11-2 Key to evens 2a) -5 2b) -3 2c) 0 4a) 0 4b) 1 4c) -2 6) -1/10 8) -5 10) 27 12) - 7/14 14) 1/8 16) 1/16 18) 0 20) 1/4 22) -1/6 24) 4 26) -1/4 28) 1 30) 0 32) 3 34) 0 36)-2/3 38)7.389

Review 4) Think through this limit :

Review 5) Think through this limit :

Definition of a Horizontal Asymptote? The line y = b is a horizontal asymptote of the function y = f(x), if y approaches b as x approaches ±∞.

Limits at infinity Find the following limits

11.3 Tangent Lines 2014

Precalculus Warm-up Find the equation of the line between the points: (2,5) (3,8) y=3x-1

Lesson One Sided Limits

Lesson Why? Because the limit from the left of 0 does not agree with the limit from the right of 0.

Because both one sided limits agree, Graph to verify

Precalculus Warm-up At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by where s is measured in feet, and t in seconds. a)When does the diver hit the water? b)What is the diver’s average rate of change on the dive? c)What is the diver’s velocity at impact? ?

Slope and the Limit Process In the warm-up we found average velocity of a graph by finding the slope of a secant line between two points. How can we take the idea of the slope of a secant line and use it to find the velocity at any point on the graph? With limits!

The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).

The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?

slope slope at The slope of the curve at the point is:

is called the difference quotient of f at x.  If you are asked to find the slope using the limit definition or using the difference quotient, this is the technique you will use.  The slope of a curve at a point is the same as the slope of the tangent line at that point.

Example: Find a formula for the slope of the graph of using the limit process.

Example Find a formula for the slope of the graph of using the difference quotient. Then find the slope at the point ? How about at ?

Derivatives

The limit we used to define the slope of a tangent line is also used to define one of 2 fundamental operations in calculus-differentiation. The derivative of a function of x is also itself a function of x. This “new” function gives the slope of the tangent line to the graph of the function at a point, provided that the graph has a tangent line at this point. So, the derivative describes the slope of a function at any point along the graph of the function.

“The derivative of f with respect to x is …” This equation finds the formula for the derivative of a function at any point.

“The derivative of at is …” There are many ways to write the derivative of This equation finds the derivative of a function at a.

“f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks uv eff uv ecks”or “the derivative of f of x”

dx does not mean d times x ! dy does not mean d times y !

does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.) does not mean times !

Example: Use the limit of the difference quotient to find the derivative of What is the derivative of the function at x = 2? What does this mean?

You must be able to do this : Use the limit process to find the derivative of f(x). Find the equation of a line tangent to the graph at (-2,-1)

Example: Use the limit process to find the derivative of and write the equation of the line tangent to the graph of f(x) at x=4.

You Try: Find the derivative of using the limit process. a) Find any values of x where the tangent line is horizontal. b) Find the equation of the line tangent at x=1.

Big Idea: Velocity (slope of a tangent line to a curve at a point) can be found by taking the limit of difference quotient as the change in x (we are calling this h) goes to 0. Back to the warm-up problem. What is the diver’s instantaneous rate of change at impact? Use the limit process to find the diver’s velocity at impact. Recall that impact occurred at t = 2. Finally:

Recap of ideas: average slope: slope at a point: average velocity: instantaneous velocity: If is the position function: These are often mixed up by Calculus students! So are these! velocity = slope 

Homework 11.3 : Day 1: pg odd Day 2: pg odd, 43, 45

Example: Find a formula for the slope of the graph of using the limit process. Use the derivative to determine any points on the graph of f at which the tangent line is horizontal and find the equation of each tangent line.

Derivatives 11.3 Day 2

Homework 11.3 : Day 1: pg odd Day 2: pg odd, 43, 45 Quiz tomorrow on 11.3