Fall 2025 Calculus 1 Final Exam Review

Congratulations on making it to the end of the semester! As we approach the finish line, it is time to consolidate everything we have learned into one final push. Attached to this post is the Final Exam Review Packet. Over the next two classes, I will be posting video lectures walking through these specific problems, but I highly encourage you to attempt them on your own first.

Here is a breakdown of the key concepts covered in this review that you should focus your study efforts on:

1. Limits and Asymptotes

We start with the foundations. You need to be comfortable finding limits both algebraically and graphically. Be sure you can identify:

  • Graphical Limits: As seen in Problem 1, determine limits at specific points (e.g., $\lim_{x \to 1} f(x)$) and behavior at infinity ($\lim_{x \to \infty} f(x)$).
  • Asymptotes: Identify vertical and horizontal asymptotes based on the graph or function definition.
  • Algebraic Limits: Review techniques for indeterminate forms, such as the limit in Problem 8: $$\lim_{x \to 1} \frac{\sin(x-1)}{x^3 + 2x - 3}$$

2. Differentiation Rules

A significant portion of the exam will test your ability to apply differentiation rules correctly. The review packet covers a mix of the Power, Product, Quotient, and Chain rules. Practice these specific functions from the review:

  • Product & Chain Rule: $f(x) = (x + 7\sqrt{x})e^x$
  • Quotient Rule: $y = \frac{2x}{9-\tan(x)}$
  • Complex Chain Rule: $n(x) = \left(\frac{x^3-2}{x^3+2}\right)^8$
  • Implicit Differentiation: Don't forget how to handle equations where $y$ is not isolated, such as finding the tangent line for $x^2 + 6xy + 12y^2 = 28$.

3. Curve Sketching and Analysis

In Problem 9, we look at the rational function $f(x) = \frac{x-4}{x^2}$. This is a classic "curve sketching" problem. You must be able to use the first and second derivatives to find:

  • Intervals where the function is increasing or decreasing.
  • Local maximums and minimums.
  • Intervals of concavity (concave up vs. concave down).

4. Applications: Optimization and Related Rates

Math is useful because it solves real-world problems! We have two major application problems in this packet:

  • Optimization: Calculating the maximum volume of an open-top box constructed from a 7 ft by 6 ft piece of cardboard.
  • Related Rates: Determining how fast the distance between two boats is changing when one travels south and the other travels east.

5. Integration

Finally, we wrap up with Integral Calculus. You will need to evaluate both definite and indefinite integrals. Pay close attention to U-Substitution, which is required for Problem 14:

$$ \int \frac{e^x}{(7-e^x)^2} dx $$

Next Steps: Download the PDF attached below and start solving! Bring your questions to our next class, and keep an eye out for the video solutions I will be posting shortly. You have put in the work all semester—finish strong!