Math 114 Calculus I - University of Wisconsin–Eau Clairepeople.uwec.edu/daviscw/oldClasses/math114Fall2015/...Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4

Math 114Calculus I

Instructor Christopher Davis

September 1, 2015

Instructor Christopher Davis Math 114 Calculus I September 1, 2015 1 / 15

Who am I?

Christopher Davis,Office: 533 Hibbard Humanities Hall

eMail: [email protected] hours: Monday, Tuesday,

Wednesday, Friday 3 - 4 PM(also by appointment,

but please give me warning)

I prefer the film Alien overthe film Predator.


Who am I?

Christopher Davis,Office: 533 Hibbard Humanities Hall

eMail: [email protected] hours: Monday, Tuesday,

Wednesday, Friday 3 - 4 PM(also by appointment,

but please give me warning)

I prefer the film Alien overthe film Predator.


Syllabus:


What is Calculus?

Any opinions?

At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)

1−.5 = (20−5)−(20−1.25).5 = −3.75

.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0

0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?

Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.

For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)

1−.5 = (20−5)−(20−1.25).5 = −3.75




What is Calculus?

Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?

p(1)−p(.5)1−.5 = (20−5)−(20−1.25)

.5 = −3.75.5 = −7.5

From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0



What is Calculus?

Any opinions?At its heart, Calculus (All three classes in the series and beyond) is whatyou get when you take approximations and make them arbitrarily precise.For example, suppose that you drop a ball off a 40 meter tall building.The flight of the ball is modeled by p(t) = 20 − 5t2. What is the averagerate of change of the ball from t0 = .5 to t1 = 1?p(1)−p(.5)

1−.5 =

(20−5)−(20−1.25).5 = −3.75




What is Calculus?


1−.5 = (20−5)−(20−1.25).5 =

−3.75.5 = −7.5




What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75

.5 = −7.5




What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75

.5 = −7.5From t0 = .9 to t1 = 1?

−9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0



What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75

.5 = −7.5From t0 = .9 to t1 = 1? −9.5

From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1? 0



What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75

.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1?

−9.95Trick question From t0 = 1 to t1 = 1? 0



What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75

.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95

Trick question From t0 = 1 to t1 = 1? 00 makes no sense. The average

velocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75

.5 = −7.5From t0 = .9 to t1 = 1? −9.5 From t0 = .99 to t1 = 1? −9.95Trick question From t0 = 1 to t1 = 1?

00 makes no sense. The average

velocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75


0 makes no sense.

The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75


0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.

Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75


0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?

Think about the three computations we just did. What do these numbersseem to get close to? 10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75


0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to?

10.The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75


0 makes no sense. The averagevelocity from t0 = 1 to t1 = 1 is undefined.Doing calculus: If you just take t0 really close to 1 then what does it looklike the average velocity gets close to?Think about the three computations we just did. What do these numbersseem to get close to? 10.

The instantaneous velocity is the Limit of the average veclocity. It is 10meters per second when t = 1 second.


What is Calculus?


1−.5 = (20−5)−(20−1.25).5 = −3.75




Limits

The core of calculus is the concept of the Limit. I will introduce it viaanother example.

Consider the function given by the for-mula

f (h) =1 − h2

1 − h

If you take h = 1 The formula becomes 00

which does not make any sense.Here is the graph of the function.

If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.

We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim

h→1f (h) = 2.


Limits

The core of calculus is the concept of the Limit. I will introduce it viaanother example.Consider the function given by the for-mula

f (h) =1 − h2

1 − h





h→1f (h) = 2.


Limits


f (h) =1 − h2

1 − h


which does not make any sense.

Here is the graph of the function.



h→1f (h) = 2.


Limits


f (h) =1 − h2

1 − h



If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim

h→1f (h) = 2.


Limits


f (h) =1 − h2

1 − h


which does not make any sense.Here is the graph of the function.If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?

Answer 2.We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim

h→1f (h) = 2.


Limits


f (h) =1 − h2

1 − h


which does not make any sense.Here is the graph of the function.If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.


h→1f (h) = 2.


Limits


f (h) =1 − h2

1 − h


which does not make any sense.Here is the graph of the function.If you take h really close to (but not quiteequal to) 1, then what is f (h) close to?Answer 2.We say that the limit of f (h) as h approaches 1 is 2. As a shorthand wewrite lim

h→1f (h) = 2.


Informal definition of limits.

Definition

We say that the limit of f (x) as x approaches a is L (abbreviatedlimx→a f (x) = L) if by taking x close to a but not equal to a we canguarantee that f (x) is as close as we like to L.

Notice that the definition of the limit does not care what value thefunction takes at x = a.Example:

Here is the graph of f (x) = x3−8x−2 − 8.

When x is close to 2 what is f (x) closeto?

What is limx→2

x3 − 8

x − 2− 8?

limx→2

x3 − 8

x − 2− 8 = 4.



Definition


Notice that the definition of the limit does not care what value thefunction takes at x = a.

Example:



What is limx→2

x3 − 8

x − 2− 8?

limx→2

x3 − 8

x − 2− 8 = 4.



Definition





What is limx→2

x3 − 8

x − 2− 8?

limx→2

x3 − 8

x − 2− 8 = 4.



Definition





What is limx→2

x3 − 8

x − 2− 8?

limx→2

x3 − 8

x − 2− 8 = 4.


More limits from graphs: How close to x need to be to a?

(1,0) 1.9 2.1(1,0)

y=3

y=5Here is the graph of f (x) = x3−8

x−2 − 8, againjust zoomed in some.

If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.



(1,0) 1.9 2.1(1,0)

y=3


x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?

Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.



(1,0) 1.9 2.1(1,0)

y=3


x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.

You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.



(1,0) 1.9 2.1(1,0)

y=3


x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.

The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.



(1,0) 1.9 2.1(1,0)

y=3


x−2 − 8, againjust zoomed in some.If x is within .1 of 2 (between 1.9 and 2.1),then according to this graph how close do youknow that f (x) is to 4?Answer: It looks like 3 < f (x) < 5. f (x) iswithin 1 of 4.You can guarantee that f (x) is within 1 of 4by insisting that x be within .1 of 2.The moral: The graph of f (x) does morethan tell you about limits. It tells you howx has to be to a for f (x) to be close to thelimit.


x=.5 x=1.5

y=2

y=4

x=.75 x=1.25

y=2.5

y=3.5

Get a guess for the value of limx→1

2x + 1. (plotted above)

Using the picture above, how close to 1 does x need to be if you want2x + 1 to be within 1 of 3?Within .5 of 3?


x=.5 x=1.5

y=2

y=4

x=.75 x=1.25

y=2.5

y=3.5



Using the picture above, how close to 1 does x need to be if you want2x + 1 to be within 1 of 3?

Within .5 of 3?


x=.5 x=1.5

y=2

y=4

x=.75 x=1.25

y=2.5

y=3.5



Using the picture above, how close to 1 does x need to be if you want2x + 1 to be within 1 of 3?Within .5 of 3?


Getting guesses for limits from tables

Let’s determine the limit of f (x) = x3−8x−2 − 8 as x approaches 2 again.

Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.

Let’s fill in this table.x 1 1.9 1.99 1.999 1.9999

x3−8x−2 − 8

-1 3.41 3.94 3.99 3.999

What number does it look like these values are approaching?

At this point we guess that limx→2

x3 − 8

x − 2− 8 = 4.




Here is our strategy, we will take choices of x which get close to 2 (butdon’t equal 2) and evaluate f (x) at these inputs.Let’s fill in this table.

x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8

-1 3.41 3.94 3.99 3.999What number does it look like these values are approaching?


x3 − 8

x − 2− 8 = 4.





x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1

3.41 3.94 3.99 3.999What number does it look like these values are approaching?


x3 − 8

x − 2− 8 = 4.





x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41

3.94 3.99 3.999What number does it look like these values are approaching?


x3 − 8

x − 2− 8 = 4.





x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94

3.99 3.999What number does it look like these values are approaching?


x3 − 8

x − 2− 8 = 4.





x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99

3.999What number does it look like these values are approaching?


x3 − 8

x − 2− 8 = 4.





x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99 3.999



x3 − 8

x − 2− 8 = 4.





x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99 3.999



x3 − 8

x − 2− 8 = 4.





x 1 1.9 1.99 1.999 1.9999x3−8x−2 − 8 -1 3.41 3.94 3.99 3.999



x3 − 8

x − 2− 8 = 4.


For You:

Pick some sample points getting close to 1 and see what 2x + 1 gets closeto for your sample points.What do you predict that limx→1 2x + 1 is?


Building more limit intuition (a preview of limit laws.)

Ready to think in the abstract?

Definition


Last page we guessed that limx→1

2x + 1 = 3.

Let’s try to justify this.

Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.




Definition



2x + 1 = 3.

Let’s try to justify this.Suppose that x is really close to 1.

If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.




Definition



2x + 1 = 3.

Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get?

Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.




Definition



2x + 1 = 3.

Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 2

2x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.




Definition



2x + 1 = 3.

Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2

What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.




Definition



2x + 1 = 3.

Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?

2x + 1 is really close to 3.




Definition



2x + 1 = 3.

Let’s try to justify this.Suppose that x is really close to 1.If we double a number that’s really close to 1 then what do we get? Anumber really close to 22x is really close to 2What happens if we add 1 to a number really close to 2?2x + 1 is really close to 3.


Compute one more limit in this intuitive way.

Let’s compute limx→2 2x2 + 3x − 1.Let’s let x be really close to 2Complete these questions.

1 If you square a number (x) close to 2 what do you get?

2 If you double a number close to (The result of 1) then what doyou get?

3 If you multiply a number close to 2 by 3 what do you get?

4 If you start with a number close to (The result of 2), add(The result of 3) and then subtract 1 what do you get?

What is limx→2 2x2 + 3x − 1?


A function without a limit

Here is the graph of the sign function:There are typo’s on this page of notes.

sign(x) =

−1 x < 00 x = 0+1 x > 0

No matter how close to x = 0 you look, sign(x) is+1 and −1 in that neighborhood.

There is no number which is super close to each of −1 and +1.There is no limit.If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)




sign(x) =

−1 x < 00 x = 0+1 x > 0


There is no number which is super close to each of −1 and +1.

There is no limit.If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)




sign(x) =

−1 x < 00 x = 0+1 x > 0


There is no number which is super close to each of −1 and +1.There is no limit.

If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)




sign(x) =

−1 x < 00 x = 0+1 x > 0


There is no number which is super close to each of −1 and +1.There is no limit.If the function approaches different things from the left and the right thenthere is no limit.

The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)




sign(x) =

−1 x < 00 x = 0+1 x > 0


There is no number which is super close to each of −1 and +1.There is no limit.If the function approaches different things from the left and the right thenthere is no limit.The moral: If the graph of a function jumps, then the function has nolimit (Later: one-sided-limits)


A terrible function

Consider the function f (x) = cos(π/x). Let’s try to compute limx→0 f (x).Fill in this table Together on the board

x 1/10 1/100 1/1000 1/10000

cos(π/x)

1 1 1 1What does it look like limx→0 cos(π/x) is?

This table predicts that it is 1

x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1

This table thinks that the limit is −1.What gives? The limit can’t be both.

Here’s a graph of cos(π/x).

It wiggles too muchsuper wiggly functions don’t have limits!


A terrible function


x 1/10 1/100 1/1000 1/10000

cos(π/x) 1

1 1 1What does it look like limx→0 cos(π/x) is?


x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1





A terrible function


x 1/10 1/100 1/1000 1/10000

cos(π/x) 1 1

1 1What does it look like limx→0 cos(π/x) is?


x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1





A terrible function


x 1/10 1/100 1/1000 1/10000

cos(π/x) 1 1 1

1What does it look like limx→0 cos(π/x) is?


x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1





A terrible function


x 1/10 1/100 1/1000 1/10000

cos(π/x) 1 1 1 1

What does it look like limx→0 cos(π/x) is?


x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1





A terrible function


x 1/10 1/100 1/1000 1/10000

cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?

This table predicts that it is 1x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1





A terrible function


x 1/10 1/100 1/1000 1/10000

cos(π/x) 1 1 1 1What does it look like limx→0 cos(π/x) is?This table predicts that it is 1

x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1





A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001

cos(π/x)

-1 -1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.




A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001

cos(π/x) -1

-1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.




A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001

cos(π/x) -1 -1

-1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.




A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001

cos(π/x) -1 -1 -1

-1This table thinks that the limit is −1.What gives? The limit can’t be both.




A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001

cos(π/x) -1 -1 -1 -1





A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001

cos(π/x) -1 -1 -1 -1This table thinks that the limit is −1.What gives? The limit can’t be both.




A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001





A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001


Here’s a graph of cos(π/x).It wiggles too much

super wiggly functions don’t have limits!


A terrible function


x 1/10 1/100 1/1000 1/10000


x 1/11 1/101 1/1001 1/10001


Here’s a graph of cos(π/x).It wiggles too muchsuper wiggly functions don’t have limits!


preview

Friday we’ll deduce some rules which make it easier to compute limits offunctions which come from polynomials.See you then!


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Math 114 Calculus I - University of Wisconsin–Eau Clairepeople.uwec.edu/daviscw/oldClasses/math114Fall2015/...Instructor Christopher Davis Math 114 Calculus I September 1, 2015 4