MATH 1131Q - Calculus 1.

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2 MATH 1131Q - Calculus 1. Álvaro Lozano-Robledo Department of Mathematics University of Connecticut Day 2 Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 2 / 30

3 An Example: A Very Surprising Rate of Change The Hubble Deep Field (HDF) is an image of a small region in the constellation Ursa Major, constructed from a series of observations by the Hubble Space Telescope.

4 An Example: A Very Surprising Rate of Change Edwin Powell Hubble (November 20, September 28, 1953) American Astronomer. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 4 / 30

5 An Example: A Very Surprising Rate of Change The Hubble Deep Field (HDF) is an image of a small region in the constellation Ursa Major, constructed from a series of observations by the Hubble Space Telescope.

6 An Example: An Expanding Universe The Volume of the Universe is expanding...

7 An Example: An Expanding Universe The Volume of the Universe is expanding... at an accelerated rate!

8 About this class: MATH 1131Q - Calculus 1 Our Website for Section Main Website for all MATH 1131Q lectures. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 7 / 30

9 An Example: What is π? Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 31 / 40

10 MATH 1131Q: Calculus 1 - Section Day 2 Some reminders... UCONNCALCPROF@GMAIL.COM Register your iclicker at HuskyCT or Remember to set your clicker frequency to DD: press and hold [power button] for two seconds. Use A-E to enter two letter frequency codes. Begin to do homework with WebAssign, via HuskyCT. Reminder: do not use Safari or Internet Explorer! Safest bet: Firefox. First online homework due tomorrow, Friday, by 11:59pm. First online quiz due Saturday, by 11:59pm. Second and third online homework sets, and two quizzes, due Tuesday, by 11:59pm. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 9 / 30

11 Clicker Question... WebAssign Choose the answer that best describes your situation: 1 I m already ahead of the game, done so much online homework,... it is crazy. 2 I ve attempted some homework problems. I can do this. 3 I looked around WebAssign, seems cool. 4 What s WebAssign again? 5 I am still having trouble and can t log into WebAssign! Help me! Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 10 / 30

12 Q Center tutoring begins on Tuesday, at 11am The Q Center offers free drop-in peer tutoring on the first floor of the Homer Babbage Library. Tutoring begins Tuesday, at 11am. Schedule in Fall 2014: Sundays 1-11pm Monday-Thursday 11am-11pm Fridays 11am-3pm Our schedule of tutors and topics we tutor in in the website:

13 I will be away next week... Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 12 / 30

14 I will be away next week... Amit Savkar will teach my classes. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 13 / 30

15 It is time for... Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 14 / 30

16 It is time for... Functions Definition A function is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 15 / 30

17 It is time for... Functions Definition A function is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. The set D is called the domain of the function f. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 16 / 30

18 It is time for... Functions Definition A function is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. The set D is called the domain of the function f. The range of f is the set of all possible values f (x) as x varies throughout the elements of D. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 16 / 30

19 Some set notation Definition Some notation: 1 The set of natural numbers 1, 2, 3, 4,... will be denoted by N. 2 The set of all integers..., 1, 0, 1, 2,... will be denoted by Z. 3 The set of all rational numbers m n denoted by Q. with m Z and n N will be 4 The set of all real numbers will be denoted by R. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 17 / 30

20 Definition Let f : R R be a function (with domain R and values in R). The graph of f is the set Example {(x, y) : y = f (x), for x R} = {(x, f (x)) : x R}. Let f (x) : R R be given by the rule f (x) = x + 1. Sketch the graph of 2 f (x) Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 18 / 30

21 Definition Let f : R R be a function. 1 We say that f is increasing if f (a) < f (b) whenever a < b, 2 We say that f is decreasing if f (a) > f (b) whenever a < b. Example Let f (x) : R R be given by the rule f (x) = x + 1. Is f (x) increasing or 2 decreasing?.

22 Catalog of elementary functions 1 Linear functions, f (x) = mx + b, for some constants m, b. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 30

23 Catalog of elementary functions 1 Linear functions, f (x) = mx + b, for some constants m, b. 2 Power functions, f (x) = x a, for some real number a. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 30

24 Catalog of elementary functions 1 Linear functions, f (x) = mx + b, for some constants m, b. 2 Power functions, f (x) = x a, for some real number a. 3 Polynomials, f (x) = a n x n + + a 2 x 2 + a 1 x + a 0, for some constants a i, and non-negative integer n. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 30

25 Catalog of elementary functions 1 Linear functions, f (x) = mx + b, for some constants m, b. 2 Power functions, f (x) = x a, for some real number a. 3 Polynomials, f (x) = a n x n + + a 2 x 2 + a 1 x + a 0, for some constants a i, and non-negative integer n. 4 Rational functions f (x) = p(x), where p and q = 0 are q(x) polynomials. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 30

26 Catalog of elementary functions 1 Linear functions, f (x) = mx + b, for some constants m, b. 2 Power functions, f (x) = x a, for some real number a. 3 Polynomials, f (x) = a n x n + + a 2 x 2 + a 1 x + a 0, for some constants a i, and non-negative integer n. 4 Rational functions f (x) = p(x), where p and q = 0 are q(x) polynomials. 5 Exponential functions, f (x) = a x, where a > 0. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 30

27 Catalog of elementary functions 1 Linear functions, f (x) = mx + b, for some constants m, b. 2 Power functions, f (x) = x a, for some real number a. 3 Polynomials, f (x) = a n x n + + a 2 x 2 + a 1 x + a 0, for some constants a i, and non-negative integer n. 4 Rational functions f (x) = p(x), where p and q = 0 are q(x) polynomials. 5 Exponential functions, f (x) = a x, where a > 0. 6 Trigonometric functions, f (x) = sin(x), cos(x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 20 / 30

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29 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x) be a fixed function, and we define a new function g(x) as follows: 1 Vertical shift of C units: g(x) = f (x) + C, where C is a constant. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 30

30 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x) be a fixed function, and we define a new function g(x) as follows: 1 Vertical shift of C units: g(x) = f (x) + C, where C is a constant. 2 Horizontal shift of C units: g(x) = f (x C). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 30

31 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x) be a fixed function, and we define a new function g(x) as follows: 1 Vertical shift of C units: g(x) = f (x) + C, where C is a constant. 2 Horizontal shift of C units: g(x) = f (x C). 3 Vertical stretch: g(x) = Cf (x), with C > 0. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 30

32 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x) be a fixed function, and we define a new function g(x) as follows: 1 Vertical shift of C units: g(x) = f (x) + C, where C is a constant. 2 Horizontal shift of C units: g(x) = f (x C). 3 Vertical stretch: g(x) = Cf (x), with C > 0. 4 Horizontal stretch: g(x) = f (Cx), with C > 0. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 30

33 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x) be a fixed function, and we define a new function g(x) as follows: 1 Vertical shift of C units: g(x) = f (x) + C, where C is a constant. 2 Horizontal shift of C units: g(x) = f (x C). 3 Vertical stretch: g(x) = Cf (x), with C > 0. 4 Horizontal stretch: g(x) = f (Cx), with C > 0. 5 Vertical reflection about x-axis: g(x) = f (x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 30

34 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x) be a fixed function, and we define a new function g(x) as follows: 1 Vertical shift of C units: g(x) = f (x) + C, where C is a constant. 2 Horizontal shift of C units: g(x) = f (x C). 3 Vertical stretch: g(x) = Cf (x), with C > 0. 4 Horizontal stretch: g(x) = f (Cx), with C > 0. 5 Vertical reflection about x-axis: g(x) = f (x). 6 Horizontal reflection about y-axis: g(x) = f ( x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 22 / 30

35 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x), g(x) be fixed functions, and we define a new function h(x) as follows: 1 Add/subtract functions, h(x) = f (x) + g(x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 23 / 30

36 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x), g(x) be fixed functions, and we define a new function h(x) as follows: 1 Add/subtract functions, h(x) = f (x) + g(x). 2 Multiply/divide functions, h(x) = f (x)g(x), h(x) = f (x)/ g(x), where g(x) = 0. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 23 / 30

37 New functions from old functions We can create new functions from elementary functions using the following transformations. Let f (x), g(x) be fixed functions, and we define a new function h(x) as follows: 1 Add/subtract functions, h(x) = f (x) + g(x). 2 Multiply/divide functions, h(x) = f (x)g(x), h(x) = f (x)/ g(x), where g(x) = 0. 3 Composition of functions, h(x) = f (g(x)). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 23 / 30

38 Inverse Functions Definition Let f : R R be a function. We say f is one-to-one (or 1-1, or injective), if it never takes the same value twice, Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 24 / 30

39 Inverse Functions Definition Let f : R R be a function. We say f is one-to-one (or 1-1, or injective), if it never takes the same value twice, i.e., f (a) = f (b) whenever a = b, Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 24 / 30

40 Inverse Functions Definition Let f : R R be a function. We say f is one-to-one (or 1-1, or injective), if it never takes the same value twice, i.e., f (a) = f (b) whenever a = b, or, equivalently, if f (a) = f (b), then a = b. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 24 / 30

41 Inverse Functions Definition Let f : R R be a function. We say f is one-to-one (or 1-1, or injective), if it never takes the same value twice, i.e., f (a) = f (b) whenever a = b, or, equivalently, if f (a) = f (b), then a = b. Definition Let f be a one-to-one function with domain A and range B. Then, its inverse function f 1 has domain B and range A, and is defined by for any y in B. f 1 (y) = x f (x) = y, Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 24 / 30

42 Inverse Functions Definition The inverse function f 1 is defined by f 1 (y) = x f (x) = y. Example Let f (x) : R R be given by the rule f (x) = x + 1. Find the inverse 2 function f 1 (x). Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 25 / 30

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44 Logarithms Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 27 / 30

45 Inverse Trigonometric Functions Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 28 / 30

46 Evaluate sin(tan 1 (3x)) Example. Álvaro Lozano-Robledo (UConn) MATH 1131Q - Calculus 1 29 / 30

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