Math 3B: Lecture 2 Noah White September 26, 2016 Last time Last - - PowerPoint PPT Presentation

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Math 3B: Lecture 2 Noah White September 26, 2016 Last time Last - - PowerPoint PPT Presentation

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Math 3B: Lecture 2 Noah White September 26, 2016 Last time Last time, we spoke about The syllabus Last time Last time, we spoke about The syllabus Problem sets, homework, and quizzes Last time Last time, we spoke about The

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Math 3B: Lecture 2

Noah White September 26, 2016

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SLIDE 2

Last time

Last time, we spoke about

  • The syllabus
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SLIDE 3

Last time

Last time, we spoke about

  • The syllabus
  • Problem sets, homework, and quizzes
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SLIDE 4

Last time

Last time, we spoke about

  • The syllabus
  • Problem sets, homework, and quizzes
  • Piazza
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SLIDE 5

Last time

Last time, we spoke about

  • The syllabus
  • Problem sets, homework, and quizzes
  • Piazza
  • Differentiation of common functions
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SLIDE 6

Last time

Last time, we spoke about

  • The syllabus
  • Problem sets, homework, and quizzes
  • Piazza
  • Differentiation of common functions
  • Product rule
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SLIDE 7

Last time

Last time, we spoke about

  • The syllabus
  • Problem sets, homework, and quizzes
  • Piazza
  • Differentiation of common functions
  • Product rule
  • Chain rule
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SLIDE 8

Quiz tomorrow

  • First quiz tomorrow and Thursday.
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SLIDE 9

Quiz tomorrow

  • First quiz tomorrow and Thursday.
  • Have a look at the graphing questions!
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SLIDE 10

Graphing using calculus: Why?

I think this is a common (and fair) question students have when they see this topic. Why learn to sketch graphs when we have computers that do it so well?

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SLIDE 11

Graphing using calculus: Why?

I think this is a common (and fair) question students have when they see this topic. Why learn to sketch graphs when we have computers that do it so well?

  • Building intuition
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SLIDE 12

Graphing using calculus: Why?

I think this is a common (and fair) question students have when they see this topic. Why learn to sketch graphs when we have computers that do it so well?

  • Building intuition
  • Understand functions qualitatively
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SLIDE 13

Graphing using calculus: Why?

I think this is a common (and fair) question students have when they see this topic. Why learn to sketch graphs when we have computers that do it so well?

  • Building intuition
  • Understand functions qualitatively
  • Better understanding of derivatives
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SLIDE 14

Building intuition

Here is an example where a good understanding of a functions behaviour is useful:

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SLIDE 15

Building intuition

Here is an example where a good understanding of a functions behaviour is useful: You are a biologist studying the population of a species of fish recently introduced to a new ecosystem. Currently they exist in small number, but you want to model their poputation over time. You know the population should increase quickly and then

  • stabalise. What functions would make good candidates for a

population model?

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SLIDE 16

Building intuition

Here is an example where a good understanding of a functions behaviour is useful: You are a biologist studying the population of a species of fish recently introduced to a new ecosystem. Currently they exist in small number, but you want to model their poputation over time. You know the population should increase quickly and then

  • stabalise. What functions would make good candidates for a

population model?

  • P(t) = t ln t

t−1

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SLIDE 17

Building intuition

Here is an example where a good understanding of a functions behaviour is useful: You are a biologist studying the population of a species of fish recently introduced to a new ecosystem. Currently they exist in small number, but you want to model their poputation over time. You know the population should increase quickly and then

  • stabalise. What functions would make good candidates for a

population model?

  • P(t) = t ln t

t−1

  • P(t) = Mt

t+1 for some large number M

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SLIDE 18

Building intuition

Here is an example where a good understanding of a functions behaviour is useful: You are a biologist studying the population of a species of fish recently introduced to a new ecosystem. Currently they exist in small number, but you want to model their poputation over time. You know the population should increase quickly and then

  • stabalise. What functions would make good candidates for a

population model?

  • P(t) = t ln t

t−1

  • P(t) = Mt

t+1 for some large number M

  • P(t) =

Mt t+et

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SLIDE 19

The ingredients

In order to sketch a function accurately we need a few ingredients

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SLIDE 20

The ingredients

In order to sketch a function accurately we need a few ingredients

  • The x and y intercepts
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SLIDE 21

The ingredients

In order to sketch a function accurately we need a few ingredients

  • The x and y intercepts
  • Horizontal asymptotes
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SLIDE 22

The ingredients

In order to sketch a function accurately we need a few ingredients

  • The x and y intercepts
  • Horizontal asymptotes
  • Vertical asymptotes
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SLIDE 23

The ingredients

In order to sketch a function accurately we need a few ingredients

  • The x and y intercepts
  • Horizontal asymptotes
  • Vertical asymptotes
  • Slanted asymptotes
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SLIDE 24

The ingredients

In order to sketch a function accurately we need a few ingredients

  • The x and y intercepts
  • Horizontal asymptotes
  • Vertical asymptotes
  • Slanted asymptotes
  • The regions of increase/decrease of the first derivative
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SLIDE 25

The ingredients

In order to sketch a function accurately we need a few ingredients

  • The x and y intercepts
  • Horizontal asymptotes
  • Vertical asymptotes
  • Slanted asymptotes
  • The regions of increase/decrease of the first derivative
  • The regions of increase/decrease of the second derivative
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SLIDE 26

Asymptotes

An asmptote is a line which the function approches. Some examples:

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SLIDE 27

Asymptotes

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SLIDE 28

Asymptotes

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SLIDE 29

Finding horizontal asymptotes

These are the easiest asymptotes to find. Suppose you have a function f (x)

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Finding horizontal asymptotes

These are the easiest asymptotes to find. Suppose you have a function f (x)

  • Calculate lim

x→∞ f (x)

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Finding horizontal asymptotes

These are the easiest asymptotes to find. Suppose you have a function f (x)

  • Calculate lim

x→∞ f (x)

  • Calculate

lim

x→−∞ f (x)

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Finding horizontal asymptotes

These are the easiest asymptotes to find. Suppose you have a function f (x)

  • Calculate lim

x→∞ f (x)

  • Calculate

lim

x→−∞ f (x)

Example

Say f (x) = x−1

1+x . In this case

lim

x→±∞

x − 1 x + 1 = 1

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Finding horizontal asymptotes

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Finding verticle asymptotes

Verticle asymptotes happen when a function "blows up", or goes to infinity as it approches a finite number. I.e. Is there a real number a so that lim

x→a+ f (x) = ±∞

  • r

lim

x→a− f (x) = ±∞

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SLIDE 35

Finding verticle asymptotes

Verticle asymptotes happen when a function "blows up", or goes to infinity as it approches a finite number. I.e. Is there a real number a so that lim

x→a+ f (x) = ±∞

  • r

lim

x→a− f (x) = ±∞

Example

f (x) = x−1

1+x , we have

lim

x→−1+

x − 1 1 + x = −∞ and lim

x→−1−

x − 1 1 + x = ∞

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SLIDE 36

Finding verticle asymptotes

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SLIDE 37

Finding slanted asymptotes

Lets come back to this. . .

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SLIDE 38

The first derivative

The first derivative tells us is the function going up or down? x y f ′(x) + − +

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SLIDE 39

The second derivative

The second derivative tells us is the function concave up or down? x y f ′′(x) + +

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SLIDE 40

Example time

. . . On the board.