Math 142, Chapter 4 Spring 2011 Zarestky Chapter 4 Reading Notes In this chapter, we will learn about a few more derivative rules and derivatives of exponential and logarithmic functions. Then, we will apply our knowledge of derivatives to products and quotients of functions. Section 4.1: The Constant e and Continuous Compound Interest By the end of this section, you should (continue to) be able to: • Solve problems involving compound interest. • Solve problems involving continuously compounded interest. • Model exponential growth and decay. We have already discussed e and continuous compound functions so nothing here should be brand new. In this section, we will apply what we have learned about limits to take another look at e and continuous compound functions. Read pages 211-212. Notice in particular the definition of the number e and Table 1. In your own words, what does the data in Table 1 show? Skim examples 1-4 to make sure you remember how to do them. Work the matched problems if you are feeling rusty. Check your understanding of this section by working a few problems from the exercises on your own. Focus on 5-10, 17-29(odd), 33-39(odd). Do you have any questions?Math 142, Chapter 4 Spring 2011 Zarestky Section 4.2: Derivatives of Exponential and Logarithmic Functions By the end of this section, you should be able to: • Use short-cut rules to find a formula for the derivative of a function. o Differentiate exponential and logarithmic functions. The first part of this section shows how to find the derivative of ex from the limit definition of the derivative. Make a note of the derivative rule and practice using it in Example 1. Work Example 1 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have? The next part of this section shows how to find the derivative of ln x from the limit definition of the derivative. Make a note of the derivative rule and practice using it in Example 2. Work Example 2 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 4 Spring 2011 Zarestky The two rules you just learned are specifically for base e exponential functions and natural logarithms. To apply the rules for bases other than e, you’ll need the rules in the box at the bottom of page 221. Make notes about these derivative rules. Work Example 3 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have? In Examples 4 and 5, you will apply your skills with derivatives of exponential and logarithmic functions to solve problems. Work Example 4. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 4 Spring 2011 Zarestky Work Example 5. Compare your answers to the book’s solution. What questions do you have? Practicing is really important! Many students feel uncomfortable with exponential and logarithmic functions. Practice is fundamental to overcoming any anxiety. Check your understanding of this section by working a few problems from the exercises on your own. Focus on 1-41(odd), 51, 53, 55, 57. What questions do you still have?Math 142, Chapter 4 Spring 2011 Zarestky Section 4.3: Derivatives of Products and Quotients By the end of this section, you should be able to: • Use short-cut rules to find a formula for the derivative of a function. o Differentiate the product or quotient of two functions. The first part of this section shows how to find the derivative of the product of two functions. Make a note of the derivative rule and practice using it in Examples 1-3. Work Example 1 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have? Work Example 2. This is also a refresher on tangent lines! Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 4 Spring 2011 Zarestky Work Example 3 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have? The next part of this section shows how to find the derivative of the quotient of two functions. Make a note of the derivative rule and practice using it in Examples 4-6. Work Example 4 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 4 Spring 2011 Zarestky Work Example 5 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have? Work Example 6. This also requires you to interpret the derivative. Refer back to Section 3.4 if you need a refresher. Compare your answers to the book’s solution. What questions do you have? Practice is really important! Check your understanding of this section by working a few problems from the exercises on your own. Focus on 1-89(odd). What questions do you still have?Math 142, Chapter 4 Spring 2011 Zarestky Section 4.4: The Chain Rule By the end of this section, you should be able to: • Use short-cut rules to find a formula for the derivative of a function. o Differentiate composite functions. The first part of this section discusses composite functions. Make a note of the definition of composite functions. If it is familiar to you skim, Examples 1 and 2. Otherwise, work through them in detail. Continue reading about the General Power Rule. You have already seen the Power Rule ddxxn= n ! xn"1 but the difference here is that the base can be a function instead of just plain x. Make a note of the General Power Rule and practice it in Example 3. Work Example 3 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 4 Spring 2011 Zarestky Continue reading about the Chain Rule. This is a more general version of the rules you have already been using. Make a note of the Chain Rule and practice it in Examples 4 and 5. Work Example 4 and the Matched Problems. Compare your answers to the book’s solution. What questions do you have? Work Example 5. Compare your answers to the book’s solution. What questions do you have?Math 142, Chapter 4 Spring 2011 Zarestky The General Derivative Rules at the bottom of page 239 are special cases of the Chain Rule. If the Chain rule makes sense to you,