Fall 2025: Chapters 3-7 & 3-8 - Class Notes

Welcome to the Fall 2025 session covering Chapters 3-7 and 3-8! In this session, we explored the applications of differentiation and exponential functions in various real-world scenarios. Let's review the key topics we covered:

Topics Covered:

  • Analyzing Particle Motion Using Differentiation: We learned how to use derivatives to analyze the motion of a particle. Given a position function $s(t)$, we can find the velocity $v(t)$ by taking the first derivative, $v(t) = s'(t)$, and the acceleration $a(t)$ by taking the second derivative, $a(t) = v'(t) = s''(t)$.
  • Example: If the position of a particle is given by $s(t) = t^3 - 6t^2 + 9t$, then the velocity is $v(t) = 3t^2 - 12t + 9$ and the acceleration is $a(t) = 6t - 12$. We also discussed finding when the particle is at rest (when $v(t) = 0$), moving forward (when $v(t) > 0$), and the total distance traveled.
  • Exponential Functions with Base e: We evaluated exponential functions with base e, which are essential for modeling real-world situations like population growth and radioactive decay.
  • Exponential Growth and Decay Word Problems: We solved word problems involving exponential growth and decay, focusing on finding final amounts using the formula $y(t) = y_0e^{kt}$, where $y(t)$ is the amount at time $t$, $y_0$ is the initial amount, and $k$ is the growth/decay constant.
  • Initial-Value Problems: We found particular solutions to initial-value problems involving growth and decay. These problems typically involve a differential equation and an initial condition, allowing us to find a unique solution.
  • Modeling Growth and Decay: We wrote and solved initial value problems to model growth or decay. For example, given that the rate of change of a quantity $y$ is proportional to $y$, we have the differential equation $\frac{dy}{dt} = ky$. With an initial condition like $y(0) = y_0$, we can solve for $y(t)$.
  • Example Problem Suppose that the velocity $v(t)$ (in m/s) of a sky diver falling near the Earth's surface is given by the following, where $t$ is measured in seconds. $v(t) = 49(1-e^{-0.21t})$ Find the velocity of the sky diver after 3 seconds and after 5 seconds.
  • Compounding interest formulas Compound interest $A=P(1+\frac{r}{n})^{nt}$ Continuous interest $A=Pe^{rt}$

Remember to practice these concepts to solidify your understanding. Keep up the great work, and don't hesitate to ask questions during our next class! Let's continue to explore the exciting world of calculus together!