AP Calculus Differentiation III Study Notes

Freshman Academy AP Calculus Differentiation III

Course Overview

• Course Level: Class 11 (2026 - 2027)

• Subject: Advanced Placement Calculus focusing on differentiation concepts.

Today's Agenda

• Introduction to Related Rates

• Solving Related Rates Problems

• Local Linearity and Linearization

• L'Hopital's Rule

Introduction to Related Rates

• Related rates problems assess the understanding of differential equations in practical contexts.

• Key points to solving related rates problems:

  • Utilize implicit differentiation to differentiate equations involving multiple variables.

  • Recognize and connect the relationships between variables that change over time.

Solving Related Rates Problems

1. Problem #1: The radius of a circle is increasing at a rate of 2 cm/s. At the moment when the radius is 5 cm, determine how fast the area is increasing.

   • Related formula: Area of a circle, $A = ext{π}r^2$.

   • Use chain rule and related rates to find $rac{dA}{dt}$.

   • Substitute $rac{dr}{dt}$ and the radius value.

2. Problem #2: A 10 m ladder leaning against a wall has its bottom sliding away from the wall at 1 m/s. Calculate how fast the top of the ladder is sliding down the wall when the bottom is 6 m from the wall.

   • Use Pythagorean theorem: $x^2 + y^2 = L^2$, where $L$ is the length of the ladder (10 m).

   • Differentiate with respect to time, applying implicit differentiation.

3. Problem #3: Air is being pumped into a spherical balloon increasing in volume at a rate of 4 cm³/s. When the radius is 3 cm, find how fast the radius is increasing.

   • Volume of a sphere: $V = rac{4}{3} ext{π}r^3$.

   • Differentiate both sides concerning time to relate $rac{dV}{dt}$ to $rac{dr}{dt}$.

Notes for Practice:

• Completing three related rates questions will take approximately 15 minutes.

Local Linearity and Linearization

• Linearization: A method of using the tangent line at a point to approximate complicated values of functions.

  • This technique is vitally useful when dealing with functions that are complex or where exact evaluations are difficult to perform.

• Example Problem: Solve a particular linearization problem demonstrating this concept.

Implicit and Explicit Differentiations

1. Problem #1: Use linearization of $f(x) = ext{sin} \, x$ at $x = 0$ to approximate $ext{sin}(0.05)$.

2. Problem #2: Find the linearization of $f(x) = 1.5$ at $x = 2$, then approximate the actual value at that point.

3. Problem #3: Use linearization to approximate the value of 4.1 in some relevant function.

4. Problem #4: Approximate $ext{ln}(1.02)$ using linearization techniques.

   • Approximately 12 minutes is necessary to work through these four questions.

L'Hopital's Rule

• L'Hopital's Rule: A powerful tool used to evaluate limits of indeterminate forms (like $0/0$ or $ext{∞}/ ext{∞}$).

  • The rule states that if you have an indeterminate form, you can take the derivative of the numerator and the derivative of the denominator separately:

  • If lim $x <br>ightarrow c$ of $rac{f(x)}{g(x)}$ is of the form $rac{0}{0}$ or $rac{ ext{∞}}{ ext{∞}}$, then:
    $ext{lim}<em>{x o c} rac{f(x)}{g(x)} = ext{lim}</em>{x o c} rac{f'(x)}{g'(x)}$

  • It's important to note that we are not taking the derivative of just the quotient, but rather of the individual components involved.

  • Illustrative examples should clarify how to apply L'Hopital's Rule in practice.