Review: 1) What is a tangent line? 2) What is a secant line? 3) What is a normal line?

Slides:

Advertisements

Similar presentations

Warm Up Page 92 Quick Review Exercises 9, 10.

Advertisements

Warm-ups 1) Find the equations of all lines tangent to y = 9 – x2 that passes through the point (1, 12).

Warmup describe the interval(s) on which the function is continuous

2.4 RATES OF CHANGE & TANGENT LINES. Average Rate of Change  The average rate of change of a quantity over a period of time is the slope on that interval.

DO NOW: Use Composite of Continuous Functions THM to show f(x) is continuous.

Rates of Change and Tangent Lines Section 2.4. Average Rates of Change The average rate of change of a quantity over a period of time is the amount of.

The derivative as the slope of the tangent line

1.4 Continuity and One-Sided Limits This will test the “Limits” of your brain!

The Derivative. Def: The derivative of a function f at a number a, denoted f’(a) is: Provided this limit exists.

2.4 Rates of Change and Tangent Lines Quick Review In Exercises 1 and 2, find the increments Dx and Dy from point A to point B. In Exercises 3 and 4,

Mrs. Rivas International Studies Charter School.Objectives: slopes and equations 1.Find slopes and equations of tangent lines. derivative of a function.

3.2 Continuity JMerrill, 2009 Review 3.1 Find: Direct substitution causes division by zero. Factoring is not possible, so what are you going to do?

Practice! 1. For the graph shown, which of these statements is FALSE? (A) f(x) is continuous at x=2 (B) (C) (D) (E) f(x) is continuous everywhere from.

AP Calculus BC Chapter Rates of Change & Limits Average Speed =Instantaneous Speed is at a specific time - derivative Rules of Limits: 1.If you.

Jeopardy Limits Limits with Trig Slope of a Curve Continuity Potpourri $100 $200 $300 $400 $500 $100 $200 $300 $400 $500.

Today in Calculus Go over homework Average Rate of Change (aka slope)

1.6 – Tangent Lines and Slopes Slope of Secant Line Slope of Tangent Line Equation of Tangent Line Equation of Normal Line Slope of Tangent =

3.1 Derivatives and Rates of Change 12.7 Derivatives and Rates of Change.

2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change.

Tangents. The slope of the secant line is given by The tangent line’s slope at point a is given by ax.

Velocity and Other Rates of Change Notes: DERIVATIVES.

1.4 Continuity  f is continuous at a if 1. is defined. 2. exists. 3.

Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6.

Tangents, Velocities, and Other Rates of Change Definition The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope.

Solve and show work! State Standard – 4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line.

TANGENT LINES Notes 2.4 – Rates of Change. I. Average Rate of Change A.) Def.- The average rate of change of f(x) on the interval [a, b] is.

Warm Up Determine a) ∞ b) 0 c) ½ d) 3/10 e) – Rates of Change and Tangent Lines.

2.1 The Derivative and The Tangent Line Problem Slope of a Tangent Line.

UNIT 1B LESSON 7 USING LIMITS TO FIND TANGENTS 1.

AP CALCULUS 1006: Secants and Tangents. Average Rates of Change The AVERAGE SPEED (average rate of change) of a quantity over a period of time is the.

1.3 – Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.

OBJECTIVES: To introduce the ideas of average and instantaneous rates of change, and show that they are closely related to the slope of a curve at a point.

Objectives Determine tangent lines to functions for a given point Compute the slope of curves Compute Instantaneous rate of change.

Section 2.1 – Average and Instantaneous Velocity.

Index FAQ The derivative as the slope of the tangent line (at a point)

Section 1.4 The Tangent and Velocity Problems. WHAT IS A TANGENT LINE TO THE GRAPH OF A FUNCTION? A line l is said to be a tangent to a curve at a point.

Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer during this trip, it might read 65 mph.

1 10 X 8/30/10 8/ XX X 3 Warm up p.45 #1, 3, 50 p.45 #1, 3, 50.

Ch. 2 – Limits and Continuity 2.4 – Rates of Change and Tangent Lines.

Ch. 5 – Applications of Derivatives 5.2 – Mean Value Theorem.

Unit 2 Lesson #3 Tangent Line Problems

3.2 Rolle’s Theorem and the

Ch. 2 – Limits and Continuity

2.4 Rates of Change and Tangent Lines

Warm Up a) What is the average rate of change from x = -2 to x = 2? b) What is the average rate of change over the interval [1, 4]? c) Approximate y’(2).

2-4 Rates of change & tangent lines

2.1 Tangents & Velocities.

Section 11.3A Introduction to Derivatives

Rate of Change.

2.1 Tangent Line Problem.

2.4 Rates of Change and Tangent Lines Day 1

2.1A Tangent Lines & Derivatives

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE (2.2)

2.4 Rates of Change & Tangent Lines

2.1 Rates of Change and Limits Day 1

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE (2.2)

Chapter 2 – Limits and Continuity

3.2 Rolle’s Theorem and the

Derivatives by Definition

2.2C Derivative as a Rate of Change

2.7/2.8 Tangent Lines & Derivatives

Derivatives: definition and derivatives of various functions

32 – Applications of the Derivative No Calculator

Drill: Find the limit of each of the following.

2.4 Rates of Change & Tangent Lines

2.4 Rates of Change and Tangent Lines

Drill: Find the limit of each of the following.

Unit 2 - Derivatives.

If {image} choose the graph of f'(x).

Presentation transcript:

Review: 1) What is a tangent line? 2) What is a secant line? 3) What is a normal line?

Answers: 3. Continuous. x ≠ 0 6. Continuous It is a composition of functions 9. x = 0 12a. Yes f(1) = 1 b. Yes lim = 2 c. No d. No 15. f(2) = Yes 21a. x = 1 b. Non-removable, jump 24a. x = 1,2 b. x = 1 is non-removable x = 2 is removable f(2) = Possible Graph 45. x ≈ and Consider f(x) = x - e -x, f is continuous, f(0) = -1, and f(1) = 1 - 1/e >.5 By the Intermediate Value Thm, for some c in (0,1), f(c) = 0 and e -c = c 63. Since |x| is continuous, all functions in absolute value are continuous.

Average Rate of Change What is the formula for the average rate of change? What does it mean?

Tangents & Secants A tangent line intersects the curve at exactly one point. The tangent line can describe the rate of change at that point. A secant line intersects the curve at two or more points. We can use it to determine the slope of the line tangent if the points are infinitesimally close.

Ex 1: Fine the average rate of change for f(x) = x 3 - x over the interval [2,4]

Ex 2: Use the points P(23, 150) and Q(45, 350) to computer the average rate of change and slope of the secant line PQ.

Ex 3: Find the slope of the parabola f(x) = x 2 at the point P(2,4). Write the equation for the tangent to the parabola at this point.

Slope of a a Point The slope of a curve f(x) at the point (a,f(a)) is m = lim f(a + h) - f(a) h 0 h The tangent line to the curve at the point is the line through the point with this slope.

Normal to the curve. The normal line to a curve at a point is the line perpendicular to the tangent at that point. f(x) tangent line normal

Ex 1: Find the slope of the curve at x = a. a) f(x) = x 2 b) f(x) = 1/x

Ex 1: Find the slope of the curve at x = a. a) f(x) = x 2 b) f(x) = 1/x

Ex 3: The equation for free fall at the surface of Jupiter is s = 11.44t 2 m with t in seconds. Assume a rock is dropped from the top of a 500 m cliff. Find the speed of the rock at t = 2.

Assignment: p92 # 1-33odd