Download presentation

Presentation is loading. Please wait.

Published byUriel Everingham Modified over 3 years ago

1
2.1 Derivatives and Rates of Change

2
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 the secant through (4,16): We could get a better approximation if we move the point closer to (1,1), i.e. (3,9): Even better would be the point (2,4): The tangent problem

3
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? The tangent problem

4
slope slope at The slope of the curve at the point is: Note: This is the slope of the tangent line to the curve at the point. The tangent problem

5
Consider a graph of displacement (distance traveled) vs. time. time (hours) distance (miles) Average velocity can be found by taking: A B The speedometer in your car does not measure average velocity, but instantaneous velocity. (The velocity at one moment in time.) The velocity problem

6
Derivatives Definition: The derivative of a function at a number a, denoted by f ′(a), is if this limit exists. Example: Find f ′(a) for f(x)=x 2 +3.

7
Equation of the tangent line The tangent line to y=f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is equal to f ′(a). Then the equation of the tangent line to the curve y=f(x) at the point (a,f(a)): Example: Find an equation of the tangent line to f(x)=x 2 +3 at (1,4). From previous slide: f ′(1)=2 1=2. Thus, the equation is y-f(1)= f ′(1)(x-1) y-4=2(x-1) or y=2x+2

8
Rates of Change: Average rate of change = Instantaneous rate of change = These definitions are true for any function. ( x does not have to represent time. )

Similar presentations

Presentation is loading. Please wait....

OK

3.4 Velocity and Rates of Change

3.4 Velocity and Rates of Change

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on law against child marriage in afghanistan Ppt on zener diode basics Ppt on series and parallel circuits physics Ppt on paintings and photographs related to colonial period of america Ppt on area of trapezium rectangle Ppt on as 14 amalgamation symbol Ppt on water conservation in india Ppt on action research in education Ppt on leadership challenges Ppt on disk formatting types