To solve this problem, let’s go through each part of the question step-by-step.

Given: $P(t) = \frac{24t + 10}{t^2 + 1}$ where $t$ is the time in hours after a toxin is introduced.

Part (a) - At what rate is the population changing when $t = 0$ and $t = 1$?

To determine the rate of change of the population, we need to find the derivative $P'(t)$ and then evaluate it at $t = 0$ and $t = 1$.

1. Find $P'(t)$: $P'(t) = \frac{d}{dt} \left( \frac{24t + 10}{t^2 + 1} \right)$ Using the quotient rule, $P'(t) = \frac{f'(t)g(t) - f(t)g'(t)}{[g(t)]^2}$, where:

   • $f(t) = 24t + 10$
   • $g(t) = t^2 + 1$

   Then:

   • $f'(t) = 24$
   • $g'(t) = 2t$

   Substituting these into the quotient rule: $P'(t) = \frac{(24)(t^2 + 1) - (24t + 10)(2t)}{(t^2 + 1)^2}$ Simplify the numerator to find $P'(t)$.

2. Evaluate $P'(0)$ and $P'(1)$: Substitute $t = 0$ and $t = 1$ into $P'(t)$ to determine the rates of change.

Part (b) - Is the rate increasing or decreasing at $t = 0$ and $t = 1$?

To determine if the rate is increasing or decreasing, analyze the sign of $P'(t)$ at $t = 0$ and $t = 1$. Positive values indicate an increasing population, while negative values indicate a decreasing population.

Part (c) - At what time does the population begin to decrease?

To find when the population begins to decrease, set $P'(t) = 0$ and solve for $t$. This gives the critical points. Then, analyze $P'(t)$ around these points to determine where $P'(t)$ changes from positive to negative.

Would you like detailed calculations for each part, or do you have questions on a specific aspect?

Follow-Up Questions for Exploration:

1. What is the interpretation of the rate of change in the context of bacterial growth?
2. How does the toxin affect the bacterial population over time?
3. Can we determine the maximum population reached by the bacteria?
4. What does the behavior of $P(t)$ tell us about the effect of the toxin?
5. How would the solution change if we used a different function for $P(t)$?

Tip: When using the quotient rule, be careful with signs and simplifications to avoid errors in calculating derivatives.