Lab: Derivatives in Context – Deep Applications of Product & Quotient Rules

1. Function and its Derivative graphs

Let:

$$f(x) = e^x + \frac{1}{e^x} + x^3$$

Step-by-Step Instructions

1. Find the derivative $f'(x)$.

2. Use GeoGebra to plot $f(x)$ and $f'(x)$.

3. Find the critical points of $f(x)$ by solving $f'(x) = 0$ graphically.

• (What are the x values for which $f'(x) = 0$?)

4. Determine the local maxima and minima of $f(x)$.

5. Determine the sign of $f'(x)$ in each interval divided by those points.

• (Whether $f'(x)$ is positive or negative in each interval decided by the critical points?)

6. Determine the intervals on which $f(x)$ is increasing or decreasing.

• (Hint: can you see the connection between the sign of $f'(x)$ and the increasing/decreasing intervals?)

Observe the graphs of $f(x)$ and $f'(x)$ to get answers and intuition for the questions above.

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2. Server Load Model

A server handles incoming data packets. The incoming data rate (in packets/sec) is modeled as:

$$f(t) = (5t + 1)(e^{-t})$$

where $t$ is time in seconds.

Tasks:

1. Compute $f'(t)$ using the product rule.

2. Interpret the meaning of $f'(2)$. Is the load increasing or decreasing at this moment?

3. Use GeoGebra to plot $f(t)$ and $f'(t)$ on the same axes.

4. Mark local max/min and points where $f'(t) = 0$.

5. Relate these turning points to real-world server behavior (e.g. peak load time, diminishing usage).

3. Pollution Density in a River

The concentration of a pollutant at a certain downstream point is modeled by:

$$C(t) = \frac{100t}{t^2 + 1}$$

where $C(t)$ is the concentration (in ppm) at time $t$ hours.

Tasks:

1. Use the quotient rule to find $C'(t)$.

2. Simplify the derivative and determine when the concentration is increasing or decreasing.

3. Find the time $t$ when the concentration is at its maximum.

4. Explain why this makes sense physically (e.g. build-up vs. dilution of the pollutant over time).

4. Cooling of a Beverage

You take a hot drink outside in the winter. Its temperature (in °C) is modeled by:

$$T(t) = \frac{100 + 10t}{t + 2}$$

where $t$ is time in minutes after exposure.

Tasks:

• Compute $T'(t)$ using the quotient rule.
• Use GeoGebra to plot $T(t)$ and $T'(t)$ on the same axes.

Label:

• Points where $|T'(t)|$ is large (i.e. the drink is cooling fastest).
• Points where $T'(t) \approx 0$ (i.e. cooling has slowed down).

Answer:

1. At what time is the drink cooling fastest? Why?

2. What does it mean when $T'(t) \to 0$? Is this realistic?

5. Visibility from a Drone: Tangents to a Hill

A drone hovers at the point (3, 20), observing a hill whose shape is modeled by the function:

$$f(x) = -x^2 + 3x + 15$$

The drone’s camera can just barely see the edges of the hill where its line of sight is tangent to the surface.

Your Task:

Find the equations of all lines that pass through the point (3, 20) and are tangent to the graph of

$$f(x) = -x^2 + 3x + 15.$$

Why It Matters:

In terrain mapping and environmental science, this kind of problem helps determine the visible range of land from an elevated observation point, such as a drone or watchtower. The tangent lines define the outer boundary of what the observer can see.