3: Applications of Differentiation

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• 3.1: Curve Sketching
This section examines some of the interplay between the shape of the graph of f and the behavior of f'. If we have a graph of f, we will see what we can conclude about the values of f'. If we know values of f', we will see what we can conclude about the graph of f. We will also utilize the information from f'' that we learning in the last section.
• 3.2: Second Derivative and Concavity
• 3.3: Optimization
Calculus provides ways of drastically narrowing the number of points we need to examine to find the exact locations of maximums and minimums, while at the same time ensuring that we haven’t missed anything important.
• 3.4: Applied Optimization
• 3.5: Implicit Differentiation and Related Rates
The key idea behind implicit differentiation is to assume that y is a function of x even if we cannot explicitly solve for y. This assumption does not require any work, but we need to be very careful to treat y as a function when we differentiate and to use the Chain Rule.

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