Calc 2.3b
- 1. Quotient rule applied to trig, finding second derivatives, third derivatives, and so on!
- 2. Say what? Why would those be the derivatives? Let’s look at tan x:
- 3. Now I want you students to figure out why the rest work. On the given sheet, show your group’s work. Step 1: Define as quotient. Step 2: Derive Step 3: Simplify Group 1: find y’ when y = csc x Group 2: find y’ when y = sec x Group 3: find y’ when y = cot x
- 4. So how to memorize? Compare and contrast and come up with some patterns.
- 5. Ex 8 p.124 Differentiating Trigonometric Functions Function Derivative y = x – tan x b. y = x sec x
- 6. Ex 9 p. 124 Different Forms of a Derivative Differentiate both forms of First form: 2 nd Form: Are these equivalent? Check it out!
- 7. Much of the work in calculus comes AFTER taking the derivative. Characteristics of a simplified form? Absence of negative exponents Combining of like terms Factored forms
- 8. Higher-Order Derivatives Just as velocity is the derivative of a position function, acceleration is the derivative of a velocity function. s(t) Position Function. . . . v(t) = s’(t) Velocity Function. . . . a(t) = v’(t) = s”(t) Acceleration Function. a(t) is the second derivative of s(t) – which is the derivative of a derivative!
- 9. Notations for higher-order derivatives:
- 10. Ex 10 p. 125 Finding Acceleration Due to Gravity Because the moon has no atmosphere, a falling object on the moon hits no air resistance. In 1971, astronaut David Scott showed that a hammer and a feather fell at the same rate on the moon. is the position function where s(t) is the height in meters and t is time in seconds. What is the ratio of the Earth’s gravitational force to the moon’s? To find acceleration due to gravity on moon, differentiate twice.
- 11. Assignment 2.3b p. 126 #45, 61, 73-78, 83-87 odd, 93-101 odd