Calculus I MATH 150/151 Course Notes PDF

Calculus I MATH 150/151 Course Notes Department. of Mathematics, SFU Copyright c 2023 Department of Mathematics, SFU S ELF P UBLISHED http://www.math.sfu.ca Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License (the “License”). You may not use this do...

Calculus I MATH 150/151 Course Notes Department. of Mathematics, SFU Copyright c 2023 Department of Mathematics, SFU S ELF P UBLISHED http://www.math.sfu.ca Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License (the “License”). You may not use this document except in compliance with the License. You may obtain a copy of the License at http://creativecommons.org/licenses/by-nc-sa/4.0/. Unless required by applicable law or agreed to in writing, software distributed under the License is dis- tributed on an “AS IS ” BASIS , WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. First printing, August 2006  Contents Preface....................................................... 7 Greek Alphabet............................................... 9 I Part One: Review of Functions and Models 1 Functions and Models........................................ 13 1.1 Review: Functions 13 1.2 Mathematical Models: A Catalog of Essential Functions 19 1.3 New Functions from Old Functions 27 1.4 Exponential Functions & Inverse Functions and Logarithms 32 1.5 Summary 39 II Part Two: Differentiation 2 Limits and Differentiation...................................... 43 2.1 The Tangent and Velocity Problems 44 2.2 The Limit of a Function 47 2.3 Calculating Limits Using the Limit Laws 52 2.4 The Precise Definition of Limit (omitted) 57 2.5 Continuity 59 2.6 Limits at Infinity: Horizontal Asymptotes 66 2.7 Derivatives and Rates of Change 70 2.8 The Derivative as a Function 78 2.9 Summary 85 3 Differentiation Rules........................................... 87 3.1 Derivatives of Polynomials and Exponential Functions 88 3.2 The Product and Quotient Rules 93 3.3 Derivatives of Trigonometric Functions 97 3.4 Chain Rule 102 3.5 Implicit Differentiation 107 3.6 Derivatives of Logarithmic Functions 112 3.7 Rates of Change in the Natural and Social Sciences 116 3.8 Exponential Growth and Decay 121 3.9 Related rates 126 3.10 Linear Approximation and Differentials 132 3.11 Summary 138 4 Applications of the Derivative................................ 139 4.1 Maximum and Minimum Values 140 4.2 The Mean Value Theorem 145 4.3 How Derivatives Affect the Shape of a Graph 150 4.4 Indeterminate Forms and L’Hospital’s Rule 154 4.5 Summary of Curve Sketching 160 4.6 Optimization Problems 165 4.7 Newton’s Method 171 4.8 Summary 175 III Part Three: Parametric Curves and Polar Coords 5 Parametric Curves and Polar Coordinates..................... 179 5.1 Curves Defined by Parametric Equations 180 5.2 Polar Coordinates 187 5.3 Summary 195 IV Exam Preparation 6 Review Materials for Exam Preparation........................ 199 6.1 End of Term Review Notes 200 6.2 Final Exam Checklist 206 6.3 Final Exam Practice Questions 210 V Appendix Solutions to Exercises........................................ 217 Bibliography................................................ 233 Articles 233 Books 233 Web Sites 233 Index....................................................... 235  Preface This booklet contains the note templates for courses Math 150/151 - Calculus I at Simon Fraser University. Students are expected to use this booklet during each lecture by following along with the instructor, filling in the details in the blanks provided. Definitions and theorems appear in highlighted boxes. Next to some examples you’ll see [link to applet]. The link will take you to an online interactive applet to accompany the example - just like the ones used by your instructor in the lecture. The link above will take you to the following url [Mul22] containing all the applets: http://www.sfu.ca/~jtmulhol/calculus-applets/html/appletsforcalculus.html Try it now. No project such as this can be free from errors and incompleteness. We will be grateful to everyone who points out any typos, incorrect statements, or sends any other suggestion on how to improve this manuscript. Veselin Jungic Simon Fraser University [email protected] Jamie Mulholland Simon Fraser University [email protected] September 3, 2023  Greek Alphabet lower capital name pronunciation lower capital name pronunciation case case α A alpha (al-fah) ν N nu (new) β B beta (bay-tah) ξ Ξ xi (zie) γ Γ gamma (gam-ah) o O omicron (om-e-cron) δ ∆ delta (del-ta) π Π pi (pie) ε E epsilon (ep-si-lon) ρ P rho (roe) ζ Z zeta (zay-tah) σ Σ sigma (sig-mah) η H eta (ay-tah) τ T tau (taw) θ Θ theta (thay-tah) υ ϒ upsilon (up-si-lon) ι I iota (eye-o-tah) φ Φ phi (fie) κ K kappa (cap-pah) χ X chi (kie) λ Λ lambda (lamb-dah) ψ Ψ psi (si) µ M mu (mew) ω Ω omega (oh-may-gah) I Part One: Review of Functions and Models 1 Functions and Models............... 13 1.1 Review: Functions Basic Sets of Numbers Four Ways to Define a Function 1.2 Mathematical Models: A Catalog of Essential Func- tions Linear Functions Some Common Functions Power, Polynomial, Rational Trigonometric Functions 1.3 New Functions from Old Functions Algebra of Functions Composition of Functions 1.4 Exponential Functions & Inverse Functions and Log- arithms Exponential Functions Inverse Functions Logarithmic Function Inverse Trig Functions 1.5 Summary  1. Functions and Models In this chapter we lay down the foundations for this course. We introduce functions, how to represent them, and how to work with them. Since our functions will take real numbers as input we start with a brief review of types of numbers. 1.1 Review: Functions 1.1.1 Basic Sets of Numbers natural numbers: the set of counting numbers N = {1, 2, 3, 4, 5,...} (Some authors include 0 in this set.) integers: the set of natural numbers with their negatives Z = {... , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5,...} rational numbers: the set of ratios of integers n a o Q= : a, b ∈ Z, b 6= 0 b real numbers R: These are more difficult to define, but we already have an intuitive idea of what they are. They include all the rational numbers Q and all the numbers which fill in all the gaps between the rational numbers.  14 Chapter 1. Functions and Models 1.1.2 Four Ways to Define a Function Definition 1.1.1 A function (or map) is a rule or correspondence that associates each element of a set X, called the domain, with a unique element of a set Y , called the codomain. The range of f is the set of all elements in Y which correspond to an element of X: range f = { f (x) : x ∈ X}. Example 1.1 The following function maps each person to their age. domain = codomain = range = 1.1 Review: Functions 15 Reminder In calculus we will only consider functions whose domain and codomain consist of real numbers. Functions can then be described in various ways: (a) verbally (word description) ex. The area of a circle is π times the radius squared. (b) algebraically (by a formula) ex. A(r) = πr2 (c) numerically (by a table of values) ex. time (s) 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 velocity (m/s) 0 0.2 0.5 0.8 1.0 0.6 0.2 0 -0.1 or by a set of ordered pairs {(0, 0), (0.2, 0.2), (0.4, 0.5), (0.6, 0.8), (0.8, 1.0), (1.0, 0.6), (1.2, 0.2), (1.4, 0), (1.6, −0.1)} (d) visually (by a graph) Example 1.2 Let f (x) = x2. √ (a) Find the following values: f (2), f (−1), f (0), f (2/3), f ( 2), f (π), f (a + h). (b) Sketch the graph of f. 16 Chapter 1. Functions and Models Example 1.3 A 10-ft wall stands 5 ft from a building and a ladder of variable length L, supported by the wall, is placed so it reaches from the ground to the building. Let y denote the vertical distance from the ground to where the tip of the ladder touches the building, and let x denote the horizontal distance from the wall to the base of the ladder. (a) Find an expression for the height y as a function of x. (b) Find an expression for the length L as a function of x. (c) Determine the domain and range of the function L(x) found in part (b). Reminder If a function is given by a formula and the domain is not stated explicitly, the convention is that the domain is the set of all numbers for which the formula makes sense and defines a real number. Example 1.4 (a) Find the domain of the function 1 g(x) =. x2 − x (b) Find the domain of the function p h(t) = 16 − t 2. What is the range? 1.1 Review: Functions 17 Reminder The graph of a function f is defined to be the set of all points (x, y) in the Cartesian plane satisfying the equation y = f (x). Example 1.5 Sketch the graphs of the following functions. (a) f (x) = x + 1 (b) g(t) = t 2 + 1 ( 2x + 3 if x ≤ 0 (c) h(x) = 2 x +3 if x > 0 (d) f (x) = |x| 18 Chapter 1. Functions and Models Reminder Vertical line test for testing whether a curve is the graph of a function: If every vertical line intersects the curve at most once then the curve is a graph of a function. Example 1.6 Which curve is the graph of a function? Exercise 1.1 Find a formula for the function f graphed in the figure:  ( 2x + 3 if x ≤ 0 Exercise 1.2 Sketch the graph of h(x) =  x2 + 3 if x > 0 Exercise 1.3 √ the graph of f (x) = |x| and write it as a piecewise defined function. (a) Sketch (b) Consider g(x) = x2. Is it true that g(x) √ = x? (c) What is the relationship between g(x) = x2 and f (x) = |x|?  Exercise 1.4 Sketch the graph of p(x) = |x| + |x + 1|   1.2 Mathematical Models: A Catalog of Essential Functions 19 1.2 Mathematical Models: A Catalog of Essential Functions 1.2.1 Linear Functions Lines (linear function): A line is determined by two bits of information: and. Or, equivalently, by. Example 1.7 Find the equation of the line in each of the following cases. (a) slope = 2, containing P = (1, 3). (b) containing the points (1, 3) and (−2, 7).  20 Chapter 1. Functions and Models 1.2.2 Some Common Functions Power, Polynomial, Rational (a) Power Functions: A power function is a function of the form f (x) = xa where a is a fixed real number. 1.2 Mathematical Models: A Catalog of Essential Functions 21 22 Chapter 1. Functions and Models (b) Polynomials: A polynomial is a function of the form f (x) = an xn + an−1 xn−1 +... + a2 x2 + a1 x + a0 where n is an integer and the ai are fixed real numbers, which are called the coefficients of f. Exercise 1.5 A polynomial of degree 0 is of the form:  Exercise 1.6 A polynomial of degree 1 is of the form:  Exercise 1.7 A polynomial of degree 2 is of the form:  Exercise 1.8 A polynomial of degree 3 is of the form:  1.2 Mathematical Models: A Catalog of Essential Functions 23 (c) Rational Functions: A rational function is the ratio of two polynomials: p(x) f (x) = q(x) where p(x) and q(x) are polynomials.  24 Chapter 1. Functions and Models 1.2.3 Trigonometric Functions Let us recall the trigonometric functions sine, cosine, and tangent. Why are there 360◦ in a full rotation? (◦ is read "degrees") Radian measure of an angle: Sketch the following angles (radians) in standard position and give the measure of the angle in degrees: π π 5π 13π a) b) c) − d) 2 4 6 3 1.2 Mathematical Models: A Catalog of Essential Functions 25 Determine the coordinates of the point where the terminal side of the angle intersects the unit circle. π π a) b) 4 3 Definition: The sin and cos of an angle: 26 Chapter 1. Functions and Models We can fill out the following table θ sin θ cos θ tan θ 0 0 1 0 √ π 1 3 1 (30◦ ) √ 6 2 2 3 π 1 1 (45◦ ) √ √ 1 4 2 2 √ π 3 1 √ (60◦ ) 3 3 2 2 π (90◦ ) 1 0 − 2 Sketching the graphs of sin and cos.  1.3 New Functions from Old Functions 27 1.3 New Functions from Old Functions Big idea: take well understood functions (eg: f (x) = x2 , g(x) = sin(x)....) and perform well understood transformations (eg: shift to the right, reflection...) to create and get graphs of new, related functions. Example 1.8 Sketch the graphs of y = x2 + 2, y = (x − 1)2 , and y = (x − 1)2 + 2. Example 1.9 Sketch the graph of y = 3x2 − 6x + 1. Vertical & Horizontal Shifts: Suppose c > 0. To obtain the graph of y = f (x) + c, shift the graph of y = f (x) a distance c units upward y = f (x) − c, shift the graph of y = f (x) a distance c units downward y = f (x − c), shift the graph of y = f (x) a distance c units to the right y = f (x + c), shift the graph of y = f (x) a distance c units to the left Example 1.10 Sketch the graph of y = sin (2x).