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This lecture covers five applications of differential calculus — related rates, Newton's method, L'Hôpital's rule, extrema of functions, and the shape of a graph (first & second derivative tests).

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📋 At a Glance

• [ ] Understand related rates and how to differentiate both sides of an equation
• [ ] Solve related rate problems by substituting known values
• [ ] Understand Newton's method as an iterative root-finding process
• [ ] Apply Newton's method formula to approximate roots
• [ ] Apply L'Hôpital's rule to indeterminate forms (0/0, ∞/∞)
• [ ] Know when to apply L'Hôpital's rule repeatedly
• [ ] Distinguish between local (relative) and global (absolute) extrema
• [ ] Find critical points where f'(x) = 0 or f'(x) is undefined
• [ ] Find global extrema on closed vs open intervals
• [ ] Apply the first derivative test (increasing/decreasing)
• [ ] Apply the second derivative test (concavity)
• [ ] Identify inflection points
• [ ] Determine parameter values for concavity conditions

1. Related Rates

The Idea

So far we've differentiated $f(x)$ with $x$ as the independent variable. But what if multiple variables are all functions of a shared variable like time $t$?