Welcome to the finish line, students! We have arrived at the Calculus 1 Final Exam. To help you prepare, I have put together a comprehensive review covering the most critical topics we have studied this semester. Below you will find the blank review sheet to practice on your own, followed by the handwritten notes from our two-part review session.

Key Topics Covered

The final exam is cumulative, but this review focuses heavily on the core pillars of Calculus:

  • Limits and Continuity: We start by analyzing graphs to find limits at infinity and identifying vertical and horizontal asymptotes. Remember, for $\lim_{x \to \infty} f(x)$, look at the end behavior of the function.
  • Differentiation Techniques: You will need to be comfortable combining rules. We review:
    • The Power Rule for functions like $g(x) = \frac{5}{8}x^2 - 8x + 17$
    • The Product and Chain Rules for functions like $f(x) = (x + 7\sqrt{x})e^x$
    • Derivatives of Inverse Trig functions, specifically involving $\tan^{-1}(x)$
  • Curve Sketching: One of the most important problems in this review involves the function $$f(x) = \frac{x-4}{x^2}$$ We go through the full analysis: finding the domain, intercepts, intervals of increase/decrease using the first derivative $f'(x)$, and concavity using the second derivative $f''(x)$.

Applications of Derivatives

Calculus isn't just about moving variables around; it's about solving real-world problems. We tackle two classic application types:

  1. Optimization: We solve for the maximum volume of an open-top box constructed from a 7 ft by 6 ft piece of cardboard. Remember to define your variables and set up the volume equation $V(x)$ before taking the derivative.
  2. Related Rates: We analyze the distance between two boats traveling in different directions (South and East) using the Pythagorean theorem $$D^2 = x^2 + y^2$$ and differentiating with respect to time $t$.

Integration

Finally, we wrap up with integration. Make sure you are comfortable with:

  • Definite integrals using the Power Rule: $\int_{1}^{16} x^{-\frac{3}{4}} dx$
  • U-Substitution: This is crucial for problems like $$\int \frac{e^x}{(7-e^x)^2} dx$$ where identifying the correct $u$ simplifies the problem immensely.

Download the files below and follow along with the video lectures. Good luck studying—you've got this!