2301107 Calculus I

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2301107 Calculus I

1. Limits and Continuity

Chapter 1:Lmits and Continuity 2

Outline

1.1. Limits

1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws

1.1.4 Mathematical definition of a limit 1.1.5 Infinite limit

1.1. Continuity

1.1.1. Motivation:Tangent

● “Tangent” derived from the Latin word, tangens, which means touching.

Tangent line in a circle, intersect a circle only once

numerically?

● Given the parabola y = x², find the tangent line to the parabola at a point P(1, 1).

P(1, 1) 0

PQ

2 3

1.5 2.5 1.1 2.1 1.01 2.01 1.001 2.001 Q(x, x²) x

slope of PQ=x²−1 x−1

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situation? Velocity problem

● Suppose a ball has been drop from ENG4, 400 m. above the ground, find velocity after 5 sec.

s(t) = 4.9 t²

● Average = distance traveled/time elapsed

● The (instantaneous) velocity after 5 sec. is 49 m/s.

Time interval Average velocity (m/s) 53.9 49.49 49.245 49.049 49.0049 5 < t < 6

5 < t < 5.1 5 < t < 5.05 5 < t < 5.01 5 < t < 5.001

numerical computation.

● Consider y = x² – x + 2 for values of x near 2.

when x is close to 2 (on either side of 2), f(x) is close to 4.

Note, f(2) is not significant here!

1 2 3 8

1.5 2.75 2.5 5.75 1.9 3.71 2.1 4.31 1.99 3.97 2.01 4.03 1.999 3.997 2.001 4.003 x f(x) x f(x)

0

(2, 4)

2 4

Mathematical notation of limit

● The limit of f(x), as x approaches a, equal L is written as

1. Guess the value of using a calculator.

Limit is _______. We do not compute value at a.

lim

xa

f x=L lim

x1

x−1 x²−1

x f(x) x f(x)

Student note

2. Estimate the value of by calculator.

Limit is _____________ . lim

t0



^t²^9−3

t²

x f(x) x f(x)

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Student note

3. Guess the value of using a calculator.

Limit is ________________.

lim

x0

sinx x

x f(x) x f(x)

Student note

4. Investigate lim

x0

sin



^^x



^.

x f(x) x f(x) x f(x)

Student note

5. Find lim

x1

∣x−1∣.

Student note

6. Investigate lim

x0

sin



^^x



^.

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Student note

7. Investigate Ht=

{

⁰¹ ^{if t}^{if t}^0^0

}

^.

x f(x)

One-Sided Limits

● Definition: We write

as the left-hand limit of f(x) as x approaches a [limit of f(x) as x approaches a from the left] is equal to L if we can make f(x) arbitrarily close to L by taking x (less than a) to be sufficiently close to a.

The right-hand limit of f(x) as x approaches a [limit of f(x) as x approaches a from the right] is equal to L if we can make f(x) arbitrarily close to L by taking x (greater than a) to be sufficiently close to a.

lim

xa⁻

f x=L

lim

xa^

f x=L

Graph and its limit

● Rule:

if and only if and

● Determine the limit of g.

lim

xa

f x=L lim

xa⁻

f x=L lim

xa^

f x=L.

lim

x3⁻

gx

lim

x3^

gx

lim

x7⁻

gx

lim

x7^

gx

lim

x0

gx

0

g 4

2

3 7

1 3

Infinite limits

● Determine if the limit exists lim

x0

1 x².

y= 1 x²

0 x

lim

x0

1 x²=∞

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Infinite limits

● Definition: Let f be a function defined on both sides of a, except possibly at a itself. Then

means f(x) can be made arbitrarily large as x sufficiently close to a, but not equal to a.

lim

xa

f x=∞

Infinite limits

● Determine if the limit exists

y=−1 x² lim

x0

− 1 x².

0 x

Infinite limits

● Definition: Let f be a function defined on both sides of a, except possibly at a itself. Then

means f(x) can be made arbitrarily small (large negative) as x sufficiently close to a, but not equal to a.

lim

xa

f x=−∞

Vertical asymptote

● Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:

lim

xa

f x=∞ lim

xa⁻

f x=∞ lim

xa^

f x=∞

lim

xa

f x=−∞ lim

xa⁻

f x=−∞ lim

xa^

f x=−∞

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Student note

8. Find lim

x3^

2 x

x−3 , lim

x3⁻

2 x x−3.

Student note

9. Find the vertical asymptotes of f(x) = tan x.

1.1.3 Limit laws

● Suppose that c is a constant and limits of the following exist

then 1. lim

xa

f xgx=lim

xa

f xlim

xa

gx 2. lim

xa

f x−gx=lim

xa

f x−lim

xa

gx 3. lim

xa

c f x=c lim

xa

f x

4. lim

xa

f xgx=lim

xa

f x⋅lim

xa

gx

5. lim

xa

f x

gx = lim

xa

f x

lim

xa

gx if lim

xa

gx≠0 lim

xa

f x, lim

xa

gx

1.1.3 Limit laws (cont.)

● Moreover, 6. lim

xa

[ f x]ⁿ=



^lim^x^a ^f ^^x^



ⁿ^{where n∈ℤ}^

7. lim

xa

c=c 8. lim

xa

x=a 9. lim

xa

xⁿ=aⁿwhere n∈ℤ^ 10. lim

xa



n ^x=



ⁿ a where n∈ℤ^ 11. lim

xa



n ^f ^^x^=



ⁿ ^lim_x_a ^f ^^x^ ^where



ⁿ^lim_x__a ^f ^^x^ ^exists

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Limit of polynomial and rational functions

● If f is a polynomial or a rational function and a is in the domain of f, then

● For example, f(x) = 3x⁴ – 5x² + x – 3, then lim

xa

f x=f a

lim

xa

3 x⁴−5 x²x−3=3 a⁴−5 a²a−3

Student note

10. Find lim

x1

x²−1 x−1 .

Student note

11. Find wherelim

x1

gx gx=

{

^^x1 ^{if x≠1}^{if x=1}

}

^.

Student note

12. Show that lim

x0

∣x∣=0 .

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Student note

13. Prove that does not exist.lim

x0

∣x∣ x

Student note

14. Determine if whether limit exists.lim

x4

f x

f x=

{

^{8−2 x}



^x−⁴ ^{if x}^{if x}^≥4^4

}

Theorem of limits

● Theorem: If f(x) < g(x) when x is near a and the limits of f and g both exist as x approaches a, then

● The Squeeze Theorem: If f(x) < g(x) < h(x) when x is near a and

then

lim

xa

f xlim

xa

gx

lim

xa

f x=lim

xa

hx=L

lim

xa

gx=L

Student note

lim

x0

x²sin1 x =0 . 15. Show that

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Student note

16. Given

find the limits that exists. If not, explain.

lim

xa

f x=−3, lim

xa

gx=0, lim

xa

hx=8

1. lim

xa

f xhx 2. lim

xa

f x² 3. lim

xa



3 ^h^^x^

Student note

17. Given

find the limits that exists. If not, explain.

lim

xa

f x=−3, lim

xa

gx=0, lim

xa

hx=8

1. lim

xa

f x

hx 2. lim

xa

1

f x 3. lim

xa

f x

gx

Student notes

18. Evaluate the limits, if they exist.

1. lim

x2

x²x−6

x−2 2. lim

x−3

t²−9 2 t²7 t3

Student notes

1. lim

x4

x²−4 x

x²−3 x−4 2. lim

x1

x³−1 x²−1

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Student notes

1. lim

h0

1h⁴−1

h 2. lim

t9

9−t 3−



^t

1.1.4 Mathematical definite of limit

● Definition: Let f be a function defined on some open interval that contains a, except possibly at a. The limit of f(x) as x approaches a is L, written

if for every number ε > 0 there is a number δ > 0,

| f(x) – L | < ε whenever 0 < | x – a | < δ lim

xa

f x=L

1.1.4 Mathematical definite of limit

● For every number ε > 0 there is a number δ > 0,

| f(x) – L | < ε whenever 0 < | x – a | < δ

lim

xa

f  x = L

Limit of f(x) as x approaches a

0 L

ε ε

δ δ

a

Alternative interpretation

● Alternatively,

If 0 < | x – a | < δ then | f(x) – L | < ε.

x a f(x) f(a)

a x L

( ) ( ) f(x)

ε ε

δ δ

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Definite of left & right-hand limit

● Definition:

| f(x) – L | < ε whenever a – δ < x < a

● Definition:

| f(x) – L | < ε whenever a < x < a + δ lim

xa⁻

f x=L

lim

xa^

f x=L

1.1.5 Infinite limit definition

● Definition:

means for every positive number M there is a number δ > 0, f(x) > M whenever 0 < | x – a | < δ

● Definition:

means for every negative number N there is a number δ > 0, f(x) < N whenever 0 < | x – a | < δ

lim

xa

f x=∞

lim

xa

f x=−∞

1.2 Continuity

● Definition: A function f is continuous at a number a if

● Note

– f(a) is defined. (that is, a is in the domain of f)

– Limit of f as x approaches a exists

– The value of the limit is the same as function value.

● If f is not continuous at a, we say that f is discontinuous at a.

lim

xa

f x=f a

Graph and continuity

● Determine from the graph g, which points of g are discontinuous and why?

0 4

2 g

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Student note

21. Where are the following functions discontinuous?

where = the largest integer that is less than or equal to x.〚x〛 1. f x=x²−x−2

x−2 2. f x=〚x〛

Student note

3. fx=

{

^x²^−x^x−2¹⁻² ^{if x≠2}^{if x=2}

}

^4. ^f ^^x^=

{

^x¹¹² ^{if x≠0}^{if x=0}

}

One-sided continuity

● Definition: A function f is continuous from the right at a number a if

● Definition: A function f is continuous from the left at a number a if

● Definition: A function f is continuous on an interval if it is continuous at every number in the interval.

lim

xa^

f x=f a

lim

xa⁻

f x=f a

Student note

22. Show that the function f(x) = is continuous on the interval [-1, 1].

1−



¹⁻^x²

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Theorem for continuity

● Theorem: If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:

1. f + g 2. f – g 3. cf 4. fg 5.

● Theorem: Any polynomial and rational function is continuous wherever it is defined.

● The following types of functions are continuous at every number in their domains: Polynomials, Rational functions, Root functions, Trigonometric functions.

f

g if ga≠0

Continuity of a composite function

● Theorem: If f is continuous at b and then

In other words,

● Theorem: If g is continuous at a and f is continuous at g(a), then the composite function f○g given by (f ○ g)(x) = f(g(x)) is continuous at a.

lim

xa

f gx=f b

lim

xa

gx=b

lim

xa

f gx= f lim

xa

gx

Student note

23. Where are the following functions continuous?

(a) h(x) = sin(x²) (b)Fx= 1



^x²^7−4

Intermediate Value Theorem

● Intermediate Value Theorem: Suppose f is continuous on [a, b] and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a, b) such that f(c) = N.

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Student note

24. Show that there is a root of the equation 4x³ – 6x² + 3x – 2 = 0 between 1 and 2.

Student note

25. If f(x) = x³ – x² + x, show that there is a number c such that f(c) = 10.

Student note

26. Explain why the function is discontinuous at the given number a. Sketch the graph of the function.

1. f x= − 1

x−1² a=1

2. f x=

{

¹²^x ^{if x}^{if x}^≠1⁼¹

}

^a=1

3. f x=

{

^x²⁻^x⁻⁵^3^x⁻¹² ^{if x}^{if x}^≠−3⁼⁻³

}

^a=−3

Student note

27. If f and g are continuous functions with f(3) = 5 and determine g(3).lim

x3

2 f x−gx=4

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Tangent line using limit

● Definition: The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope

provided that this limit exists.

m=lim

xa

f x−f a x−a

0

P(a, f(a)) Q Q Q Q

Student note

28. Find an equation of the tangent line to the parabola y = x² at the point P(1, 1).

Tangent line

● Alternatively, m=lim

h0

f ah− f a

h .

Student note

29. Find an equation of the tangent line to the hyperbola y = 3/x at the point (3, 1).

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Student note

30. Find the slopes of the tangent lines to the graph of the function f(x) = at the points (1, 1), (4, 2) and (9, 3).



^x

Velocities

● The function f that describes the motion is called the position function of the object, s = f(t).

● The change in position from t = a to t = a+h is f(a + h) – f(a)

● The average velocity over this time interval is

● We define the (instantaneous) velocity v(a) at time t = a to be the limit of the average velocities:

average velocity= f ah−f a h

va=lim

h0

f ah−f a h

Rates of change

● Suppose y depends on x, y = f(x).

● The change in x from x₁ to x₂ is called the increment of x is ∆x = x₂ - x₁

and the corresponding change in y is

∆y = f(x₂) – f(x₁)

● The difference quotient

is called the average rate of change of y with respect to x over the interval [x₁, x₂].

 y

x= f x₂− f x₁ x₂−x₁

Rates of change

● Instantaneous rate of change lim

x0

y

x=lim

x2x1

f x₂−f x₁ x₂−x₁

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Student note

31. Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m. above the ground. What is the velocity of the ball after 5 seconds?

How fast is the ball traveling when it hits the ground?

Student note

32. The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 4t³+6t+2, where t is measured in seconds. Find the

velocity of the particle at times t = a, t = 1, t = 2 and t = 3.

Student note

33. Find an equation of the tangent line to the curve at the given point.

1. y=12 x−x³,1, 2

2. y=



^{2 x1 ,}^{4, 3}

3. y=x−1

x−2 ,3, 2 2 x

Student note

34. Find the slope of the tangent to the parabola y = 1 + x + x² at the point where x = a. Find the slopes of the tangent lines at the points whose x-coordinate are (i) -1, (ii) -½, (iii) 1.

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Student note

35. If a ball is thrown into the air with a velocity of 40 ft/s, its height (in feet) after t seconds is given by y = 40t – 16t². Find the velocity when t = 2.

Student note

36. The displacement (in meters) of a particle moving in a straight line is given by s = t²–8t + 18, where t is

measured in seconds.

1. Find the average velocity over each time interval:

(i) [3, 4], (ii) [3.5, 4], (iii) [4, 5]

Student note

2. Find the instantaneous velocity when t = 4.