Math 11 - Section 1.6

Professor Monte begins by introducing the concept of product and quotient rules in calculus. He explains that the product rule is used when multiplying two functions, while the quotient rule applies to division. The need for these rules is demonstrated through an example involving the derivative of a function expressed as the product of two powers of x. The professor shows that simple multiplication of the derivatives does not yield the correct result, hence the necessity for the product rule. A similar example is shown for the quotient rule.

The professor then focuses on the product rule, illustrating how to correctly derive the derivative of a product of two functions. He simplifies the formula provided in the textbook into a more digestible form, emphasizing the process of multiplying out the functions and their derivatives in a specific order. The product rule is applied to several examples to demonstrate its utility and to ensure understanding.

The application of the product rule is further explored with more complex functions, including polynomials and expressions involving square roots. The professor shows how to multiply out the derivatives and combine like terms to find the derivative of the product. He also emphasizes the importance of practice and checking one's work against the provided solutions.

The professor moves on to the quotient rule, which is more complex and requires careful application. He introduces a memorable phrase, 'low D high minus high D low, square the bottom and away we go,' to help remember the formula. The rule is demonstrated with an example involving monomials, emphasizing the importance of following the order of operations due to the minus sign in the formula.

The professor applies the quotient rule to a more complex example, showing how to handle expressions that cannot be simplified before taking the derivative. He demonstrates the process of multiplying out the terms, combining like terms, and simplifying the expression to find the derivative. The quotient rule is shown to be effective, even for more complicated functions.

The professor applies the quotient rule to a real-world problem involving average cost. He explains the concept of average cost and how it is derived from total cost. The problem involves finding the rate at which average cost is changing when a certain number of units have been produced. The professor uses the quotient rule to find the derivative of the average cost function and then evaluates it at the given number of units.

In the final example, the professor applies the derivative concept to model the rate of change in a person's temperature over time during an illness. He finds the derivative of the temperature function with respect to time and then evaluates it at specific time points to determine the rate of temperature change. The professor interprets the results to understand the trend in the temperature changes over time.

The professor concludes the session by emphasizing the importance of understanding the product and quotient rules, as they will be used throughout the course. He encourages students to practice these rules and not to be discouraged if they find them challenging. The professor motivates students by stating that with the right amount of effort, everyone can pass the class.