We are given a function for the number of visitors to a website in a week over a 1-year period:

$P(t) = 115 + (t - 85)e^{0.02t}$ where $t$ represents the week and $1 \leq t \leq 52$.

Part (a) & (b) - Increasing and Decreasing Intervals:

To find where the number of visitors is increasing or decreasing, we need to analyze the first derivative of $P(t)$, $P'(t)$.

Step 1: Find the derivative $P'(t)$

We use the product and chain rules to differentiate $P(t)$: $P(t) = 115 + (t - 85)e^{0.02t}$ $P'(t) = \frac{d}{dt}\left((t - 85)e^{0.02t}\right)$ Applying the product rule: $P'(t) = (t - 85) \cdot \frac{d}{dt}\left(e^{0.02t}\right) + e^{0.02t} \cdot \frac{d}{dt}(t - 85)$ Since $\frac{d}{dt}\left(e^{0.02t}\right) = 0.02e^{0.02t}$ and $\frac{d}{dt}(t - 85) = 1$, we get: $P'(t) = (t - 85) \cdot 0.02e^{0.02t} + e^{0.02t}$ Factor out $e^{0.02t}$: $P'(t) = e^{0.02t}\left(0.02(t - 85) + 1\right)$ Simplify inside the parentheses: $P'(t) = e^{0.02t}(0.02t - 1.7 + 1)$ $P'(t) = e^{0.02t}(0.02t - 0.7)$

Step 2: Analyze the sign of $P'(t)$

We now solve for when $P'(t) = 0$: $e^{0.02t}(0.02t - 0.7) = 0$ Since $e^{0.02t} \neq 0$ for any $t$, we solve: $0.02t - 0.7 = 0$ $0.02t = 0.7$ $t = \frac{0.7}{0.02} = 35$

Thus, the critical point occurs at $t = 35$.

Step 3: Determine the intervals of increase and decrease

• For $t < 35$, $0.02t - 0.7 < 0$, so $P'(t) < 0$ and the number of visitors is decreasing.
• For $t > 35$, $0.02t - 0.7 > 0$, so $P'(t) > 0$ and the number of visitors is increasing.

Conclusion for parts (a) and (b):

• The number of visitors is decreasing on the interval $(1, 35)$.
• The number of visitors is increasing on the interval $(35, 52)$.

Part (c) - Critical Point:

The critical point is given as $(35, 14.312)$. This means:

• At week $t = 35$, the number of visitors reaches a minimum of approximately 14.312.
• This critical point signifies the week during the year where the visitor count stops decreasing and starts increasing again.

Follow-up questions:

1. How does the behavior of $P(t)$ change as $t$ approaches the endpoints $t = 1$ and $t = 52$?
2. What happens to the number of visitors as $t \to \infty$, based on the nature of the exponential term?
3. Can you find the total number of visitors throughout the year by integrating $P(t)$?
4. How sensitive is the function to changes in $t$, particularly near the critical point $t = 35$?
5. How would the intervals change if the growth rate in the exponential term was larger or smaller?

Tip: Critical points occur where the derivative equals zero or is undefined. Always check these points for potential maxima, minima, or points of inflection.