Welcome back to Professor Baker's Math Class! In this session, we are bridging the gap between abstract calculus rules and the real physical world. We are covering material from Chapters 3-7, 3-8, and 3-9, which focuses on how derivatives describe rates of change in physics, economics, and dynamic geometric systems. Let's break down the key concepts from the class notes.

1. Particle Motion (Physics Applications)

One of the most intuitive uses of the derivative is analyzing linear motion. If we have a position function $s = f(t)$, we can determine the particle's entire behavior through differentiation:

  • Velocity ($v$): The rate of change of position. $$v(t) = s'(t)$$
  • Acceleration ($a$): The rate of change of velocity. $$a(t) = v'(t) = s''(t)$$

Example from Class:
Given the position function $s(t) = t^3 - 6t^2 + 9t$, we found the velocity to be $v(t) = 3t^2 - 12t + 9$.

To analyze the motion, we look for critical moments:

  • At Rest: We set $v(t) = 0$. In our example, this occurred at $t=1$ and $t=3$.
  • Total Distance vs. Displacement: Remember, displacement is just $s_{final} - s_{initial}$, but total distance requires us to break the journey into intervals where the particle changes direction. As seen in the notes (Page 4), we calculated the absolute distances traveled between turning points: $|s(1)-s(0)| + |s(3)-s(1)| + |s(5)-s(3)|$.

2. Rates of Change in Natural and Social Sciences

Calculus isn't just for physics! We also explored how these rates of change apply to other fields:

  • Economics (Marginal Cost): If $C(x)$ is the cost to produce $x$ items, the derivative $C'(x)$ is the marginal cost function. It predicts the cost of producing the next item (the $(x+1)^{th}$ unit). As we saw in the notes, $C'(100)$ gave us a very close approximation to the actual cost of producing the 101st pair of jeans.
  • Natural Rhythms (Tides): We modeled tide depths using trigonometric functions like $D(t) = 7 + 5\cos[0.503(t - 6.75)]$. Finding the rate at which the tide is rising involves using the Chain Rule carefully.

3. Related Rates (The Strategy)

Chapter 3-9 introduces "Related Rates," which is often considered one of the most challenging topics in Calculus I. In these problems, two or more variables are changing with respect to time ($t$), and we must find the relationship between their rates.

The Golden Rule: Differentiate with respect to $t$!

From the class notes (Page 9), here is the essential strategy for solving these problems:

  1. Read carefully and draw a diagram.
  2. Assign symbols to all quantities that change with time.
  3. Write an equation that relates the variables (often Pythagorean theorem, Trig functions, or Volume formulas).
  4. Differentiate both sides with respect to $t$ (Implicit Differentiation).
  5. Substitute known values and solve for the unknown rate.

Classic Example: The Sliding Ladder

We looked at a 10ft ladder sliding down a wall. By setting up the relationship $x^2 + y^2 = 10^2$, we derived the related rates equation:

$$2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$$

This allowed us to plug in the speed of the bottom of the ladder ($ rac{dx}{dt}$) to find how fast the top was falling ($ rac{dy}{dt}$).

Remember, the key to related rates is to never substitute your constants too early. Only plug in the specific values (like "when $x=6$") after you have taken the derivative.

Keep practicing these diagrams and implicit differentiation techniques. You've got this!