Section 2.6 Slopes of tangents  SWBAT:  Find slopes of tangent lines  Calculate velocities.

Slides:



Advertisements
Similar presentations
2.7 Tangents, Velocities, & Rates of Change
Advertisements

2.1 Derivatives and Rates of Change. The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of.
LIMITS AND DERIVATIVES 2. The idea of a limit underlies the various branches of calculus.  It is therefore appropriate to begin our study of calculus.
LIMITS AND DERIVATIVES
2 Derivatives.
LIMITS 2. In this section, we will learn: How limits arise when we attempt to find the tangent to a curve or the velocity of an object. 2.1 The Tangent.
Copyright © 2011 Pearson Education, Inc. Slide Tangent Lines and Derivatives A tangent line just touches a curve at a single point, without.
Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.
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.
AP CALCULUS 1005: Secants and Tangents. Objectives SWBAT determine the tangent line by finding the limit of the secant lines of a function. SW use both.
Limits Section 15-1.
THE DERIVATIVE AND THE TANGENT LINE PROBLEM
DO NOW: Use Composite of Continuous Functions THM to show f(x) is continuous.
DERIVATIVES Derivatives and Rates of Change DERIVATIVES In this section, we will learn: How the derivative can be interpreted as a rate of change.
LIMITS 2. LIMITS The idea of a limit underlies the various branches of calculus.  It is therefore appropriate to begin our study of calculus by investigating.
1 Instantaneous Rate of Change  What is Instantaneous Rate of Change?  We need to shift our thinking from “average rate of change” to “instantaneous.
M 112 Short Course in Calculus Chapter 2 – Rate of Change: The Derivative Sections 2.2 – The Derivative Function V. J. Motto.
10/26/20151 A Rates of Change Calculus - Santowski.
LIMITS AND DERIVATIVES 2. The problem of finding the tangent line to a curve and the problem of finding the velocity of an object both involve finding.
2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.
2.1 The Tangent and Velocity Problems 1.  The word tangent is derived from the Latin word tangens, which means “touching.”  Thus a tangent to a curve.
 As we saw in Section 2.1, the problem of finding the tangent line to a curve and the problem of finding the velocity of an object both involve finding.
Limits and Derivatives 2. Derivatives and Rates of Change 2.6.
Chapter 3: Derivatives 3.1 Derivatives and Rate of Change.
Section 2.6 Tangents, Velocities and Other Rates of Change AP Calculus September 18, 2009 Berkley High School, D2B2.
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.
DERIVATIVES 3. DERIVATIVES In this chapter, we begin our study of differential calculus.  This is concerned with how one quantity changes in relation.
 The derivative of a function f(x), denoted f’(x) is the slope of a tangent line to a curve at any given point.  Or the slope of a curve at any given.
Copyright © Cengage Learning. All rights reserved. 2 Derivatives.
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.
Warm Up Determine a) ∞ b) 0 c) ½ d) 3/10 e) – Rates of Change and Tangent Lines.
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.
2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming.
Section 2.1 How do we measure speed?. Imagine a ball being thrown straight up in the air. –When is that ball going the fastest? –When is it going the.
Derivative Notation and Velocity. Notation for the Derivative.
Rate of Change. What is it? A slope is the rate at which the y changes as the x changes Velocity is the rate the position of an object changes as time.
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.
Section 2.4 Rates of Change and Tangent Lines Calculus.
1 10 X 8/30/10 8/ XX X 3 Warm up p.45 #1, 3, 50 p.45 #1, 3, 50.
Calc Tangents, Velocities and Other Rates of Change Looking to calculate the slope of a tangent line to a curve at a particular point. Use formulas.
Graphs of a falling object And you. Objective 1: Graph a position –vs- time graph for an object falling from a tall building for 10 seconds Calculate.
Copyright © Cengage Learning. All rights reserved.
2-4 Rates of change & tangent lines
2.1 Tangents & Velocities.
Copyright © Cengage Learning. All rights reserved.
Activity 5-2: Understanding Rates of Change
Rate of Change.
LIMITS AND DERIVATIVES
Rate of change and tangent lines
2.4 Rates of Change & Tangent Lines
Sec 2.7: Derivative and Rates of Change
Definition of the Derivative
The Tangent and Velocity Problems
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
2 Derivatives.
Derivatives and Rates of Change
THE DERIVATIVE AND THE TANGENT LINE PROBLEM
Rate of Change and Instantaneous Velocity
The Tangent and Velocity Problems
2.2C Derivative as a Rate of Change
2.7/2.8 Tangent Lines & Derivatives
Packet #4 Definition of the Derivative
Section 2.1 Limits, Rates of Change, and Tangent Lines
Copyright © Cengage Learning. All rights reserved.
Drill: Find the limit of each of the following.
2.4 Rates of Change & Tangent Lines
Sec 2.7: Derivative and Rates of Change
2 Derivatives.
Presentation transcript:

Section 2.6 Slopes of tangents  SWBAT:  Find slopes of tangent lines  Calculate velocities

Review of Secant lines  Remember this!?! the slope of the secant line PQ :the slope of the secant line PQ :

Concept: take the limit x approaches a.

Slope of a Tangent to f(x) at point ( a, f(a) ) is:

Example  Start with y = x 2 at the point P(2, 4).  Use our definition of the slope of the tangent at a point.  Need a hint:

Example (cont’d)

 Using the point-slope form of the equation of a line y – 4 = 4(x – 2) So,y = 4x – 4

Another Expression (cont’d)

Another Expression  There is another way to define slope of a tangent: We let h = x – aWe let h = x – a Then x = a + hThen x = a + h Thus the slope of secant line PQ isThus the slope of secant line PQ is

A second definition is:

Example 2:  Use to calculate the slope of f(x) at a.

Example 3. Find the equation of the tangent line to the hyperbola y = 3/x at the point (3, 1)

Example (cont’d)

 Solution Let f(x) = 3/x. Then the slope of the tangent at (3, 1) is – ⅓.  Therefore an equation of the tangent line is y – 1 = – ⅓ (x – 3),  or y= ⅓ x+2.

Example (cont’d)

Velocity The average velocity over a time interval h equals the slope of the secant line PQ.

Velocity (cont’d)

Taking it to the limit we get... Instantaneous Velocity  This means that the velocity at time t = a is equal to thevelocity at time t = a is equal to the slope of the tangent line at P.slope of the tangent line at P.

Example  (Calculator active)  Suppose a ball is dropped from the top of a tower 450m. Tall. Find the velocity at 5 seconds.  Position function f(t) = 4.9t 2

Example (cont’d) a) The velocity after 5 s is v(t)=9.8t v(5) = (9.8)(5) = 49 m/s.

Part deux:  How fast is the ball going when it hits the ground?

Example (cont’d)  It hits the ground when it traveled 450m  The ball will hit the ground when, 4.9t 2 = 450.  Solving for t gives t ≈ 9.6 s.  The velocity of the ball as it hits the ground is therefore v(t) = 9.8t ≈ 94 m/s

Assignment 11  P odd

Other Rates of Change  In general, we can have other average rates of change:  These can lead to instantaneous rates of change.

Other Rates (cont’d)

Example  Temperature readings T (in °C) were recorded every hour starting at midnight on a day in Whitefish, Montana.  The time x is measured in hours from midnight.  The data are given in the table on the next slide:

Example (cont’d)

a) Find the average rate of change of temperature with respect to time i.from noon to 3 P.M. ii.from noon to 2 P.M. iii.from noon to 1 P.M. b) Estimate the instantaneous rate of change at noon.

Solution to (a) i. From noon to 3 P.M. the temperature changes from 14.3 °C to 18.2 °C, so ∆T = T(15) – T(12) = 18.2 – 14.3 = 3.9 °C while the change in time is ∆x = 3 h. Therefore, the average rate of change of temperature with respect to time is

Solution to (a) (cont’d) ii. From noon to 2 P.M. the average rate of change is iii. From noon to 1 P.M. the average rate of change is

Solution to (b)  We plot the given data on the next slide and sketch a smooth curve of the temperature function.  Then we draw the tangent at the point P where x = 12.  By measuring the sides of triangle ABC, we estimate the slope of the tangent line to be 10.3/5.5 ≈ 1.9.

Solution to (b) (cont’d)

 Therefore the instantaneous rate of change of temperature with respect to time at noon is about 1.9 °C.

Review  Use of limits to define Tangent linesTangent lines Instantaneous velocitiesInstantaneous velocities Other rates of changeOther rates of change