of a Polynomial Function in a Real-World Context Mathematics • Second Year of Secondary School

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A populationβs size after π‘ days is given by π(π‘) = 11π‘Β² + 35,923. Find the instantaneous rate of change in the population when π‘ = 12.

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### Video Transcript

A populationβs size after π‘ days is given by π of π‘ is equal to 11π‘ squared plus 35,923. Find the instantaneous rate of change in the population when π‘ equals 12.

In this question, we are asked to find the instantaneous rate of change in the population. And we are asked to calculate this when π‘ is equal to 12. This means the derivative of the population function π of π‘ with respect to π‘ when π‘ is 12.

We know by definition that π prime of π is equal to the limit as β approaches zero of π of π plus β minus π of π all divided by β. In this question, we need to calculate π prime of 12 as shown. We will begin by using our expression for π of π‘ to find an expression for π of 12 plus β. Replacing π‘ with 12 plus β, we have 11 multiplied by 12 plus β squared plus 35,923. Expanding 12 plus β squared gives us 144 plus 24β plus β squared. And multiplying through by 11, we have π of 12 plus β is equal to 1,584 plus 264β plus 11β squared plus 35,923.

Next, we can find an expression for π of 12. This is equal to 11 multiplied by 12 squared plus 35,923. And 11 multiplied by 12 squared is equal to 1,584. We can now substitute these into our expression for π prime of 12. The two constants on the numerator will cancel. And we are left with the limit as β approaches zero of 264β plus 11β squared all divided by β. Since β is approaching zero and will therefore never be equal to zero, we can divide the numerator and denominator by β. This gives us the limit as β approaches zero of 264 plus 11β.

As we now have a polynomial in terms of β, we can use direct substitution such that π prime of 12 is equal to 264. And we can therefore conclude that the instantaneous rate of change in the population when π‘ is equal to 12 is 264.

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