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