Introduction to the Derivative - Weebly
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Introduction to the Derivative
Ethan Zell
University of Michigan
Ethan Zell Introduction to the Derivative
A Scenario
Say you are running the 100m dash. You get off the blocks quicklyand manage your best time ever, 11.3 seconds. You and yourcoach do some thinking and want to know how you can improve,so you come up with a list of questions...
Ethan Zell Introduction to the Derivative
A Scenario
Your coach asks, “What was your average velocity?”
This one you know. We look at the change in distance over thechange in time:
100 − 0
11.3 − 0≈ 8.85 m/s
“How fast were you running halfway through the race?”
This one you don’t yet know. Answering this question requiresknowing the definition of instantaneous velocity, an application ofthe derivative.
Ethan Zell Introduction to the Derivative
A Scenario
Your coach asks, “What was your average velocity?”
This one you know. We look at the change in distance over thechange in time:
100 − 0
11.3 − 0≈ 8.85 m/s
“How fast were you running halfway through the race?”
This one you don’t yet know. Answering this question requiresknowing the definition of instantaneous velocity, an application ofthe derivative.
Ethan Zell Introduction to the Derivative
A Scenario
Your coach asks, “What was your average velocity?”
This one you know. We look at the change in distance over thechange in time:
100 − 0
11.3 − 0≈ 8.85 m/s
“How fast were you running halfway through the race?”
This one you don’t yet know. Answering this question requiresknowing the definition of instantaneous velocity, an application ofthe derivative.
Ethan Zell Introduction to the Derivative
A Scenario
Your coach asks, “What was your average velocity?”
This one you know. We look at the change in distance over thechange in time:
100 − 0
11.3 − 0≈ 8.85 m/s
“How fast were you running halfway through the race?”
This one you don’t yet know. Answering this question requiresknowing the definition of instantaneous velocity, an application ofthe derivative.
Ethan Zell Introduction to the Derivative
Instantaneous Velocity
If d(t) is a function of your distance travelled dependent on time,we define the instantaneous velocity at time t to be:
limh→0
d(t + h) − d(t)
h
if the limit exists.
Ethan Zell Introduction to the Derivative
What does this look like?
Ethan Zell Introduction to the Derivative
Key Takeaways
So, the instantaneous velocity is the slope of the distance curve ata point.
However, the average velocity over a period of time [a, b] is givenby the slope passing of the line passing through (a, d(a)) and(b, d(b)).
Ethan Zell Introduction to the Derivative
Key Takeaways
So, the instantaneous velocity is the slope of the distance curve ata point.
However, the average velocity over a period of time [a, b] is givenby the slope passing of the line passing through (a, d(a)) and(b, d(b)).
Ethan Zell Introduction to the Derivative
Example
We want to know the instantaneous velocity at t = 2, when thedistance function is d(t) = t2.
limh→0
(2 + h)2 − 4
h= lim
h→0
h2 + 4h + 4 − 4
h= lim
h→0
h2 + 4h
h=
limh→0
(h + 4) = 4.
Ethan Zell Introduction to the Derivative
Example
We want to know the instantaneous velocity at t = 2, when thedistance function is d(t) = t2.
limh→0
(2 + h)2 − 4
h= lim
h→0
h2 + 4h + 4 − 4
h= lim
h→0
h2 + 4h
h=
limh→0
(h + 4) = 4.
Ethan Zell Introduction to the Derivative
Example
We want to know the instantaneous velocity at t = 2, when thedistance function is d(t) = t2.
limh→0
(2 + h)2 − 4
h= lim
h→0
h2 + 4h + 4 − 4
h= lim
h→0
h2 + 4h
h=
limh→0
(h + 4) = 4.
Ethan Zell Introduction to the Derivative
Estimation
If you can’t solve a limit algebraically, use estimation.
From theprevious example, we could we would use the following values forh :
−0.1,−0.01,−0.001, . . . and 0.1, 0.01, 0.001, . . .
In this way, we try to approximate approaching the limit from bothsides. This usually lets us guess what the limit might beapproaching or if it even exists at all.
Ethan Zell Introduction to the Derivative
Estimation
If you can’t solve a limit algebraically, use estimation. From theprevious example, we could we would use the following values forh :
−0.1,−0.01,−0.001, . . . and 0.1, 0.01, 0.001, . . .
In this way, we try to approximate approaching the limit from bothsides. This usually lets us guess what the limit might beapproaching or if it even exists at all.
Ethan Zell Introduction to the Derivative
Estimation
If you can’t solve a limit algebraically, use estimation. From theprevious example, we could we would use the following values forh :
−0.1,−0.01,−0.001, . . . and 0.1, 0.01, 0.001, . . .
In this way, we try to approximate approaching the limit from bothsides. This usually lets us guess what the limit might beapproaching or if it even exists at all.
Ethan Zell Introduction to the Derivative
Tough Limit Example
If your velocity, is v(t) = πet3, it might be tough to compute the
instantaneous velocity at 1:
limh→0
πe(1+h)3 − πe13
h.
We can use our calculator to get more info:
Input, h -0.1 -0.01 -0.001 0.001 0.01 0.1
Output, f (h) 20.272 24.991 25.555 25.683 26.272 33.507
where the output is calculated by evaluating f (h) = πe(1+h)3−πe13h
at the input values for h.
Ethan Zell Introduction to the Derivative
Tough Limit Example
If your velocity, is v(t) = πet3, it might be tough to compute the
instantaneous velocity at 1:
limh→0
πe(1+h)3 − πe13
h.
We can use our calculator to get more info:
Input, h -0.1 -0.01 -0.001 0.001 0.01 0.1
Output, f (h) 20.272 24.991 25.555 25.683 26.272 33.507
where the output is calculated by evaluating f (h) = πe(1+h)3−πe13h
at the input values for h.
Ethan Zell Introduction to the Derivative
Tough Limit Example
If your velocity, is v(t) = πet3, it might be tough to compute the
instantaneous velocity at 1:
limh→0
πe(1+h)3 − πe13
h.
We can use our calculator to get more info:
Input, h -0.1 -0.01 -0.001 0.001 0.01 0.1
Output, f (h) 20.272 24.991 25.555 25.683 26.272 33.507
where the output is calculated by evaluating f (h) = πe(1+h)3−πe13h
at the input values for h.
Ethan Zell Introduction to the Derivative
Tough Limit Example
If your velocity, is v(t) = πet3, it might be tough to compute the
instantaneous velocity at 1:
limh→0
πe(1+h)3 − πe13
h.
We can use our calculator to get more info:
Input, h -0.1 -0.01 -0.001 0.001 0.01 0.1
Output, f (h) 20.272 24.991 25.555 25.683 26.272 33.507
where the output is calculated by evaluating f (h) = πe(1+h)3−πe13h
at the input values for h.
Ethan Zell Introduction to the Derivative
Questions?
Any questions?
Ethan Zell Introduction to the Derivative
Challenge 1!
Write a formula for the instantaneous velocity of f (t) = (4 + t)t att = 1.
Use a calculator to create a table in order to estimate the value ofyour formula.
Ethan Zell Introduction to the Derivative
Exit Ticket
A particle moves at a varying velocity along a line and s = f (t)represents the particle’s distance from a point as a function oftime, t. Sketch a possible graph for f if the average velocity of theparticle between t = 2 and t = 6 is the same as the instantaneousvelocity at t = 5.
Ethan Zell Introduction to the Derivative