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Math 137 Calculus 1 for Honours Mathematics Course Notes Barbara A. Forrest and Brian E. Forrest

Version 1.5

c Barbara A. Forrest and Brian E. Forrest. Copyright  All rights reserved. January 1, 2020 All rights, including copyright and images in the content of these course notes, are owned by the course authors Barbara Forrest and Brian Forrest. By accessing these course notes, you agree that you may only use the content for your own personal, non-commercial use. You are not permitted to copy, transmit, adapt, or change in any way the content of these course notes for any other purpose whatsoever without the prior written permission of the course authors. Author Contact Information: Barbara Forrest ([email protected]) Brian Forrest ([email protected])

i

QUICK REFERENCE PAGE 1 Right Angle Trigonometry

sin θ =

opposite hy potenuse

jacent cos θ = hyadpotenuse

csc θ =

1 sin θ

sec θ =

1 cos θ

tan θ =

opposite ad jacent

cot θ =

1 tan θ

Radians Definition of Sine and Cosine

The angle θ in radians equals the length of the directed arc BP, taken positive counter-clockwise and negative clockwise. Thus, π radians = 180◦ or 1 rad = 180 . π

For any θ, cos θ and sin θ are defined to be the x− and y− coordinates of the point P on the unit circle such that the radius OP makes an angle of θ radians with the positive x− axis. Thus sin θ = AP, and cos θ = OA.

The Unit Circle

ii

QUICK REFERENCE PAGE 2 Trigonometric Identities

cos2 θ + sin2 θ = 1

Pythagorean Identity

−1 ≤ cos θ ≤ 1

Range

−1 ≤ sin θ ≤ 1 cos(θ ± 2π) = cos θ

Periodicity

sin(θ ± 2π) = sin θ cos(−θ) = cos θ

Symmetry

sin(−θ) = − sin θ

Sum and Difference Identities cos(A + B) = cos A cos B − sin A sin B cos(A − B) = cos A cos B + sin A sin B sin(A + B) = sin A cos B + cos A sin B sin(A − B) = sin A cos B − cos A sin B

Complementary Angle Identities cos( π2 − A) = sin A sin( 2π − A) = cos A

Double-Angle

cos 2A = cos2 A − sin2 A

Identities

sin 2A = 2 sin A cos A

Half-Angle

cos2 θ =

1+cos 2θ 2

Identities

sin2 θ =

1−cos 2θ 2

Other

1 + tan2 A = sec2 A

iii

QUICK REFERENCE PAGE 3

f (x) = x2

f (x) = x3

f ( x) = | x|

f (x) = cos(x)

f ( x) = sin( x)

f (x) = tan( x)

f (x) = sec( x)

f ( x) = csc( x)

f ( x) = cot(x)

1 1

f (x) = ex

f (x) = ln( x)

iv

QUICK REFERENCE PAGE 4 Table of Antiderivatives

Differentiation Rules

xn+1 xn dx = +C n+1 R 1 dx = ln(| x |) + C x R ex dx = ex + C R sin( x) dx = − cos( x) + C R cos( x) dx = sin( x) + C R sec2 (x) dx = tan( x) + C R 1 dx = arctan(x) + C 1 + x2 R 1 dx = arcsin(x) + C √ 1 − x2 R −1 dx = arccos(x) + C √ 1 − x2 R sec( x) tan(x) d x = sec( x) + C R x ax a dx = +C ln(a)

R

Function

Derivative

f (x) = cxa , a , 0, c ∈ R

f ′ (x) = cax a−1

f (x) = sin(x)

f ′ ( x) = cos( x)

f (x) = cos(x)

f ′ (x) = − sin( x)

f (x) = tan(x)

f ′ ( x) = sec2 ( x)

f (x) = sec(x)

f ′ ( x) = sec(x) tan( x)

f (x) = arcsin(x)

f ′ (x) = √

1

1 − x2 1 f ′ (x) = − √ 1 − x2 1 f ′ (x) = 1 + x2

f (x) = arccos(x) f (x) = arctan(x) f (x) = ex

f ′ (x) = ex

f (x) = a x with a > 0

f ′ (x) = a x ln(a) 1 f ′ (x) = x

f (x) = ln( x) for x > 0

n-th degree Taylor polynomial for f centered at x = a n (k) P f (a ) T n,a (x) = (x − a)k k! k=0

= f (a) + f ′ (a)(x − a) +

f ′′(a) (x 2!

− a)2 + · · · +

f (n) (a ) (x n!

− a)n

Linear Approximations ( L0 (x)) and Taylor Polynomials (T n,0 (x))

f (x) = ex

L0 ( x) = T 1,0 (x) = f (0) + f ′ (0)(x − 0) = e0 + e0 (x) = 1 + x T 2,0 (x) = f (0) + f ′ (0)(x − 0) + T 3,0 (x) = 1 + x + T 4,0 (x) = 1 + x +

f (x) = sin(x)

x2 2 x2 2

+

x3 6

+

x3 6

+

x4 24

L0 ( x) = T 1,0 (x) = x T 2,0 (x) = x T 3,0 (x) = x − T 4,0 (x) = x −

f (x) = cos(x)

x3 6 x3 6

L0 ( x) = T 1,0 (x) = 1 T 2,0 (x) = 1 − T 3,0 (x) = 1 − T 4,0 (x) = 1 −

x2 2 x2 2 x2 2

4

x + 24

v

f ′′(0) 2! (x

− 0)2 = e0 + e0 (x) +

e0 2! (x

− 0)2 = 1 + x +

x2 2

Table of Contents Page 1 Sequences and Convergence 1.1 Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Inequalities Involving Absolute Values . . . . . . . . . . . 1.2 Sequences and Their Limits . . . . . . . . . . . . . . . . . . . . . 1.2.1 Introduction to Sequences . . . . . . . . . . . . . . . . . 1.2.2 Recursively Defined Sequences . . . . . . . . . . . . . . 1.2.3 Subsequences and Tails . . . . . . . . . . . . . . . . . . 1.2.4 Limits of Sequences . . . . . . . . . . . . . . . . . . . . . 1.2.5 Divergence to ±∞ . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Arithmetic for Limits of Sequences . . . . . . . . . . . . . 1.3 Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Monotone Convergence Theorem . . . . . . . . . . . . . . . . . 1.5 Introduction to Series . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Geometric Series . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Divergence Test . . . . . . . . . . . . . . . . . . . . . . .

1 3 7 7 10 16 17 27 28 37 39 45 49 51

2 Limits and Continuity 2.1 Introduction to Limits for Functions . . . . . . . . . . . . . . . . . 2.2 Sequential Characterization of Limits . . . . . . . . . . . . . . . . 2.3 Arithmetic Rules for Limits of Functions . . . . . . . . . . . . . . 2.4 One-sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Fundamental Trigonometric Limit . . . . . . . . . . . . . . . 2.7 Limits at Infinity and Asymptotes . . . . . . . . . . . . . . . . . . 2.7.1 Asymptotes and Limits at Infinity . . . . . . . . . . . . . . 2.7.2 Fundamental Log Limit . . . . . . . . . . . . . . . . . . . 2.7.3 Vertical Asymptotes and Infinite Limits . . . . . . . . . . . 2.8 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Types of Discontinuities . . . . . . . . . . . . . . . . . . . 2.8.2 Continuity of Polynomials, sin(x), cos( x), ex and ln(x) . . . 2.8.3 Arithmetic Rules for Continuous Functions . . . . . . . . 2.8.4 Continuity on an Interval . . . . . . . . . . . . . . . . . . 2.9 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . 2.9.1 Approximate Solutions of Equations . . . . . . . . . . . . 2.9.2 The Bisection Method . . . . . . . . . . . . . . . . . . . . 2.10 Extreme Value Theorem . . . . . . . . . . . . . . . . . . . . . . . 2.11 Curve Sketching: Part 1 . . . . . . . . . . . . . . . . . . . . . . .

56 56 66 70 76 78 82 86 87 93 98 103 105 107 110 113 115 119 123 126 130

vi

1

3 Derivatives 3.1 Instantaneous Velocity . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Tangent Line . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Differentiability versus Continuity . . . . . . . . . . . . . . 3.3 The Derivative Function . . . . . . . . . . . . . . . . . . . . . . . 3.4 Derivatives of Elementary Functions . . . . . . . . . . . . . . . . 3.4.1 The Derivative of sin(x) and cos(x) . . . . . . . . . . . . . 3.4.2 The Derivative of ex . . . . . . . . . . . . . . . . . . . . . 3.5 Tangent Lines and Linear Approximation . . . . . . . . . . . . . . 3.5.1 The Error in Linear Approximation . . . . . . . . . . . . . 3.5.2 Applications of Linear Approximation . . . . . . . . . . . . 3.6 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Arithmetic Rules of Differentiation . . . . . . . . . . . . . . . . . 3.8 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Derivatives of Other Trigonometric Functions . . . . . . . . . . . 3.10 Derivatives of Inverse Functions . . . . . . . . . . . . . . . . . . 3.11 Derivatives of Inverse Trigonometric Functions . . . . . . . . . . 3.12 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Local Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.1 The Local Extrema Theorem . . . . . . . . . . . . . . . . 3.14 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 135 137 138 141 143 145 147 150 154 157 161 166 170 175 177 183 189 196 199 202

4 The Mean Value Theorem 4.1 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . 4.2 Applications of the Mean Value Theorem . . . . . . . . . . . . . 4.2.1 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Increasing Function Theorem . . . . . . . . . . . . . . . . 4.2.3 Functions with Bounded Derivatives . . . . . . . . . . . . 4.2.4 Comparing Functions Using Their Derivatives . . . . . . . 4.2.5 Interpreting the Second Derivative . . . . . . . . . . . . . 4.2.6 Formal Definition of Concavity . . . . . . . . . . . . . . . 4.2.7 Classifying Critical Points: The First and Second Derivative Tests . . . . . . . . . . . 4.2.8 Finding Maxima and Minima on [a, b] . . . . . . . . . . . 4.3 L’Hoˆ pital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Curve Sketching: Part 2 . . . . . . . . . . . . . . . . . . . . . . .

208 208 213 213 218 221 223 226 228 232 236 239 247

5 Taylor Polynomials and Taylor’s Theorem 5.1 Introduction to Taylor Polynomials and Approximation . . . . . . . 5.2 Taylor’s Theorem and Errors in Approximations . . . . . . . . . . 5.3 Big-O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Calculating Taylor Polynomials . . . . . . . . . . . . . . .

259 259 271 279 286

vii

Chapter 1 Sequences and Convergence It is often the case that in order to solve complex mathematical problems we must first replace the problem with a simpler version for which we have appropriate tools to find a solution. In doing so our solution to the simplified problem may not work for the original question, but it may be close enough to provide us with useful information. Alternatively, we may be able to design an algorithm that will generate successive approximate solutions to the full problem in such a manner that if we apply the process enough times, the result will eventually be as close as we would like to the actual solution. For example, there is no algebraic method to solve the equation ex = x + 2. However, we can graphically show that there are two distinct solutions for this equation and that the two solutions are close to -2 and 1, respectively. One process we could use to solve this equation is a type of binary search algorithm that is based on the fact that the function f ( x) = ex − ( x + 2) is continuous. We could also use an alternate process which relies on the very useful fact that if a function is differentiable at a point x = a, then its tangent line is a very good approximation to the graph of a function near x = a. In fact, approximation will be a theme throughout this course. But for any process that involves approximation, it is highly desirable to be able to control how far your approximation is from the true object. That is, to control the error in the process. To do so we need a means of measuring distance. In this course, we will do this by using the geometric interpretation of absolute value.

1.1 Absolute Values One of the main reasons that mathematics is so useful is that the real world can be described using concepts such as geometry. Geometry means “earth measurement” and so at its heart is the notion of distance. We may view the number line as a ruler that extends infinitely in both directions. The point 0 is chosen as a reference point. We can think of the distance between 0 and 1 as our fixed unit of measure. It then 1

Chapter 1: Sequences and Convergence

2

follows that the√point π is located 3.141592 . . . units to the right of the reference point 0. The point − 2 is located 1. 41421 . . . units to the left of 0. Consequently, we can think of the non-zero real numbers as being quantities that are made up of two parts. First there is a sign, either positive or negative, depending on the point’s orientation with respect to zero, and a magnitude that represents the distance that the point is from 0. This magnitude is also a real number, but it is always either positive or 0. This magnitude is called the absolute value of x and is denoted by | x |. It is common to think of the absolute value as being a mechanism that simply drops negative signs. In fact, this is what it does provided that we are careful with how we represent a number. For example, we all know that | 5 |= 5 and that | −3 |= 3. However, what if x was some unknown quantity, would | −x |= x? It is easy to see that this is not true if the mystery number x actually turned out to be −3. Since the absolute value plays an important role for us in this course, we will take time to give it a careful definition that will remove any ambiguity.

DEFINITION

Absolute Value

For each x ∈ R, define the absolute value of x by ( x if x ≥ 0 | x |= −x if x < 0

REMARK If we use this definition, then it is easy to see that | x |=| −x | . So far we have only considered the distance between a fixed number and 0. However, it makes perfect sense to consider the distance between any two arbitrary points. For example, we would assume that the distance between the points 2 and 3 should be 1.

The distance between −3 and 2 is slightly more complicated. To get from −3 to 2 we can first travel from −3 to 0, a distance of 3 units, and then travel from 0 to 2, an additional 2 units. This makes for a total of 5 units. We observe that | 3 − 2 |=| 2 − 3 |= 1 (from the first example) and that | −3 − 2 |=| 2 − (−3) |= 5.

Calculus 1

(B. Forrest)2

Section 1.1: Absolute Values

3

REMARK These examples illustrate an important use of the absolute value, namely that given any two points a, b on the number line, the distance from a to b is given by | b − a |. Note that the distance is also | a−b | since | b−a |=| −(b−a) |=| a−b |. Geometrically, this last statement corresponds to the fact that the distance from a to b should be equal the distance from b to a.

1.1.1 Inequalities Involving Absolute Values One of the fundamental concepts in Calculus is that of approximation. Consequently, we are often faced with the question of “when is an approximation close enough to the exact value of the quantity?” Mathematically, this becomes an inequality involving absolute values. These inequalities can look formidable. However, if you keep distances and geometry in mind, it will help you significantly. One of the most fundamental inequalities in all of mathematics is the Triangle Inequality. In the two-dimensional world, this inequality reduces to the familiar statement that the sum of the length of any two sides of a triangle exceeds the length of the third. This means that if you have three points x, y and z, it is always at least as long to travel in a straight line from x to z and then from z to y as it is to go from x to y directly. x |x − y| |x − z| z Calculus 1

y |z − y| (B. Forrest)2

Chapter 1: Sequences and Convergence

4

If we use this last statement as our guide, and recognize that the exact same principle applies on the number line, we are led to the following very important theorem:

THEOREM 1

Triangle Inequality

Let x, y and z be any real numbers. Then | x − y |≤| x − z | + | z − y |

This theorem essentially says: The distance from x to y is less than or equal to the sum of the distance from x to z and the distance from z to y. While we will try to avoid putting undue emphasis on formal proofs, it is often enlightening to convince ourselves of the truth of a mathematical assertion. To do this for the triangle inequality, we first note that since | x − y |=| y − x |, we could always rename the points so that x ≤ y. With this assumption, we have three cases to consider:

PROOF Case 1 : z < x. In this case, it is clear from the picture below that the distance from

z to y exceeds the distance from x to y. That is, | x − y | 3, but in all cases the sequence will rapidly approach 3. Problem: What happens if a1 = 3?

In the next example we will present an algorithm for calculating square roots that has its historical origins going back to the Babylonians. The algorithm itself was first presented explicitly by the Greek mathematician Heron of Alexandria.

Calculus 1

(B. Forrest)2

Chapter 1: Sequences and Convergence

EXAMPLE 4

14

Heron’s Algorithm for Finding Square Roots

Consider the following recursively defined sequence: a1 = 4

17 1 and an+1 = (an + ). 2 an

Let’s see what this sequence looks like by calculating the first 10 terms. You will notice that up to ten decimal places the sequence actually stabilizes from n = 4 onwards. In fact the terms do change as n increases but the difference between successive terms is so small as to be almost undetectable very quickly. In particular, like our previous example, the terms of this sequence seem to rapidly approach a certain limiting value α which we would guess to be very close to 4.1231056256. So it is now worth asking, what is the significance of this value α?

n

an

1 2 3 4 5 6 7 8 9 10

4 4.125 4.1231060606 4.1231056256 4.1231056256 4.1231056256 4.1231056256 4.1231056256 4.1231056256 4.1231056256

√ In fact it turns out that to show that α = 17. In particular, we √ we will later be able −14 can show that a4 − 17 is roughly 2.31 × 10 which means √ √ that a4 is a√remarkably accurate approximation to 17. Even a3 is very close to 17 with a3 − 17  4.35 × 10−7 . It is also worth noting that despite the fact that in the table above we represented the terms in the sequence with decimal expansions, the calculations certainly produce , a3 = 2177 and a4 = 9478657 . rational values for each an . In particular, a1 = 4, a2 = 33 8 2298912 √ 528 So this algorithm not only gives an approximate value for 17, but it also generates very accurate rational approximations to this irrational number. More generally, if α > 0 is any positive real number and we have a1 chosen so that √ a1 is a real number that is reasonably close to α, then the recursive sequence with initial term a1 and α 1 an+1 = (an + ) 2 an √ will generate a sequence which will very rapidly approach the value α. If both α and a1 are rational numbers, then so is an for each n. Problem: Let α = 198 and a1 = 14. Find the rational expressions for a2 and a3 using

the algorithm above. Using a calculator calculate the decimal expression for a3 and √ for 198 to 8 decimal places? Are they the same? Before we leave this example, let’s take a closer look at this algorithm. For example, √ suppose we want to find 17. This is the same as finding the unique positive solution to the equation x2 − 17 ...