Overview
This lecture introduces related rates problems in calculus, focusing on using implicit differentiation and the chain rule to relate the rates of change of different quantities over time in real-world scenarios.
Introduction to Related Rates
- Related rates problems involve finding how different quantities change over time in relation to each other.
- These problems often use measurements of one rate to find an unknown rate.
Sphere (Balloon) Example
- Consider a sphere (balloon) with volume increasing at a known rate, dV/dt.
- The radius (r) and volume (V) are related by the formula: ( V = \frac{4}{3}\pi r^3 ).
- To find dr/dt, differentiate both sides with respect to time (t) using the chain rule.
- The derivative gives: ( \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} ).
- Solving for ( \frac{dr}{dt} ): ( \frac{dr}{dt} = \frac{dV/dt}{4\pi r^2} ).
- Example calculation: dV/dt = 100 cm³/s and r = 25 cm yields ( \frac{dr}{dt} = \frac{1}{25\pi} ) cm/s.
Ladder Against a Wall Example
- A 10-foot ladder slides away from a wall at 1 ft/s; x (distance from wall) and y (height on wall) both change over time.
- The relationship: ( x^2 + y^2 = 100 ) (Pythagorean theorem).
- Differentiate both sides with respect to time: ( 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 ).
- Solving for ( \frac{dy}{dt} ): ( \frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt} ).
- With x = 6 ft, y = 8 ft, ( \frac{dx}{dt} = 1 ) ft/s, ( \frac{dy}{dt} = -\frac{3}{4} ) ft/s.
- Negative sign indicates height y is decreasing as the ladder slides down.
General Approach to Related Rates Problems
- Draw a diagram to visualize the problem.
- Write equations relating the variables involved.
- Differentiate with respect to time (using the chain rule as needed).
- Substitute known values and solve for the desired rate.
Key Terms & Definitions
- Related Rates — Problems involving the simultaneous rates of change of two or more related variables over time.
- Chain Rule — A calculus rule used to differentiate composite functions; essential when variables themselves depend on time.
- Implicit Differentiation — Differentiating equations involving several variables without solving explicitly for one variable.
Action Items / Next Steps
- Practice related rates problems involving spheres, ladders, and other geometric scenarios.
- Review the use of implicit differentiation and the chain rule in these contexts.