Important websites:Course website: Notes, written and reading assignments, etc.

zdaugherty.ccnysites.cuny.edu/teaching/m201f18/My Math Lab (MML): Online assignments.

www.pearson.com/mylab(See main course website for instructions.)

Upcoming deadlines:Due Sunday 9/2

∗ From MML: Orientation AssignmentDue Tuesday 9/4

∗ From MML: Section 1.1, Section 1.2∗ From course website: Homework 0 email

Due Thursday 9/6∗ From MML: Section 1.3, Section 1.5∗ From course website: summaries

First quiz: In class, Tuesday 9/4.

Trigonometric functions, step one: similar triangles

θa

b

c

θC

B

A

Two similar triangles have the same setof angles, and have the properties that

A

B=a

b,

B

C=b

c, and

A

C=a

c.

Define

cos(θ) =b

cand sin(θ) =

a

c.

Then let

tan(θ) =sin(θ)

cos(θ)=a

b, csc(θ) =

1

sin(θ)=c

a,

sec(θ) =1

cos(θ)=c

b, cot(θ) =

1

tan(θ)=b

a.

Trigonometric functions, step one: similar triangles

θa

b

c

θC

B

A

Two similar triangles have the same setof angles, and have the properties that

A

B=a

b,

B

C=b

c, and

A

C=a

c.

Define

cos(θ) =b

cand sin(θ) =

a

c.

Then let

tan(θ) =sin(θ)

cos(θ)=a

b, csc(θ) =

1

sin(θ)=c

a,

sec(θ) =1

cos(θ)=c

b, cot(θ) =

1

tan(θ)=b

a.

Trigonometric functions, step one: similar triangles

θa

b

c

θC

B

A

Two similar triangles have the same setof angles, and have the properties that

A

B=a

b,

B

C=b

c, and

A

C=a

c.

Define

cos(θ) =b

cand sin(θ) =

a

c.

Then let

tan(θ) =sin(θ)

cos(θ)=a

b, csc(θ) =

1

sin(θ)=c

a,

sec(θ) =1

cos(θ)=c

b, cot(θ) =

1

tan(θ)=b

a.

Easy angles:

isosceles right triangle: equilateral triangle cut in half:

√2

1

1

π/4

1

1/2

1

π/3

π/6

h

h =√1− (1/2)2 =

√3/2

cos(θ) sin(θ) tan(θ) sec(θ) csc(θ) cot(θ)

π/4

1√2

1√2

1√2

√2 1

π/3

1

2

√3

2

√3 2

2√3

1√3

π/6

√3

2

1

2

1√3

2√3

2√3

Easy angles:

isosceles right triangle: equilateral triangle cut in half:

√2

1

1

π/4

1

1/2

1

π/3

π/6

h

h =√1− (1/2)2 =

√3/2

cos(θ) sin(θ) tan(θ) sec(θ) csc(θ) cot(θ)

π/41√2

1√2

1√2

√2 1

π/31

2

√3

2

√3 2

2√3

1√3

π/6

√3

2

1

2

1√3

2√3

2√3

Step two: the unit circle

-1 1

-1

1

θ

(x , y)For 0 < θ < π

2 ...

cos(θ) =x

1= x

sin(θ) =y

1= y

Use this idea to extend trig functions to any θ...

Step two: the unit circle

-1 1

-1

1

θ

(x , y)

y

x

1

For 0 < θ < π2 ...

cos(θ) =x

1= x

sin(θ) =y

1= y

Use this idea to extend trig functions to any θ...

Step two: the unit circle

-1 1

-1

1

θ

(x , y)

y

x

1

For 0 < θ < π2 ...

cos(θ) =x

1= x

sin(θ) =y

1= y

Use this idea to extend trig functions to any θ...

Definecos(θ) = x sin(θ) = y,

where θ is defined by...

0 ≤ θ ≤ 2π all θ ≥ 0 θ < 0

-1 1

-1

1

θ

(x , y)

-1 1

-1

1

θ

(x , y)

-1 1

-1

1

θ

(x , y)

Sidebar: In calculus, radians are king. Where do they come from?Circumference of a unit circle: 2π

Arclength of a wedge with angle θ:θ

360◦ ∗ 2π (if in degrees) or θ2π ∗ 2π = θ (if in radians)

Definecos(θ) = x sin(θ) = y,

where θ is defined by...

0 ≤ θ ≤ 2π all θ ≥ 0 θ < 0

-1 1

-1

1

θ

(x , y)

-1 1

-1

1

θ

(x , y)

-1 1

-1

1

θ

(x , y)

Sidebar: In calculus, radians are king. Where do they come from?Circumference of a unit circle: 2π

Arclength of a wedge with angle θ:θ

360◦ ∗ 2π (if in degrees) or θ2π ∗ 2π = θ (if in radians)

Reading off of the unit circle

-1 1

-1

1

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ)

1 0 −1 0√32

1√2

12

sin(θ)

0 1 0 −1 12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ)

-12 - 1√2

-√32 -

√32 - 1√

2-12

12

1√2

√32

sin(θ)

√32

1√2

12 -12 - 1√

2-√32 -

√32 - 1√

2-12

Reading off of the unit circle

-1 1

-1

1 π/2

π

3π/2

0

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ) 1 0 −1 0

√32

1√2

12

sin(θ) 0 1 0 −1

12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ)

-12 - 1√2

-√32 -

√32 - 1√

2-12

12

1√2

√32

sin(θ)

√32

1√2

12 -12 - 1√

2-√32 -

√32 - 1√

2-12

Reading off of the unit circle

-1 1

-1

1

π/4

π/3

π/6

π/2

π

3π/2

0

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ) 1 0 −1 0√32

1√2

12

sin(θ) 0 1 0 −1 12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ)

-12 - 1√2

-√32 -

√32 - 1√

2-12

12

1√2

√32

sin(θ)

√32

1√2

12 -12 - 1√

2-√32 -

√32 - 1√

2-12

Reading off of the unit circle

-1 1

-1

1

π/4

π/3

π/6

π/2

π

3π/2

0θθ

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ) 1 0 −1 0√32

1√2

12

sin(θ) 0 1 0 −1 12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ)

-12 - 1√2

-√32 -

√32 - 1√

2-12

12

1√2

√32

sin(θ)

√32

1√2

12 -12 - 1√

2-√32 -

√32 - 1√

2-12

Reading off of the unit circle

-1 1

-1

1

π/4

π/3

π/6

π/2

π

3π/2

0θθ

5π/6

3π/42π/3

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ) 1 0 −1 0√32

1√2

12

sin(θ) 0 1 0 −1 12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ) -12 - 1√2

-√32

-√32 - 1√

2-12

12

1√2

√32

sin(θ)√32

1√2

12

-12 - 1√2

-√32 -

√32 - 1√

2-12

Reading off of the unit circle

-1 1

-1

1

π/4

π/3

π/6

π/2

π

3π/2

0θ-θ

5π/6

3π/42π/3

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ) 1 0 −1 0√32

1√2

12

sin(θ) 0 1 0 −1 12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ) -12 - 1√2

-√32

-√32 - 1√

2-12

12

1√2

√32

sin(θ)√32

1√2

12

-12 - 1√2

-√32 -

√32 - 1√

2-12

Reading off of the unit circle

-1 1

-1

1

π/4

π/3

π/6

π/2

π

3π/2

0θ-θ

5π/6

3π/42π/3

7π/6

5π/44π/3

5π/3

11π/67π/4

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ) 1 0 −1 0√32

1√2

12

sin(θ) 0 1 0 −1 12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ) -12 - 1√2

-√32 -

√32 - 1√

2-12

12

1√2

√32

sin(θ)√32

1√2

12 -12 - 1√

2-√32 -

√32 - 1√

2-12

Reading off of the unit circle

-1 1

-1

1

π/4

π/3

π/6

π/2

π

3π/2

0

5π/6

3π/42π/3

7π/6

5π/44π/3

5π/3

11π/67π/4

cos(π − θ) = − cos(θ) sin(π − θ) = sin(θ)

cos(−θ) = cos(θ) sin(−θ) = − sin(θ)

cos(2πn+ θ) = cos(θ) sin(2πn+ θ) = sin(θ)

x2 + y2 = 1 =⇒ cos2(θ) + sin2(θ) = 1

0 π2 π 3π

2π6

π4

π3

cos(θ) 1 0 −1 0√32

1√2

12

sin(θ) 0 1 0 −1 12

1√2

√32

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6

cos(θ) -12 - 1√2

-√32 -

√32 - 1√

2-12

12

1√2

√32

sin(θ)√32

1√2

12 -12 - 1√

2-√32 -

√32 - 1√

2-12

Plotting on the θ-y axis

Graph of y = cos(θ):

-1

1

π

2π

-π

-2π

A = Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

Graph of y = sin(θ):

-1

1

π

2π

-π

-2π

A=Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

You try: Transform the graph of sin(θ) into the graph of2 sin(12θ + π/6)− 1, one step at a time. (See notes)What is the amplitude of 2 sin

(12θ +

π6

)− 1? What is the period?

Plotting on the θ-y axis

Graph of y = cos(θ):

-1

1

π

2π

-π

-2π

A = Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

Graph of y = sin(θ):

-1

1

π

2π

-π

-2π

A=Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

You try: Transform the graph of sin(θ) into the graph of2 sin(12θ + π/6)− 1, one step at a time. (See notes)What is the amplitude of 2 sin

(12θ +

π6

)− 1? What is the period?

Plotting on the θ-y axis

Graph of y = cos(θ):

-1

1

π

2π

-π

-2π

A

A = Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

Graph of y = sin(θ):

-1

1

π

2π

-π

-2π

A

A=Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

You try: Transform the graph of sin(θ) into the graph of2 sin(12θ + π/6)− 1, one step at a time. (See notes)What is the amplitude of 2 sin

(12θ +

π6

)− 1? What is the period?

Plotting on the θ-y axis

Graph of y = cos(θ):

-1

1

π

2π

-π

-2πT

A = Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

Graph of y = sin(θ):

-1

1

π

2π

-π

-2πT

A=Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

You try: Transform the graph of sin(θ) into the graph of2 sin(12θ + π/6)− 1, one step at a time. (See notes)What is the amplitude of 2 sin

(12θ +

π6

)− 1? What is the period?

Plotting on the θ-y axis

Graph of y = cos(θ):

-1

1

π

2π

-π

-2πT

A = Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

Graph of y = sin(θ):

-1

1

π

2π

-π

-2πT

A=Amplitude = 12 length of the range = 1

T=Period = time to repeat = 2π

You try: Transform the graph of sin(θ) into the graph of2 sin(12θ + π/6)− 1, one step at a time. (See notes)What is the amplitude of 2 sin

(12θ +

π6

)− 1? What is the period?

Trig identities to know and love:

Even/odd:

cos(−θ) = cos(θ) (even) sin(−θ) = − sin(θ) (odd)

Pythagorean identity:

cos2(θ) + sin2(θ) = 1

Angle addition:

cos(θ + φ) = cos(θ) cos(φ)− sin(θ) sin(φ)

sin(θ + φ) = sin(θ) cos(φ) + cos(θ) sin(φ)

(in particular cos(2θ) = cos2(θ)− sin2(θ) and sin(2θ) = 2 sin(θ) cos(θ))

Other trig functions

y = cos(θ) y = sin(θ)

-2

-1

1

2

π-π

-2

-1

1

2

π-π

sec(θ) = 1/ cos(θ) csc(θ) = 1/ sin(θ)

-2

-1

1

2

π-π

-2

-1

1

2

π-π

Other trig functions

y = cos(θ) y = sin(θ)

-2

-1

1

2

π-π

-2

-1

1

2

π-π

tan(θ) = sin(θ)/ cos(θ) cot(θ) = cos(θ)/ sin(θ)

-2

-1

1

2

π-π

-2

-1

1

2

π-π

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 225 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m (an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2

25 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m (an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 225 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m (an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 225 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m

(an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 225 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m

(an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 225 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m (an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 225 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m (an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Exponential functions

The basics: Let n and m be positive integers, and a be a real number.

an = a · a · · · · · a︸ ︷︷ ︸n

(MML: a∧n)

Examples:

25 = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 225 ∗ 23 = (2 ∗ 2 ∗ 2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 28

(23)5 = (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) ∗ (2 ∗ 2 ∗ 2) = 215

235

= 2243 >> (23)5

= 215

23 ∗ 53 = (2 ∗ 2 ∗ 2) ∗ (5 ∗ 5 ∗ 5) = (2 ∗ 5) ∗ (2 ∗ 5) ∗ (2 ∗ 5) = (2 ∗ 5)3

Some identities:an ∗ am = an+m (an)m = an∗m

(Notice: amn

means a(mn), since (am)

ncan be written another way)

an ∗ bn = (a ∗ b)n

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0

, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1

, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a

, so a1/n = n√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0

, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1

, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a

, so a1/n = n√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1

, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a

, so a1/n = n√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1

, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a

, so a1/n = n√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a

, so a1/n = n√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a

, so a1/n = n√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a, so a1/n = n

√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

Pushing it further...

Take for granted: If n and m are positive integers,

an = a · a · · · · · a︸ ︷︷ ︸n

, an ∗ am = an+m, (an)m = an∗m.

Notice:

1. What is a0?an = an+0 = an ∗ a0, so a0 = 1 .

2. What is ax if x is negative?

an ∗ a−n = an−n = a0 = 1, so a−n = 1/(an) .

3. What is ax if x is a fraction?

(an)1/n = an∗1n = a1 = a, so a1/n = n

√a

and am/n = n√am = ( n

√a)m

.

Example: 85/3 =(

3√8)5

= 25 = 32 or 85/3 =3√85 = 3

√32,768 = 32

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

x = 1, 2, 3, . . .

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

x = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

x = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

x = n/2, for n = 0,±1,±2,±3, . . .

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

x = n/2 and n/3, for n = 0,±1,±2,±3, . . .

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

x = n/2, n/3, . . . , n/15, for n = 0,±1,±2,±3, . . .

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

x = n/2, n/3, . . . , n/100, for n = 0,±1,±2,±3, . . .

What is ax for all x?

If a > 1:

(e.g. a = 2)

-3 -2 -1 1 2 3

2

4

6

8

y = ax

What is ax for all x?

If 0 < a < 1:

(e.g. a = 12)

-3 -2 -1 1 2 3

2

4

6

8

x = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

What is ax for all x?

If 0 < a < 1:

(e.g. a = 12)

-3 -2 -1 1 2 3

2

4

6

8

x = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

What is ax for all x?

If 0 < a < 1:

(e.g. a = 12)

-3 -2 -1 1 2 3

2

4

6

8

x = n/2, n/3, n/4, n/5, for n = 0,±1,±2,±3, . . .

What is ax for all x?

If 0 < a < 1:

(e.g. a = 12)

-3 -2 -1 1 2 3

2

4

6

8

x = n/2, n/3, . . . , n/100, for n = 0,±1,±2,±3, . . .

What is ax for all x?

If 0 < a < 1:

(e.g. a = 12)

-3 -2 -1 1 2 3

2

4

6

8

y = ax

What is ax for all x?

If 0 > a:

(e.g. a = −2)

-3 -2 -1 1 2 3

-8

-6

-4

-2

2

4

6

8

x = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

What is ax for all x?

If 0 > a:

(e.g. a = −2)

-3 -2 -1 1 2 3

-8

-6

-4

-2

2

4

6

8

x = n/3, for n = 0,±1,±2,±3, . . .

What is ax for all x?

If 0 > a:

(e.g. a = −2)

-3 -2 -1 1 2 3

-8

-6

-4

-2

2

4

6

8

x = n/3 and n/2, for n = 0,±1,±2,±3, . . .

What is ax for all x?If 0 > a:

(e.g. a = −2)

-3 -2 -1 1 2 3

-8

-6

-4

-2

2

4

6

8

x = n/2, n/3, . . . , n/100, for n = 0,±1,±2,±3, . . .OH NO!

The function ax:

1 < a: 0 < a < 1:

11

D: (−∞,∞), R: (0,∞) D: (−∞,∞), R: (0,∞)

a = 1: a = 0:

1

D: (−∞,∞), R: {1} D: (0,∞), R: {0}

(If a < 0, then ax is not defined as a function on the real numbers.)

Properties:

ab ∗ ac = ab+c (ab)c = ab∗c a−x = 1/ax ac ∗ bc = (ab)c

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

a=1.1

a=1.5

a=2a=3

a=10

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

a=1.1

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

a=1.5

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

a=2

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :a=3

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

a=10

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

Q: Is there an exponential function whose slope at (0,1) is 1?

A: ex is the exponential function whose slope at (0,1) is 1.(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

a=2

a=3

Q: Is there an exponential function whose slope at (0,1) is 1?

A: ex is the exponential function whose slope at (0,1) is 1.(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

a=2.71828183...

Q: Is there an exponential function whose slope at (0,1) is 1?

A: ex is the exponential function whose slope at (0,1) is 1.(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

e=2.71828183...

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.

Our favorite exponential function:Look at how the function is increasing through the point (0, 1):

y = ax :

e=2.71828183...

Q: Is there an exponential function whose slope at (0,1) is 1?A: ex is the exponential function whose slope at (0,1) is 1.

(e = 2.71828183 . . . is to calculus as π = 3.14159265 . . . is to geometry)

Read: “Exponential growth and decay”, examples 3 and 4.