Finite Differences and Exponential/Logarithmic Functionsmrjtaylor.weebly.com/uploads/3/9/7/2/39723380/mhf_4ui_unit_8... · Clearly label all asymptotes. a) fx 2 1 x ... Determine

MHF 4UI Unit 8 Day 1

Finite Differences and Exponential/Logarithmic Functions

1. Finite Differences

a) b)

________________________________

________________________________ __________________________________________

________________________________ __________________________________________

________________________________ __________________________________________

c)

______________________________________

______________________________________

______________________________________

______________________________________

d)

_____________________________

_____________________________

_____________________________

_____________________________

x x y 1st

Diff

2nd

Diff

-2 -17 -1 -9

0 -3

1 1

2 3

x x y 1st

Diff

2nd

Diff

3rd

Diff

-3 -5

-2 0

-1 -3

0 -8

1 -9

2 0

x x y 1st

Diff

2nd

Diff

3rd

Diff

4th

Diff

-4 225

-3 80

-2 27

-1 12

0 5

1 0

2 15

x x y 1st

Diff

2nd

Diff

3rd

Diff

4th

Diff

5th

Diff

-5 4608

-4 1372

-3 216

-2 0

-1 64

0 108

1 72

2 16

1

1 1 1

8

6

4

2

-2

-2

-2

1

1 1 1

5

-3

-5

-1

-8

-2

4

1

9

10

6

6

6

1

2080

940

1

-36

-3236

-20

-80

1

1

-1156

-216

64

280

1

44

1

1

-1140

1

-56

-20

-660

-300

-60

60

480

360

240

120

-120

-120

-120

1

1

1

-53

-15

-7

92

38

8

1

-5

-30

-6

1

1

15

-145

2

20

1

-54

18

24

24

24

x

y

2. Key Features of Exponential Functions

Sketch the graph of the function in the grid provided. Determine the key properties of the

function. Clearly label the key properties on the graph.

1023 1

xy

Describe the transformations to the

base function xy 2

Domain ______________________

Range _______________________

Increasing or decreasing function?

End behaviours:

For x-int , let y = 0 (state an exact answer, then state answer

accurate to one decimal place)

For y-int , let x = 0

Finite differences:

x x y y Ratio

-2

-1

0

1

2

3

x

y

3. Key Features of Logarithmic Functions

Sketch the graph of the function in the grid provided. Determine the key properties of the


12

y 2log 4x 8 5

Describe the transformations to the

base function 12

y log x

Domain ______________________

Range _______________________

Increasing or decreasing function?

End behaviours:

For x-int , ______________ (state an exact answer, then state answer

accurate to one decimal place)

For y-int , _______________

Finite differences:

Ratio x x y y

-1.75

-1.5

-1

0

2


Sketch 32

12

xx

xf

using BIG/LITTLE CONCEPT

x

y

Key Features of Rational Functions

4. Sketch the graph of each of the functions in the grids provided. Determine the key properties

of each function. Clearly label all asymptotes.

a) 2

1f x

x 2x 3

The sketch of 2y x 2x 3 is given

on the grid at the right.

With 2

1f x

x 2x 3

:

For x-int, ____________

For y-int, _____________

Vertical asymptote(s) ___________________

Horizontal asymptote ___________________

Linear oblique asymptote _________________

Domain ______________________________

Pos. Intervals__________________________

Neg. Intervals_________________________

Incr. Intervals_________________________

Decr. Intervals________________________

x

y4. b) 2

2

2x 2x 12f x

x 6x 9

For y-int, ____________

For x-int, _____________

Vertical asymptote(s) ___________________

Horizontal asymptote ___________________

Linear oblique asymptote _________________

Domain ______________________________

Range _______________________________

Pos. Intervals__________________________

Neg. Intervals_________________________

Incr. Intervals_________________________

Decr. Intervals________________________

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

x

y

4. c) 3 2

2

x 3x 6x 8f x

x x 12

Vertical asymptote(s) __________________

Horizontal asymptote __________________

Linear oblique asymptote

____________________

For y-int, ____________

For x-int, _____________

Domain ______________________________


Key Features of Trigonometric Functions

5. Given g x 6sin 2x 3

a) Determine amplitude, period, vertical shift and phase shift.

b) Determine domain and range. c) Determine the y-intercept.

d) Determine the x-intercept(s) for one period.

e) Determine all zeroes.

6. Sketch the graph of the function in the grid provided. Determine the key properties of the


a) f x cos 3 x 14

vert. shift _____________

period_________________

phase shift _____________

amplitude _______

min value ____ max value ____

Range ___________________

Domain __________________

For y-int, ______________

For x-ints for one period, ____________

For all x-ints, __________________________________________________

6. b) f x 2sin x 16

vert. shift _____________

period______

phase shift _____________

amplitude _______

min value ____ max value ____

Range ___________________

Domain __________________

For y-int, ______________

For x-ints for one period, ____________

For all x-ints, __________________________________________________


Applications of Trigonometric Functions

7. A small windmill has its centre 6 m above the ground and blades 2 m long. In a steady wind,

a point P at the tip of one blade makes a complete revolution in 12 seconds.

a) Determine a function that gives the height of P above the ground at any time t.

Assume the rotation starts at the highest possible point.

b) When is the blade 7 m above the ground, during the first revolution?


8. In the Bay of Fundy, the water around the harbour changes from 1.5 m at low tide at 02:00

h to 15.5 m at high tide at 08:00 h.

a) If the tidal cycle is sinusoidal, determine a function to represent the depth of the

water in the harbour.

b) It is safe to enter the harbour when there is at least 3.5 m of water. During what

times is it safe to enter the harbour, over a 24-hour period?


Sum and Difference of Functions

If f and g are functions,

the sum function f g x is defined by: ___________________________________

the difference function f g x is defined by: ______________________________

The domain of f g and f g is the set of all real numbers that are in the domain of both f and

g.

1. Given f x x 2 and 2

g x 9 x .

a) State the domain of f. b) State the domain of g.

c) Find f g x and state the domain. d) Find f g x and state the domain.

e) Evaluate f g 1 . State the exact answer, than state the answer accurate to two

decimal places.


2. Given 4

f x log x 1 and 2

4g x log x 9 .


c) Find g f x and state the domain.

d) Find f g x and state the domain.


3. Given x 1

f x 2 3

and 2x 3

g x 4 3

.


c) Find f g x and state the domain.

4. Given f x sec x6

and g x cot x

6

.

a) State the domain of f.

b) State the domain of g.




Product and Quotient of Functions

If f and g are functions,

the product function fg x is defined by: ________________________________

the quotient function f

xg

is defined by: ______________________________

The domain of fg is the set of all real numbers that are in the domain of both f and g.

The domain of f

g is the set of all real numbers that are in the domain of both f and g, such

that g x 0 .

1. Given f x 3x and 2

g x x 1 .


c) Find fg x and state the domain. d) Find f

xg

and state the domain.


2. Given x

f x 2

and 5x 1

g x 3 2

.


c) Find fg x and state the domain.

d) Find g

xf



3. Given 2

f x log x 1 and 2

2g x log 16 x .


c) Find fg x and state the domain.

d) Find g

xf


4. Given f x tanx and g x cscx .




c) Determine fg x and state the domain.

d) Determine g

xf



Composite Functions

1. Composite Functions

Another way of combining functions is to use the output of one function as the input of

another. This is called the composite function, gf . It is defined as:

To evaluate gf , we must first apply the function g to x. Then, apply the function f to

the result.

Example 1: If xf(x) and 5xg(x) , find the following:

a) f(36) b) g(-4) c) g(20)f

d) 16fg e) xgf f) xfg

g) 81ff h) f)(-3)(g

Note:


Example 2: Given x f(x) and 5x g(x) , determine the domain of f and the domain

of g.

To determine the domain of the composite function gf :

Determine the domain of the “inside” (input) function, g.

Keep these restrictions.

Determine the domain of the new function after performing the composition.

Add these restrictions to answer from part a) above.

a) Determine the domain of gf using the functions f and g from Example 2 above.

b) Determine the domain of fg using the functions f and g from Example 2 above.

Example 3: Given x1

xf and 4xg 2 x


c) Determine xgf and state the domain.


d) Determine xfg and state the domain.

Example 4: Given 5x

2xxf

and

x1

xg

a) State the domain of f and of g

b) Determine xgf and state the domain

c) Determine xfg and state the domain


2. Writing a Function as a Composite

In Calculus, it is often necessary to write a function as the composition of two simpler

functions.

Example 5: For the given function y, determine functions f and g such that xgfy

a) 73 3 xy

b) 12

32

x

y

c) 523 752 xxy


Warm Up

1. Using finite differences, identify the function as polynomial (cubic, quartic, quintic),

exponential or logarithmic. Justify your answer.

a)

b)

c)

x y

-1 -2.875

0 -2.5

1 -1

2 5

3 29

4 125

x y

-3 37

-2 3

-1 -7

0 -5

1 -3

2 -13

x y

1.5 1

2 4

3 7

5 10

9 13

17 16


Sum/Difference/Product/Quotient of Functions Recap

1. Given 2

f x log x 4 and 2

g x log 3 x .


c. Find f g x and state the domain. d) Find f g x and state the domain.

e) Find f

xg


f) Find g

xf



2. Given f x x 1 and 2

g x 4 x .



d) Find f

xg



3. Given f x tan x4

and g x sec x

4

.




c) Determine g f x and state the domain.

d) Determine g

xf


Documents

Finite Differences and Exponential/Logarithmic Functionsmrjtaylor.weebly.com/uploads/3/9/7/2/39723380/mhf_4ui_unit_8... · Clearly label all asymptotes. a) fx 2 1 x ... Determine