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Unformatted text preview: 1 calculus with free online interactive materials c b na developed in XIMERA This document was typeset on December 14, 2018. GIT COMMIT: 890313068376fb7bbd1ce7a764c9a589f6882158 Copyright © 2018 Jim Fowler and Bart Snapp This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit If you distribute this work or a derivative, include the history of the document. The source code is available at: This text contains source material from several other open-source texts: • Single and Multivariable Calculus: Early Transcendentals. Guichard. Copyright © 2015 Guichard, Creative Commons Attribution-NonCommercial-ShareAlike License 3.0. • APEX Calculus. Hartman, Heinold, Siemers, Chalishajar, Bowen (Ed.). Copyright © 2014 Hartman, Creative Commons Attribution-Noncommercial 3.0. • Elementary Calculus: An Infinitesimal Approach. Keisler. Copyright © 2015 Keisler, Creative Commons AttributionNonCommercial-ShareAlike License 3.0. This book is typeset in the STIX and Gillius fonts. We will be glad to receive corrections and suggestions for improvement at: [email protected] 6.3 Contents 1 2 3 4 5 6 Understanding functions . . . . . . . . . . . 6 7 Horizontal asymptotes . . . . . . . . . 67 Continuity and the Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . 71 1.1 Same or different? . . . . . . . . . . . . 7 7.1 Roxy and Yuri like food . . . . . . . . . 72 1.2 For each input, exactly one output . . . 8 7.2 Continuity of piecewise functions . . . 73 1.3 Compositions of functions . . . . . . . 12 7.3 The Intermediate Value Theorem . . . . 76 1.4 Inverses of functions . . . . . . . . . . 14 An application of limits . . . . . . . . . . . . 79 Review of famous functions . . . . . . . . . . 19 8.1 Limits and velocity . . . . . . . . . . . 80 8.2 Instantaneous velocity . . . . . . . . . 81 Definition of the derivative . . . . . . . . . . 85 8 2.1 How crazy could it be? . . . . . . . . . 20 2.2 Polynomial functions . . . . . . . . . . 21 2.3 Rational functions . . . . . . . . . . . . 23 9.1 Slope of a curve . . . . . . . . . . . . . 86 2.4 Trigonometric functions . . . . . . . . 25 9.2 The definition of the derivative . . . . . 87 2.5 Exponential and logarithmic functions . 32 Derivatives as functions . . . . . . . . . . . 92 What is a limit? . . . . . . . . . . . . . . . . 35 9 10 10.1 Wait for the right moment . . . . . . . 93 3.1 Stars and functions . . . . . . . . . . . 36 10.2 The derivative as a function . . . . . . 94 3.2 What is a limit? . . . . . . . . . . . . . 37 10.3 Differentiability implies continuity . . 97 3.3 Continuity . . . . . . . . . . . . . . . . 42 Limit laws . . . . . . . . . . . . . . . . . . . . 45 11 Rules of differentiation . . . . . . . . . . . . 100 11.1 Patterns in derivatives . . . . . . . . . 101 11.2 Basic rules of differentiation . . . . . . 103 4.1 Equal or not? . . . . . . . . . . . . . . 46 4.2 The limit laws . . . . . . . . . . . . . . 47 4.3 The Squeeze Theorem . . . . . . . . . 51 11.3 The derivative of the natural exponential function . . . . . . . . . . . . 107 (In)determinate forms . . . . . . . . . . . . . 54 11.4 The derivative of sine . . . . . . . . . 109 Product rule and quotient rule . . . . . . . 111 5.1 Could it be anything? . . . . . . . . . . 55 5.2 Limits of the form zero over zero . . . . 56 12.1 Derivatives of products are tricky . . . 112 5.3 Limits of the form nonzero over zero . . 60 12.2 The Product rule and quotient rule . . 113 Using limits to detect asymptotes . . . . . . . 63 12 13 Chain rule . . . . . . . . . . . . . . . . . . . 116 6.1 Zoom out . . . . . . . . . . . . . . . . 64 13.1 An unnoticed composition . . . . . . . 117 6.2 Vertical asymptotes . . . . . . . . . . . 65 13.2 The chain rule . . . . . . . . . . . . . 118 13.3 14 15 16 17 20 21 Concepts of graphing functions . . . . . . . 184 21.1 What’s the graph look like? . . . . . . 185 21.2 Concepts of graphing functions . . . . 186 14.1 Rates of rates . . . . . . . . . . . . . . 124 14.2 Higher order derivatives and graphs . . 125 14.3 Concavity . . . . . . . . . . . . . . . 128 22.1 Wanted: graphing procedure . . . . . 189 14.4 Position, velocity, and acceleration . . 130 22.2 Computations for graphing functions . 190 Implicit differentiation . . . . . . . . . . . . 131 22 23 Computations for graphing functions . . . . 188 Mean Value Theorem . . . . . . . . . . . . 197 15.1 Standard form . . . . . . . . . . . . . 132 23.1 Let’s run to class . . . . . . . . . . . . 198 15.2 Implicit differentiation . . . . . . . . . 133 23.2 The Extreme Value Theorem . . . . . 199 15.3 Derivatives of inverse exponential functions . . . . . . . . . . . . . . . . 139 23.3 The Mean Value Theorem . . . . . . . 203 24 Logarithmic differentiation . . . . . . . . . 141 Linear approximation . . . . . . . . . . . . 206 24.1 Replacing curves with lines . . . . . . 207 Linear approximation . . . . . . . . . 208 16.1 Multiplication to addition . . . . . . . 142 24.2 16.2 Logarithmic differentiation . . . . . . 143 24.3 Explanation of the product and chain rules215 Derivatives of inverse functions . . . . . . . 146 25 Optimization . . . . . . . . . . . . . . . . . 218 We can figure it out . . . . . . . . . . 147 25.1 A mysterious formula . . . . . . . . . 219 17.2 Derivatives of inverse trigonometric functions . . . . . . . . . . . . . . 148 25.2 Basic optimization . . . . . . . . . . . 220 17.3 19 120 Higher order derivatives and graphs . . . . 123 17.1 18 Derivatives of trigonometric functions 26 Applied optimization . . . . . . . . . . . . . 223 The Inverse Function Theorem . . . . 152 26.1 Volumes of aluminum cans . . . . . . 224 More than one rate . . . . . . . . . . . . . . 154 26.2 Applied optimization . . . . . . . . . 225 18.1 A changing circle . . . . . . . . . . . 155 18.2 More than one rate . . . . . . . . . . . 156 27.1 A limitless dialogue . . . . . . . . . . 232 Applied related rates . . . . . . . . . . . . . 161 27.2 L’Hôpital’s rule . . . . . . . . . . . . 233 27 L’Hôpital’s rule . . . . . . . . . . . . . . . . 231 19.1 Pizza and calculus, so cheesy . . . . . 162 19.2 Applied related rates . . . . . . . . . . 163 28.1 Jeopardy! Of calculus . . . . . . . . . 239 Maximums and minimums . . . . . . . . . 171 28.2 Basic antiderivatives . . . . . . . . . . 240 28.3 Falling objects . . . . . . . . . . . . . 248 20.1 More coffee . . . . . . . . . . . . . . 172 20.2 Maximums and minimums . . . . . . 173 28 29 Antiderivatives . . . . . . . . . . . . . . . . 238 Approximating the area under a curve . . . 250 30 31 29.1 What is area? . . . . . . . . . . . . . . 251 29.2 Introduction to sigma notation . . . . . 252 29.3 Approximating area with rectangles . . 256 Definite integrals . . . . . . . . . . . . . . . 264 30.1 Computing areas . . . . . . . . . . . . 265 30.2 The definite integral . . . . . . . . . . 266 Antiderivatives and area . . . . . . . . . . . 274 31.1 Meaning of multiplication . . . . . . . 275 31.2 Relating velocity, displacement, antiderivatives and areas . . . . . . . 276 32 First Fundamental Theorem of Calculus . . 281 32.1 What’s in a calculus problem? . . . . . 282 32.2 The First Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . 283 33 Second Fundamental Theorem of Calculus . 287 33.1 A secret of the definite integral . . . . 288 33.2 The Second Fundamental Theorem of Calculus . . . . . . . . . . . . . . 289 33.3 34 35 36 A tale of three integrals . . . . . . . . 292 Applications of integrals . . . . . . . . . . . 294 34.1 What could it represent? . . . . . . . . 295 34.2 Applications of integrals . . . . . . . . 296 The idea of substitution . . . . . . . . . . . 302 35.1 Geometry and substitution . . . . . . . 303 35.2 The idea of substitution . . . . . . . . 304 Working with substitution . . . . . . . . . . 308 36.1 Integrals are puzzles! . . . . . . . . . 309 36.2 Working with substitution . . . . . . . 310 36.3 The Work-Energy Theorem . . . . . . 315 Index . . . . . . . . . . . . . . . . . . . . . . . . 319 Understanding functions 1 Understanding functions After completing this section, students should be able to do the following. • State the definition of a function. • Find the domain and range of a function. • Distinguish between functions by considering their domains. • Determine where a function is positive or negative. • Plot basic functions. • Perform basic operations and compositions on functions. • Work with piecewise defined functions. • Determine if a function is one-to-one. • Recognize different representations of the same function. • Define and work with inverse functions. • Plot inverses of basic functions. • Find inverse functions (algebraically and graphically). • Find the largest interval containing a given point where the function is invertible. • Determine the intervals on which a function has an inverse. 6 Same or different? Break-Ground: 1.1 Multiple Choice: Same or different? Check out this dialogue between two calculus students (based on a true story): Devyn: Riley, I have a pressing question. Riley: Tell me. Tell me everything. Devyn: Think about the function () = 2 − 3 + 2 . −2 Riley: OK. Devyn: Is this function equal to () = − 1? Riley: Well if I plot them with my calculator, they look the same. Devyn: I know! Riley: And I suppose if I write 2 − 3 + 2 −2 ( − 1)( − 2) = −2 =−1 () = (a) These are the same function although they are represented by different formulas. (b) These are different functions because they have different formulas. Problem 5. Let () = sin2 () and () = sin2 (). The domain of each of these functions is all real numbers. Which of the following statements are true? Multiple Choice: (a) There is not enough information to determine if = . (b) The functions are equal. (c) If ≠ , then ≠ . (d) We have ≠ since uses the variable and uses the variable . = (). Devyn: Sure! But what about when = 2? In this case (2) = 1 but (2) is undefined! Riley: Right, (2) is undefined because we cannot divide by zero. Hmm. Now I see the problem. Yikes! Problem 1. In the context above, are and the same function? Problem 2. Suppose and are functions but the domain of is different from the domain of . Could it be that and are actually the same function? Problem 3. Can the same function be represented by different formulas? √ Problem 4. Are () = || and () = 2 the same function? 7 For each input, exactly one output Dig-In: 1.2 For each input, exactly one output Life is complex. Part of this complexity stems from the fact that there are many relationships between seemingly unrelated events. Armed with mathematics, we seek to understand the world. Perhaps the most relevant “real-world” relation is the position of an object with respect to time. Our observations seem to indicate that every instant in time is associated to a unique positioning of the objects in the universe. You may have heard the saying, you cannot be two places at the same time, and it is this fact that motivates our definition for functions. Definition. A function is a relation between sets where for each input, there is exactly one output. Question 1. If our function is the “position with respect to time” of some object, then the input is Multiple Choice: (a) position (b) time (c) none of the above and the output is Multiple Choice: (a) position (b) time 8 (c) none of the above Something as simple as a dictionary could be thought of as a relation, as it connects words to definitions. However, a dictionary is not a function, as there are words with multiple definitions. On the other hand, if each word only had a single definition, then a dictionary would be a function. Question 2. Which of the following are functions? Select All Correct Answers: (a) Mapping words to their definition in a dictionary. (b) Given an object traveling through space, mapping time to the position of the object in space. (c) Mapping people to their birth date. (d) Mapping mothers to their children. What we are hoping to convince you is that the following are true: (a) The definition of a function is well-grounded in a real context. (b) The definition of a function is flexible enough that it can be used to model a wide range of phenomena. Whenever we talk about functions, we should explicitly state what type of things the inputs are and what type of things the outputs are. In calculus, functions often define a relation from (a subset of) the real numbers (denoted by ℝ) to (a subset of) the real numbers. Definition. We call the set of the inputs of a function the domain, and we call the set of the outputs of a function the range. Example 1. Consider the function that maps from the real numbers to the real numbers by taking a number and mapping For each input, exactly one output it to its cube: 3 2 1↦1 1 −2 ↦ −8 1.5 ↦ 3.375 −2 and so on. This function can be described by the formula () = 3 or by the graph shown in the plot below: 5 −1.5 −1 −0.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 −1 −2 Observe that here we have multiple inputs that give the same output. This is not a problem! To be a function, we merely need to check that for each input, there is exactly one output, and this condition is satisfied. −2 −1.5 0.5 1 1.5 2 Question 3. Compute: ⌊2.4⌋ −5 Question 4. Compute: Warning. A function is a relation (such that for each input, there is exactly one output) between sets. The formula and the graph are merely descriptions of this relation. • A formula describes the relation using symbols. • A graph describes the relation using pictures. The function is the relation itself, and is independent of how it is described. Our next example may be a function that is new to you. It is the greatest integer function. Example 2. Consider the greatest integer function. This function maps any real number to the greatest integer less than or equal to . People sometimes write this as () = ⌊⌋, where those funny symbols mean exactly the words above describing the function. For your viewing pleasure, here is a graph of the greatest integer function: ⌊−2.4⌋ Notice that both the functions described above pass the socalled vertical line test. Theorem 1. The curve = () represents as a function of at = if and only if the vertical line = intersects the curve = () at exactly one point. This is called the vertical line test. Sometimes the domain and range are the entire set of real numbers, denoted by ℝ. In our next examples we show that this is not always the case. Example 3. Consider the function that maps non-negative real numbers to their positive square root. This function can be described by the formula √ () = . The domain is 0 ≤ < ∞, which we prefer to write as [0, ∞) in interval notation. The range is [0, ∞). Here is a graph of = (): 9 For each input, exactly one output Finally, we will consider a function whose domain is all real numbers except for a single point. 4 2 Example 5. Are −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 () = −2 2 − 3 + 2 ( − 2)( − 1) = −2 ( − 2) and −4 () = − 1 To really tease out the difference between a function and its description, let’s consider an example of a function with two different descriptions. Example 4. Explain why √ 2 = ||. the same function? Let’s use a series of steps to think about this question. First, what if we compare graphs? Here we see a graph of : √ Although 2 may appear to simplify to just , let’s see what happens when we plug in some values. √ √ √ √ (−3)2 = 9 32 = 9 and = 3. = 3, √ Let () = 2 . We see that for any negative √ , = () = (−||) = (−||)2 = −||, and, therefore, √ √ 2 2 √|| = ||. Hence ≠ . Rather √ we see that 2 = ||. The domain of () = 2 is (−∞, ∞) and the range is [0, ∞). For your viewing pleasure we’ve included a graph of = (): 3 2 1 −2 −1 1 2 −1 −2 −3 3 On the other hand, here is a graph of : 2 1 −3 10 −2 −1 1 2 3 3 4 For each input, exactly one output 3 2 1 −2 −1 1 2 3 4 −1 −2 −3 Second, what if we compare the domains? We cannot evaluate at = 2. This is where is undefined. On the other hand, there is no value of where we cannot evaluate . In other words, the domain of is (−∞, ∞). Since these two functions do not have the same graph, and they do not have the same domain, they must not be the same function. However, if we look at the two functions everywhere except at = 2, we can say that () = (). In other words, () = − 1 when ≠ 2. From this example we see that it is critical to consider the domain and range of a function. 11 Compositions of functions and so Dig-In: 1.3 Compositions of functions (()) = ( + 7) Given two functions, we can compose them. Let’s give an example in a “real context.” Example 6. Let = ( + 7)2 + 5( + 7) + 4. Now let’s try an example with a more restricted domain. () = the amount of gas one can buy with dollars, Example 8. Suppose we have: () = 2 √ () = and let () = how far one can drive with gallons of gas. What does (()) represent in this setting? With (()) we first relate how far one can drive with gallons of gas, and this in turn is determined by how much money one has. Hence (()) represents how far one can drive with dollars. Composition of functions can be thought of as putting one function inside another. We use the notation ( ◦)() = (()). This means that √ the domain of ◦ is 0 ≤ < ∞. Next, we substitute for each instance of found in () = 2 and so √ (()) = ( ) (√ )2 = . {the range of } is contained in or equal to {the domain of } () = 2 + 5 + 4 for −∞ < < ∞, () = + 7 for −∞ < < ∞. Find (()) and state its domain. The range of is −∞ < < ∞, which is equal to the domain of . This means the domain of ◦ is −∞ < < ∞. Next, we substitute + 7 for each instance of found in () = 2 + 5 + 4 12 for 0 ≤ < ∞. Find (()) and state its domain. The domain of is 0 ≤ < ∞. From this we can see that the range of is 0 ≤ < ∞. This is contained in the domain of . Warning. The composition ◦ only makes sense if Example 7. Suppose we have for −∞ < < ∞, We can summarize our results as a piecewise function, which looks somewhat interesting: ⎧ ⎪ if 0 ≤ < ∞ ( ◦)() = ⎨ ⎪undefined otherwise. ⎩ Example 9. Suppose we have: √ () = for 0 ≤ < ∞, () = 2 for −∞ < < ∞. Compositions of functions Find (()) and state its domain. While the domain of is −∞ < < ∞, its range is only 0 ≤ < ∞. This is exactly the domain of . This means that the domain of ◦ is −∞ < < ∞. Now we may substitute 2 for each instance of found in √ () = and so (()) = (2 ) √ = 2 , = ||. Compare and contrast the previous two examples. We used the same functions for each example, but composed them in different ways. The resulting compositions are not only different, they have different domains! 13 Inverses of functions So, we could rephrase these conditions as Dig-In: 1.4 ( −1 ()) = Inverses of functions If a function maps every “input” to exactly one “output,” an inverse of that function maps every “output” to exactly one “input.” We need a more formal definition to actually say anything with rigor. Definition. Let be a function with domain and range : and −1 ( ()) = . These two simple equations are somewhat more subtle than they initially appear. Question 5. Let be a function. If the point (1, 9) is on the graph of , what point must be the the graph of −1 ? Warning. This notation can be very confusing. Keep a watchful eye: −1 () = the inverse function of evaluated at . 1 ()−1 = . () Question 6. Which of the following is notation for the inverse of the function sin() on the interval [−∕2, ∕2]? Let be a function with domain and range : Multiple Choice: (a) sin−1 () (b) sin()−1 We say that and are inverses of each other if (()) = for all in , and also ( ()) = for all in . Sometimes we write = −1 in this case. ◦ −1 Question 7. Consider the graph of = () below 2 1.5 1 and 0.5 −1 ◦ 0.5 14 1 1.5 2 Inverses of functions Is () invertible at = 1? that So far, we’ve only dealt with abstract examples. Let’s see if we can ground this in a real-life context. since we solved for −1 in our calculation. On the other hand, (( ) ) 9 −1 ( ()) = −1 + 32 5 (( ) ) 5 9 + 32 − 32 = 9 5 = Exampl...
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