An Introduction to Derivatives

An Interactive PresentationED 205-16

By: Lindsay Schrauger12th grade- Calculus

Main Slide

• Applications for derivatives• Average Velocity into Instantaneous Velocity• Slope of secant line into slope of tangent line• What is a derivative?• Example of finding a derivative• Practice Problems• Answers to Practice Problems• Video • Resources• Author’s Slide

Quit

Applications for Derivatives

Throughout our study of derivatives, we will find that derivatives have several applications. There is a list below of the different applications. We will look at several of these applications throughout the presentation.

The list includes:• Rate of Change• Instantaneous Velocity• Slope of the Tangent Line

Quit

Average Velocity to Instantaneous Velocity

• Average Velocity over an interval [1,1+h]s(1+h)-s(h) (1+h)-h

• Instantaneous Velocity at time t=1 hour lim s(1+h)-s(1) h->0 h

• Instantaneous Velocity is the Average Velocity as time intervals get smaller and smaller.

Quit

Slope of the Secant Line to the Slope of the Tangent Line

As we can see in the picture on the right, the slope of the secant line consists of two points P and Q on f(x).• The slope of the secant line is found by finding the rate of change between (x0, f(x0) and (x0+Δx,f(x0+Δx)).• The slope of the tangent line finds the rate of change at a specific point P. Therefore, the slope of the tangent is the slope at (x0, f(x0)).

Quit

Definition of a Derivative

The derivative of a function f, as written, f’(a) is defined as

f’(a)=lim f(a+h)-f(a)

h->0 h

Quit

Example of finding the derivative

Find the derivative of y=8x2+3 at x=a.

y’=lim y(a+h)-y(a) = lim 8(a+h)2+3 - (8a2-3) h->0 h h->0 h= lim 8(a2+16ah+h2) + 3 – 8a2+3 h->0 h= lim (16ah+8h2) h->0 h= lim 16a+8h = 16a h->0

Quit

Practice Problems

1. Use the definition of the derivative to find the derivative of the function f(x)=x2+x at x=a.

2. Find the instantaneous velocity at t=2 when the function is given by s(t)=1/t

3. Find the derivative of the function s(t)=t2+8t+2 at t=a and t=3.

Quit

Answers to Practice Problems

1. 2a2. -1/43. 3a2+8 and 35

Quit

Video

• In the video clip provided below, you are able to watch a tutorial of introducing derivatives, as well as view further applications of derivatives.

• Teaching Derivatives

Quit

Resources• PicturesSlide 1 http://schoolweb.dysart.org/TeacherSites/uploads/2445/ Audio/books_-_math.gifSlide 2 http://www.ebr.lib.la.us/teens/teenzone/math.png Slide 3 http://www.intmath.com/Differentiation-transcendental/deriv-ex1.gifSlide 5 http://marauder.millersville.edu/%7Ebikenaga/calculus/tangent/tangent7.pngSlide 6 http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/ img001.

GIFSlide 8

http://wwwdelivery.superstock.com/WI/223/1647/PreviewComp/SuperStock_1647R-51628.jpg

Slide 9 http://eppsnet.com/images/math-problems.gifSlide 10

http://s3.amazonaws.com/mmc-digi-beta-production/assets/201/math_teacher_626_article.jpg

• Videohttp://www.youtube.com/watch?v=zgiNciB3qhwQuit

Author’s SlideMy name is Lindsay Schrauger and I am a sophomore at Grand Valley State University. I am going into Secondary Education with a Mathematics major and Spanish minor. While I have always planned on teaching high school students, I have been gaining a passion to reach out to middle school students. I also plan on working with inner city students, where I can use my Spanish-speaking skills to reach out to Spanish-speaking parents. I hope you enjoyed this tutorial on Derivatives and if you have any questions or comments, please email me.

Quit