Today’s Agenda • Turn in Diagnostic Assessment • Upcoming Homework • Section 1.1: Functions and their representations • Section 1.2: A catalog of essential functions • Section 1.3: The limit of a function Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 1 / 19
Upcoming Homework • WeBWorK #1: Sections 1.1 & 1.2 due 8/26/2015 • WeBWorK #2: Section 1.3 due 8/28/2015 • WeBWorK #3: Section 1.4 due 8/31/2015 (this one is really long!) • Written HW A: Section 1.3, #6 (please use graph paper) and #18 (your table of values should include at least 10 entries - use powers of 10); Section 1.4, #10, 24, 26, 38, 44 (for #24, 26, and 38, describe in a sentence or two the method you used to find the limit). Due 9/2/2015. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 2 / 19
Section 1.1 Definition 1.1.1 A function f 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 on which f is defined is called the domain of f , and the set E = f ( D ) = { y : y = f ( x ) for some x ∈ D } is called the range of f . In other words, the range of f is the set of all output values that result from inputs in the domain of f . In our class, the codomain of a function will always be R (the set of real numbers), and we clarify the domain of a function by writing f : D → R . Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 3 / 19
Section 1.1 Example 1.1.2 What are the domain, codomain, and range of the following functions? � 5 − x � f ( x ) = g ( x ) = | x | − 5 5 + x Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 4 / 19
Section 1.1 Example 1.1.2 What are the domain, codomain, and range of the following functions? � 5 − x � f ( x ) = g ( x ) = | x | − 5 5 + x The domain of f is D = ( − 5 , 5]. The domain of g is ( −∞ , − 5] ∪ [5 , ∞ ). The codomain of both f and g is R . The range of both f and g is [0 , ∞ ). Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 4 / 19
Section 1.1 Definition 1.1.3 A piecewise defined function is a function that is defined by different formulas in different parts of its domain. Example 1.1.4 � 1 − x , x ≤ 1 f ( x ) = x 2 , x > 1 Example 1.1.5 � x , x ≥ 0 g ( x ) = − x , x < 0 Do you recognize the function in Example 1.1.5? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 5 / 19
Section 1.1 Definition 1.1.6 A function f is called increasing on an interval I if f ( x 1 ) < f ( x 2 ) whenever x 1 < x 2 in I , and it is called decreasing if f ( x 1 ) > f ( x 2 ) whenever x 1 < x 2 in I . Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 6 / 19
Section 1.1 Example 1.1.7 f : ( − π/ 2 , π/ 2) → R defined by f ( x ) = tan x is an increasing function. (Pay close attention to the domain.) Example 1.1.8 g : R → R defined by g ( x ) = 5 − x 3 is a decreasing function. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 7 / 19
Section 1.1 Definition 1.1.9 If a function f satisfies f ( x ) = f ( − x ) for every number x in its domain, then f is called an even function . If f satisfies f ( − x ) = − f ( x ), then f is called an odd function . Note: a function does not have to be either even or odd. Most functions are neither. Example 1.1.10 The function f ( x ) = sin x is odd, while g ( x ) = cos x is even. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 8 / 19
Section 1.2 Vertical and Horizontal Shifts Suppose c > 0. Given a function f : D → R , we obtain the graph of y = f ( x ) + c by shifting upward by c y = f ( x ) − c by shifting downward by c y = f ( x − c ) by shifting to the right by c y = f ( x + c ) by shifting to the left by c Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 9 / 19
Section 1.2 Vertical and Horizontal Stretching, Shrinking, and Reflecting Suppose c > 1. Given a function f : D → R , we obtain the graph of y = cf ( x ) by stretching vertically by a factor of c y = 1 c f ( x ) by shrinking vertically by a factor of c y = f ( cx ) by shrinking horizontally by a factor of c y = f ( x / c ) by stretching horizontally by a factor of c y = − f ( x ) by reflecting about the x -axis y = f ( − x ) by reflecting about the y -axis Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 10 / 19
Section 1.2 Example 1.2.1 Given that f : R → R is defined by f ( x ) = sin x , how to we obtain the graph of y = 1 5 sin( x − 3) + 7? Pay close attention to the order of operations. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 11 / 19
Section 1.2 Example 1.2.1 Given that f : R → R is defined by f ( x ) = sin x , how to we obtain the graph of y = 1 5 sin( x − 3) + 7? Pay close attention to the order of operations. First, we shift the graph to the right by 3. Then we shrink the graph vertically by a factor of 5, and finally we translate the graph vertically upward by 7. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 11 / 19
Section 1.2 Definition 1.2.2 Given two functions f and g , the composite function f ◦ g (pronounced ” f of g ” and also sometimes called the composition of f and g ) is defined by ( f ◦ g )( x ) = f ( g ( x )) . Example 1.2.3 Given f , g : R → R defined by f ( x ) = x 2 and g ( x ) = x − 3, what are f ◦ g and g ◦ f ? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 12 / 19
Section 1.2 Definition 1.2.2 Given two functions f and g , the composite function f ◦ g (pronounced ” f of g ” and also sometimes called the composition of f and g ) is defined by ( f ◦ g )( x ) = f ( g ( x )) . Example 1.2.3 Given f , g : R → R defined by f ( x ) = x 2 and g ( x ) = x − 3, what are f ◦ g and g ◦ f ? ( g ◦ f )( x ) = x 2 − 3 ( f ◦ g )( x ) = ( x − 3) 2 and Note that f ◦ g � = g ◦ f , so the order of composition matters! Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 12 / 19
Section 1.3 The concept of a limit is the main concept of calculus. Let’s take a moment to discuss what a limit is intuitively before we write down a formal definition. Suppose you have a piece of notebook paper and a pair of magic scissors that can cut anything into smaller pieces (even electrons or quarks – we’re playing pretend here). Now, cut your piece of notebook paper in half and throw it away. Then cut the remaining half in half and throw it away (now you have 1/4 of the original). Keep cutting the remaining halves in half with your magic scissors. Will you ever throw away the entire paper? Why or why not? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 13 / 19
Now let’s examine a table of values and try to make sense of what ”the limit” might be. x f(x) x f(x) 1.0 2.000000 3.0 8.000000 1.5 2.750000 2.5 5.750000 1.8 3.440000 2.2 4.640000 1.9 3.710000 2.1 4.310000 1.95 3.852500 2.05 4.152500 1.99 3.970100 2.01 4.030100 1.995 3.985025 2.005 4.015025 1.999 3.997001 2.001 4.003001 What is x approaching? Does it ever get there? What is f ( x ) approaching? Does it ever get there? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 14 / 19
Section 1.3 Definition 1.3.1 Suppose a function f : D → R is defined when x is near the number a (this usually means that f is defined on some open interval that contains a , except possibly at a itself). Then we write x → a f ( x ) = L lim and say, ”the limit of f ( x ), as x approaches a , equals L ” if we can make the values of f ( x ) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a ) but not equal to a . For example, in the previous slide, we would say that the limit of f ( x ) as x approaches 2 equals 4, or x → 2 f ( x ) = 4 . lim Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 15 / 19
Section 1.3 The key point when discussing limits is that we are talking about what happens as we approach something very closely, but never actually get there. In the paper example, we had smaller and smaller pieces of paper, but we never actually got to the point where we had no paper left. In this case, we could say that as time approaches ∞ , the amount of paper left approaches 0. Also recall that in the table above, our x values approached 2 from either side, but the value x = 2 was never in the table. Similarly, f ( x ) approached 4, but f ( x ) = 4 was never actually in the table, so for all we know, f ( x ) wasn’t even defined at x = 2! Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 16 / 19
Section 1.3 Take a few moments to discuss the following problems with the person next to you. You can use a table on your graphing calculator, or an actual graph of the function in question, to help you determine the limits. 1 x − 1 lim x 2 − 1 x → 1 √ 2 t 2 + 9 − 3 lim t 2 t → 0 3 � π � x → 0 sin lim x Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Monday, 24 August 2015 17 / 19
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