An Interactive Presentation
         ED 205-16
  By: Lindsay Schrauger
    12th grade- Calculus
Main Slide
•   Applications for derivatives
•   Average Velocity into Instantaneous Velocity
•   Slope of secant line into...
Applications for Derivatives
Throughout our study of derivatives, we will find that
  derivatives have several application...
Average Velocity to
           Instantaneous Velocity
• Average Velocity over an interval [1,1+h]
s(1+h)-s(h)
            ...
Slope of the Secant Line to the
          Slope of the Tangent Line
As we can see in the picture on the
right, the slope o...
Definition of a Derivative
The derivative of a function f, as written,f’(a) is
   defined as
f’(a)=limf(a+h)-f(a)
h->0    ...
Example of finding the derivative
Find the derivative of y=8x2+3 at x=a.

y’=limy(a+h)-y(a) = lim8(a+h)2+3 - (8a2-3)
h->0 ...
Practice Problems
1. Use the definition of the derivative to find
    the derivative of the function f(x)=x2+x at
    x=a....
Answers to Practice Problems
1. 2a
2. -1/4
3. 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...
Resources
• Pictures
Slide 1 http://schoolweb.dysart.org/TeacherSites/uploads/2445/ Audio/books_-
    _math.gif
Slide 2 ht...
Author’s Slide
My name is Lindsay Schrauger and I
am a sophomore at Grand Valley State
University. I am going into Seconda...
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Interactive Power Point Presentation

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This is an Interactive Power Point that gives an introduction to Derivatives in Calculus.

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Interactive Power Point Presentation

  1. 1. An Interactive Presentation ED 205-16 By: Lindsay Schrauger 12th grade- Calculus
  2. 2. 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
  3. 3. 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
  4. 4. 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 lims(1+h)-s(1) h->0 h • Instantaneous Velocity is the Average Velocity as time intervals get smaller and smaller. Quit
  5. 5. 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
  6. 6. Definition of a Derivative The derivative of a function f, as written,f’(a) is defined as f’(a)=limf(a+h)-f(a) h->0 h Quit
  7. 7. Example of finding the derivative Find the derivative of y=8x2+3 at x=a. y’=limy(a+h)-y(a) = lim8(a+h)2+3 - (8a2-3) h->0 hh->0 h = lim8(a2+16ah+h2) + 3 – 8a2+3 h->0 h = lim(16ah+8h2) h->0 h = lim 16a+8h = 16a h->0 Quit
  8. 8. 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
  9. 9. Answers to Practice Problems 1. 2a 2. -1/4 3. 3a2+8 and 35 Quit
  10. 10. 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
  11. 11. Resources • Pictures Slide 1 http://schoolweb.dysart.org/TeacherSites/uploads/2445/ Audio/books_- _math.gif Slide 2 http://www.ebr.lib.la.us/teens/teenzone/math.png Slide 3 http://www.intmath.com/Differentiation-transcendental/deriv-ex1.gif Slide 5http://marauder.millersville.edu/%7Ebikenaga/calculus/tangent/tangent7.png Slide 6 http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/ img001.GIF Slide 8 http://wwwdelivery.superstock.com/WI/223/1647/PreviewComp/SuperStock_164 7R-51628.jpg Slide 9http://eppsnet.com/images/math-problems.gif Slide 10 http://s3.amazonaws.com/mmc-digi-beta- production/assets/201/math_teacher_626_article.jpg • Video http://www.youtube.com/watch?v=zgiNciB3qhw Quit
  12. 12. Author’s Slide My 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
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