Chapter 3B More Derivative fun 

Section 3.5 Notes: Derivatives of Trigonometric Functions Objectives: Finding derivatives of trigonometric functions, Exploring simple harmonic motion, Defining and applying the concept of “jerk”

Ex. 1: a. b.

Jerk Jerk is the derivative of acceleration (or the 3rd derivative of displacement). If a body’s position at time t is s(t), the body’s jerk at time t is: ** This is the concept that mainly describes motion sickness… or why some people cannot read in a moving vehicle.

Example 2:

Ex. 3: Find equations for the lines that are tangent and normal to the graph of at x = 2.

Ex. 4: Find y” if y = sec x.

3.5 Homework 3, 6, 9, 17, 21, 27, 31, 44-49

Section 3.8 Notes: Derivatives of Inverse Trig Functions Objectives: Evaluating and using the derivatives of the inverse trig functions WARM UP for today: Please complete examples 4 and 5 from section 3.5 while I come around to check homework!

Ex. 4: Find y” if y = sec x.

Quick Quiz Time You will have one minute to write the derivatives of each trig function. Do not turn your paper over until I say go 

Derivatives of Inverse Functions MEMORIZE THESE!!!!       

Calculator Conversion Identities

Ex.1:

Ex. 2: A particle moves along the x-axis so that its position at any time is . What is the velocity of the particle when t = 16?

Ex. 3:

Ex. 4 (YOU TRY IT!): Find an equation for the line tangent to the graph of at x = -1.

Homework assignment MEMORIZE YOUR DERIVATIVES FOR MONDAY!!!! Section 3.8 book work: #1, 3, 5, 9, 14, 26, 35-40 Please use class time wisely!

Today’s Agenda Mad Minute Monday Order of Operations Turn in Take Home Quiz  3.8 Inverse Trig quick quiz Section 3.9 – Derivatives of exponential and logarithmic functions

What is the difference between csc x and sin-1x? Explain! Warm up What is the difference between csc x and sin-1x? Explain!

Section 3.9 Notes: Derivatives of Exponential and Log functions Objectives: Evaluating and using the derivatives of the various exponential functions and logarithmic functions.

Derivative of ex

Ex. 1: The spread of a flu in a certain school is modeled by the equation where P(t) is the total number of students infected t days after the flu was first noticed. Many of them may already be well again in time t.

Estimate the initial number of students infected with the flu. How fast is the flu spreading after 3 days? When will the flu spread at its maximum rate? What is this rate?

Derivative of ax For a > 0, and a 1,

Example: Find d/dx (12x)

Ex. 2: At what point on the graph of the function does the tangent line have a slope of 21?

Please complete the exploration with your table partners!

Derivative of ln x

Ex. 3: Find

Derivative of

Ex. 4: Find dy/dx if .

Ex. 5: Find the derivatives of the following functions: a.   b.

Example 6: Find d/dx (ln(x-3)). State the domain of the derivative.

3.9 Homework #3, 5, 12, 15, 16, 21, 41, 51, 57-62

Warm up: Find the points on the graph of y = cos x, where the tangent is parallel to the line 3y + 2x = 7.

Section 3.7 Notes: Implicit Differentiation Objectives: Differentiating implicitly defined functions, Investigating lenses, tangents, and normal lines. Finding derivatives of higher order implicitly, and Exploring the rational powers of differentiable functions

Implicitly Defined Functions When we need to differentiate a function in terms of 2 or more variables, we need to use a process called implicit differentiation.

Implicit Differentiation Process: Differentiate both sides of the equation with respect to x. Collect the terms with dy/dx on one side of the equation. Factor out dy/dx. Solve for dy/dx.

Ex. 1 Find dy/dx if y2 = x. Step 1: Differentiate both sides of the equation with respect to x. Step 2: Collect the terms with dy/dx on one side of the equation. Step 3: Factor out dy/dx. Step 4: Solve for dy/dx. Because “y” is not in terms of x, we attach the “dy/dx” to make it so! TADA!!!!!!!!!!

Ex. 2: Find the slope of the circle x2 + y2 = 25 at the point (3, -4). Step 1: Differentiate both sides of the equation with respect to x. Step 2: Collect the terms with dy/dx on one side of the equation. Step 3: Factor (NONE NEEDED ON THIS PROBLEM). Step 4: Solve for dy/dx. Derivative! Remember: Derivative of 25 is ZERO! Answer the question: Slope at (3, -4) =

Ex. 3: Show that the slope dy/dx is defined at every point on the graph of 2y = x2 + sin y. Step 1: Step 2: Step 3: Step 4: Answer the question: Can (2 – cos y) ever equal 0? OR… can cos y ever equal 2? NO! Therefore, the slope is defined at every point.

Lenses, Tangents, and Normal Lines In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry. This line is called the normal to the surface at the point of entry. Same stuff as last chapter… Tangent Line… Need a slope and a point… Normal Line… Need an opposite reciprocal slope and a point…

Ex. 4: WARNING: PRODUCT RULE!!! Find the tangent and normal to the ellipse x2 – xy + y2 = 7 at (-1, 2). Step 1: Step 2: Step 3: Step 4:

Assignment: We will finish the rest of the notes on Thursday!  (I know… you can HARDLY wait!) Please complete the following problems tonight: #3, 15, 17, 18 YOU NEED TO PRACTICE THIS TYPE OF PROBLEM!! They take a lot of work and a lot of getting used to!

Find the derivative of x2 +3xy + 4y2 = 2x at (3, 1). Warm Up Find the derivative of x2 +3xy + 4y2 = 2x at (3, 1).

Derivatives of Higher Order Ex. 5: Find d2y/dx2 if 2x3 – 3y2 = 8. Replace dy/dx! TADA!

Rational Powers of Differentiable Functions Rule: Power Rule for Rational Powers of x If n is any rational number, then If n < 1, then the derivative does not exist at x = 0. Think about y = x2/3 (1st grade seagull graph). What happens at x = 0? Cusp!

Ex. 6: Find the derivatives of the following functions: a. b. c.

Assignment: 3.7 bookwork problems: #27, 33, 36, 59-64

Warm Up A parachuter is dropped from an airplane flying at 18,000 feet. After collecting data from the area during its descent, the parachuter eventually lands safely on the ground. (Because of the motion of the plane, the fall will not be strictly vertical, but the elapsed time will be the same as that for a vertical path.) If the plane is flying at 625 miles per hour, how far will the parachuter move horizontally during its fall to the ground? HINT: Our standard “displacement” function stems from s(t) = - 16t2. Think about how the starting value affects your equation. Some conversions may be necessary once you find your answer. SET UP ONLY! DO NOT SOLVE: An object falls 50 feet from the rafters of a barn and plummets toward the ground below. One second later, a person throws his coffee cup into the air with a velocity of 30 ft/sec. If the cup is launched from a height of 6 feet, how much time elapses between the landing of the object and the splash of the coffee? (We optimistically assume that no one gets burned by the coffee!)

Agenda – 10/30/15 Intro to project Work with your table partner (your CSI partner) I will hand back quizzes and check your 3.9 while you are working!