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
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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
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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.
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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)).
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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
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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
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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.
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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
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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.
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