1-Limits Student Notes 2021 PDF

Preview

Summary

practice for limits derivaiteives urve sketching...

Description

Math 31:

Limits

Factoring Review Factor the following expressions fully. a) 24m3n 2 + 18m2 n3 - 12m2 n 2 + 6mn 2

b) 5m 2t - 10m 2 + t 2 - 2t .

c) x 2 + 7 x - 18

d) 4 y 2 - y - 3

e) x 4 - 3x 3 - 18x 2

f) 12 x 3 + 2 x 2 - 30x - 5

g) x 3 - 6 x 2 + 5x + 12

h) x 4 - 8 x 2 - 9

i) x 2 - 9

j) x 2 - 3

k) x - 3

l) x 4 - 1

2

Sum and Difference of Cubes

(a 3 - b 3 ) = (a - b )(a 2 + ab + b 2 ) (a 3 + b 3 ) = (a + b )(a 2 - ab + b 2 ) Factor the following expressions fully. a) x 3 - 27

b) 8 x3 + 1

c) x8 - 16

d) x - 1

Factoring Rational Exponents Review Factor the following expressions fully. -1

3

6

a) 2 x + 3x

5

-2

-1

d) x 2 + 8 x 2

g)

-4 1 53 x +x3 8

c) x + 5 + 6 x-1

b) x 2 - x 2

7

3

1

-1

1

h)

-1

f) ( x2 + 1) 2 + 3( x2 + 1) 2

e) 3x4 - 3x 4 - 6 x 4

1

1 2

5

-1

( x +1 )4 (3 x - 7 )2 + ( x + 1 )4 (3 x - 7 ) 2

Assignment: Textbook page 3 (#1-4) and the worksheet on the next page 3

Math 31 Review of Factoring Worksheet Factor the following completely. 1. 2 x 2 y 3 - 6 xy 2

x2 - 9

2. 3.

4 x 3 y 2 z 5 + 24 x 2 yz 6

4. 5.

3x2 + 2 x - 5 4 x- 3 y 2z- 4 + 8x- 5 y- 3z 2

6.

3xyz - 21x2 y2 z2 + 33x3 y3 z3

7.

x 3 - 27

8.

49 x 2 - 100 x2 - x - 6

9.

10. 3x3 + 5x2 - 6x - 10 11. 5 x2 - x - 18 12. x 6 + 64 5

1

13. x 2 - x -2 3

14. x

2

-8 3

- 2x

15. 2( x +1)( x -1)

-1

2

-2

- ( x +1) ( x -1) + 3( x + 1)( x - 1)

-2

8 16. x - 1 3

2

17. 4 x + 4 x - x - 1 3

1

-12

18. 3x 2 - 9 x 2 + 6 x - 23

19.

(1+ x )

20.

(x

2

1

x + (1+ x ) 3

2

+ 1) - 7 ( x2 + 1) + 10

3 6 21. 8x + y

22.

(x

2

+ 2x + 1) - y2

23. x 3n + y 9 n 24. x 3 - 7 x 2 + 14 x - 8 7

25.

(x - 1 )

3

2

- (x - 1 ) 2

4

Answers: 1. 2 xy2 ( xy - 3)

4.

( x - 3 )( x + 3 ) 4 x2 yz5 ( xy + 6 z ) (3 x + 5 )( x - 1)

5.

4 x -5 y - 3z - 4 ( x 2 y 5 + 2z 6 )

6.

3xyz ( 1- 7 xyz + 11x 2 y 2 z 2 )

7.

( x - 3 ) ( x2 + 3x + 9 )

2. 3.

8. 9.

(7 x - 10 )(7 x + 10 ) ( x - 3 )( x + 2 )

(

)

2 10. x - 2 (3x + 5 )

11. (5 x + 9 )( x - 2 )

(

)(

2 4 2 12. x + 4 x - 4 x + 16

13. x 14. x

1

2

-8

)

( x - 1)(x + 1) 3

(x

2

- 2)

15. x( x + 1)( x - 1)

(

-2 2

)(

4

16. (x - 1 )(x + 1 ) x + 1 x + 1

)

17. (2 x - 1 )(2 x + 1 )( x + 1 ) 1

18. 3 x- 2 ( x - 1)( x - 2) -2 3

19. (1+ x ) 20.

( 2 x + 1)

( x - 2 )( x + 2 )(x - 1 )( x + 1 )

(

)(4x

2 21. 2 x + y

2

- 2 xy 2 + y 4 )

22. (x + 1 - y )(x + 1 + y )

(

n 3n 23. x + y

)(x

2n

- x ny 3 n + y 6 n )

24. ( x - 1 )(x - 4 )(x - 2 ) 25. x ( x - 1)

3 2

( x - 2)

5

Rationalizing Numerators and Denominators

Rationalize the denominator of each of the following. a)

3 -2 1- 5

x b)

x +1 -1

Rationalize the numerator of the following. a)

x +9 +3 x

b)

x -1 x -1

Simplify and rationalize the expression

1 -2 x 4x -1

Assignment: Textbook page 4 (#1,2) 6

Review of Linear Functions Recall: A linear function can be written in •

Slope y-intercept form o y = mx + b , where m is the slope of the line and b is the y-intercept

•

General form o Ax + By + C = 0; A, B ,C Î I and A ³ 0

To find the equation of a line we need slope and a point Slope: The measure of how steep a line is. To find the slope we need two points on the line

Slope =

Rising to the right

Falling to the right

m Recall: Para

m s have the same s

Horizontal Line

m

Vertical Line

m

e

have slopes that are

ls

a) Find the equation of a line that passes through the point (-2, 2) and (6,-3) in slope yintercept form.

7

b) Determine the equation of a line parallel to the line 2 x - 3 y + 1 = 0 and passing through the point (2, 3) in general form.

c) A line has a slope of

4 . If x changes by 6, what is the change in y? 3

Assignment: Textbook page 9 (#1-5) and Math 31 Pre-Calculus Review Worksheet

8

Math 31 – Pre-Calculus Review 1. Simplify the following and write them with positive exponents.

(-18 x 4 y -3 z -8 ) (3 x-7 y2 z- 2 )

(a) (-4 x3 y6 z-2 )(2 x-2 y3 z- 3 )

(b)

(c) (3x -6 + 8 x3 )

(d) 6 x- 5 + 9 x- 7

3

-4

(e) (3x - 1) ( x + 6 ) =

(f)

5x 5 + 2x 3 + x 2 - 2 x

2. Factor the following completely: (a) x 2 - 5

(b) 8x3 + 64

(c) 8 x4 - 6 x2 - 27

7

(d) 3 x3 - 8 x 2 + 3 x + 2

3

1 4

(e) 3x4 - 12 x4 + 9 x

9

4

1 -

(f) (2x -1)5 ( x + 6) 4 - (2x - 1) 5 ( x + 6) 3

11

(g) 3x 3 +

5

1 3 x (x - 1) = 2

3. Rationalize the denominator and simplify: (a)

14 6 3 2+ 3

(b)

x2 - 64 8-x

(c)

x- 3 2 x+ 3 - 3

4. Find the equation of the line (standard form): (a) with slope - 2 3 and y - intercept 4

(b) passing through (-4, 2) and perpendicular to the line passing through (5,-2)and (1,1)

(c) passing through (-3,-5) and parallel to the line 2 x - 3y + 10 = 0

10

5. Find a common denominator and simplify:

1 2 (a) + 2+ x 3

(c)

1 3 + = 3 x -1 x - 1

3 3 (b) x + 4 29 x -5 2

2 3 = (d) x - 1 2 x + 2 2

11

Pre-Calculus Review Answer Key 1. (a) (e)

- 8xy9 z5

(b)

( 3x - 1)3 ( x + 6) 4

(f) 5 x4 + 2 x2 + x -

(

)(

2. (a) x + 5 x - 5

- 6x11 y5 z6

(c)

-

1 4

(d)

3 (2x 2 +3) x7

2 x

)

(b) (2 x + 4 )( 4 x 2 - 8 x + 16)

(c) (2 x + 3 )(2 x - 3 )( 2 x 2 + 3 ) (e) 3x

3 + 8x 9 x6

(d) (3x + 1)( x - 2 )( x - 1 )

( x - 3)( x - 1)

(f)

-

1

(2 x -1 ) 5 (x + 6 )3 (2x 2 +11x - 7 )

1 53 x ( 3x - 1)( 2 x + 1) 2 28 3 -14 2 2x + 3 + 3 3. (a) (b) - ( x + 8 ) 8 - x (c) 5 2 4. (a) 2x + 3 y - 12 = 0 (b) 4 x - 3 y + 22 = 0 (c) 2 x - 3 y - 9 = 0 -3 ( x + 5 ) 2x + 7 10 x - 4 5. (a) (b) (c) 2 x x 3( + 2) 29 ( x + 4 ) ( 3 - 1)( x - 1) (g)

(

(d)

)

- 3x + 7 2 ( x -1 )( x +1 )

12

Note: Calculus is really based on two main concepts slope and area. Discuss the importance of being able to calculate the slope of a tangent line at any point along a graph (Rates of Change, etc.) y -y Recall: m = 2 1 x2 - x1 In order to calculate the slope of a line we need: ________________ Finding the slope of a tangent line Tangent Line:

Problem: Determine the equation of the line tangent to y = x2 , at the point (2, 4).

*Problem: ______________________

Solution: à à Secant line: Recall : m = Not accurate: ________________________

General Case For the graph of y = x2 two points could be represented as ( a, a2 ) and ( x , x 2 ) Slope =

f ( x ) - f (a ) x- a 13

Try where x = 3:

y = _________ à_________ à m = _________________ When x = 2.5 à y = ______________à m = ______________ When x = 2.1 à y = ___________à m = ________________ When x = 2.01 à y = ___________à m = __________________ The slope gets closer to ____ as we pick coordinates closer and closer to the one we were given. Question: Will the slope ever equal to 4? What if we pick coordinates on the other side and repeat the process? When x = 1.5 à m = _________ When x = 1.9 à m = _________ When x = 1.99 à m = _________ à

The slope of the tangent line is the ________________of the slopes of the secant lines. R

:

Problem Determine the slope of the tangent of y = x when x = 4. Solution: Number Crunch with charts Left side x Slope 3.9 3.99 3.999

Right Side x Slope 4.01 4.001 4.0001

limit of the slope of the secants = _________________= slope of the tangent Assignment Page 9 #7, 9, 10 14

What is a limit?

lim f ( x) = L . à Read as “the limit of f ( x ) as x approaches a equals L”

x® a

Example Given f ( x ) = 2 x + 1 determine lim f ( x) . x® 3

*Note: A limit is not concerned with x = 3 , but when x is close to 3.* Number Crunch Left x 2.9 2.99 2.999

Right x 3.1 3.01 3.001

f ( x)

f ( x)

Why don’t we just plug 3 into the equation?

= lim 2 x +1

Lim left side = ________ = Lim right side ________

= =

As x approaches 3 the function approaches _________.

x® 3

Example: Determine the limit of each of the following. a) lim 7 - 2 x b) lim x2 + 2 x x® 3

d) lim x x® 4

x®4

c) lim 2 x 2 - 3 x + 4 x® 2

e) lim 7 x®2

15

x2 - 9 . x® 3 x - 3

f) lim

Note: Properties of limits page 13 in the textbook

1 1 g) lim x 2 x® 2 x - 2

i) lim

x® 4

x-4 x -2

x3 - 8 x®2 x2 - 3 x + 2

h) lim

x3 - 8 x® 2 x - 1

j) lim

16

2 -2 k) lim 1+ x x x® 0

l) lim

x +3

x® 3 x 2 - 9

Assignment: Page 18 #1-6 and Limits Worksheet

17

Math 31 Limits Worksheet Evaluate each of the following limits, if they exist.

1. lim(5x 2 - 2 x + 3) x®4

2. lim (x 3 + 2 x 2 + 6 ) x ®-3

x -2 2 x + 4x - 3

5. lim

7. lim

3

4n + 3n 3 3n + 10

8. lim

n ®-2

5

n ®2

2

4. lim

x ®-1

3. lim( n 2 - n + 1)

x 3 + 2x + 7

x ®-1

6. lim(n 4 - 7 n + 4 ) 3 n® 3

1

2h 1 1 h® 2 h + h

9. lim t

-

t®16

3

(t 2

- 14t ) 5

x 4 - 16 x- 2

x 2 - x + 12 x ®-3 x+3

11. lim

13. lim

9- n n ®9 3 - n

14. lim

2n ù é n2 + 16. lim ê ú n ®-2 n + 2 n +2 û ë

17. lim

2 ù é 1 19. limê - 2 n ®1 ë n - 1 n - 1úû

é x 2 - 81ù 20. limê ú x®9 ë x -3û

1ù é 1 22. limê - ú t ®0 t 1 + t tû ë

23. lim

2 25. lim 3 n ®¥ n

2n + 1 26. lim n ®¥ 3n - 2

3n 2 + n - 1 27. lim n ®¥ 10 n + 40

é 3n 2 5n ù + 29. limê 2 ú n ®¥ 2n + 1 n + 3û ë

30. lim

2 (1 + h )- - 1 32. lim h® 0 h

x2 - x - 2 33. lim 2 x ®-1 x + 3x + 2

35. lim n - 3

36. lim

3. 243 9. 2 15. 1/3 21. -1/9 27. dne ( ¥ ) 33. -3

5. 2 11. -7 17. 1/4 23. -1/2 29. 6.5 35. 3

10. lim

t ®1

2n 2 n ®¥ n - 4

( a + h) 3 - a 3

h ®0

34. lim t ®2

12. lim x ®2

3

3

28. lim

31. lim

x 2 - x - 12 x ®-3 x+ 3

2

h

t2 + t - 6 t2 - 4

t t -1 4

h® 0

x®2

15. lim x® 1

1+ h -1 h

2x - x x-2

n ®12

x-1 x -1

x+8 x ®-8 3 x + 2

18. lim

21. lim

( 3 + h ) -1 - 3-1 h

h ®0

x 24. lim x®0

n ®2

1 + 3x - 1

n 2 + 2n - 8 n 4 - 16

1 x ®-3 x + 3

Answer Key 1. 75 7. -2 13. 6 19. 1/2 25. 0 31. 3a2

2. -3 8. 2/5 14. dne 20. 108 26. 2/3 32. -2

4. 1/2 10. dne 16. -2 22. -1/2 28. 0 34. 5/4

6. 16 12. 32 18. 12 24. 2/3 30. 3/16 36. dn

18

Recall: Basically 3 types of limits Plug in Do some math Does not exist Review Evaluate each of the following limits if they exist.

x-1 - 2-1 a) lim x® 2 x - 2

4 (x - 1) - 1 b) lim x® 0 x

c) lim

m -8 -2

m ®8 3 m

19

Example Given f ( x) = x - 2 find lim f ( x) . x ®2

Try plugging it in.

lim x - 2

x® 2

= = = Recall: In order for the limit to exist, the limit of left side must be equal to the limit of the right side. Try number crunching. Right Side Left Side x f(x) 1.9 1.99 1.99

x f(x) 2.1 2.01 2.001 2.0001

Left limit is ____________

Right limit = ___________________

Wouldn’t ask the left limit à would ask the right limit

lim f ( x) = dne

lim f ( x) = 0

x®2 -

x®2 +

Notation: om left ha de

In summary: If lim- f ( x) ¹ lim+ f ( x) , then x® a

x® a

If lim- f ( x) = L = lim+ f ( x) , then x® a

x® a

20

Example Given the graph of a function below, determine: (a) lim- f ( x ) =

(b) lim+ f ( x ) =

(c) lim f ( x) =

(d) lim- f ( x) =

(e) lim+ f ( x) =

(f) lim f ( x) =

x ®-1

x ®-1

x ®3

(g) lim f ( x) = x®6

(i) f (6) =

x ®-1

x ®3

x®3

(h) f (-1) = (j) f (3) =

In order for a function to be continuous the following situations must be satisfied: • •

f(a) exists lim f (x ) exists (meaning lim- f ( x) = L = lim+ f ( x) )

•

lim f (x ) = f (a )

x ®a

x ®a

x ®a

x ®a

Otherwise, the function is _____________________.

(k) State whether the above function is continuous or discontinuous at the following numbers. i)

x = -1

ii) x = 3

iii) x = 6

Assignment: Page 27 - #1-4, 7

21

Lesson #4 Drawing

Discontinuous Graphs

Recall: In order for a function to be continuous the following situations must be satisfied: • •

f(a) exists lim f (x ) exists (meaning lim- f ( x) = L = lim+ f ( x) )

•

lim f (x ) = f (a )

x ®a

x ®a

x ®a

x ®a

Otherwise, the function is discontinuous. Example:

ì x2 + 2 Given f ( x) = í î3 x + 3

x£0 x> 0

Draw a sketch of the graph, find the following limits, and determine if f ( x ) is continuous or where it is discontinuous?

ì x2 + 2 f ( x) = í î3 x + 3

f ( x) = x 2 + 2

f ( x ) = 3x + 3

a) lim- f ( x )

b) lim + f ( x )

c) f (0 ) =

d) lim+ f ( x )

e) lim- f ( x )

f) lim f (x )

x®2

x®0

x®-2

x® 0

x£0 x> 0

x®0

22

Example: Determine whether the following functions are continuous or discontinuous.

ì x 2 + 1 if x < 0 ï if x = 0 (a) f ( x) = í 0 ï x 2 - 1 if x > 0 î

ì x + 1 if x ¹ 2 (b) g (x ) = í if x = 2 î1

23

ìx 2 ï (c) f ( x) = í1 ï2x- 1 î

Assignment:

if x < - 1 if - 1 £ x < 1 if x > 1

phs)

24

Consider an infinite sequence 2, 4, 8, … getting bigger and bigger ∞à

Example For each of the following functions, determine the limit as x ® ±¥ 4 (a) (b) f ( x ) = (c) f ( x) = 2x + 1 x

lim f ( x) =

lim f ( x) =

x®¥

æ 2ö f ( x) = ç ÷ è 3ø

lim f ( x) =

x®¥

x®-¥

x

x®¥

x2 + 1

lim f ( x) =

x®-¥

(d)

3x 2

lim f ( x) =

x®¥

lim f ( x) =

f ( x) =

x

æ3 ö (e) f ( x) = ç ÷ è 2ø

lim f ( x) =

x®¥

lim f ( x) =

x®-¥

2

(f) f ( x) = - ( x - 3) + 5

lim f ( x) =

x®¥

lim f ( x) =

x®-¥

25

Summary: There are four cases when x ® ¥ : 1. Degree of the top is greater than the degree of the bottom Consider:

2. Degree of the top is less than the degree of the bottom. Consider:

3. Degree of the top is equal to the degree of the bottom. Need to divide by the highest degree. Consider:

4. Variable is in the exponent. Consider:

Example Evaluate each of the following limits if they exist: x 2 + 5x 3x + 5 (a) lim (b) lim x®¥ x®¥ 2 7x

(c) lim 3n n®¥

26

x 2+5x x®¥ 7x 2

(d) lim

(e) lim

x 2 + 3x + 5 x®¥ x+ 7

(f) lim

x®¥

3x2 + 5x + 6 x2 + 4

(g) lim

n®¥

n 2 -n 2n 2 + 1

Assignment: Limits Worksheet and Page 50 #3 27

ences and Series Recall: Arithmetic Sequence: general term formula Sum of an Arithmetic Series:

Sn =

n (a + t n ) 2

Geometric Sequence: general term formula

Sum of a Geometric Series: S n =

a( r n -1) r -1

tn = a + ( n - 1)d or S n =

n [ 2a + (n - 1)d ] 2

tn = arn -1 or Sn =

a(1 - r n ) 1- r

Infinite Sequence: Ex.

Note: sequences can be arithmetic, geometric or other

Example For the sequence defined by tn = 1 -

2 n

(a) list the first six terms

(b) What is t100?

(c) What is t1000?

(d) What do ...