The Limit Concept. Introduction to Limits. Definition of Limit. Example 1. Example 2. Example 3 4/7/2015

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1 4/7/015 The Limit Concept Introduction to Limits Precalculus 1.1 The notion of a it is a fundamental concept of calculus. We will learn how to evaluate its and how they are used in the two basic problems of calculus: the tangent line problem and the area problem. Eample 1 Definition of Limit You are given 3 inches of wire and are asked to form a rectangle whose area is as large as possible. Determine the dimensions of the rectangle that will produce a maimum area. Eample Use the table to estimate numerically the it: (5 3) f()? Eample 3 Use the table to estimate numerically the it: f()? 1

2 4/7/015 Estimate the it: Eample Eample 5 Find the it of f() as approaches 5. 1, 5 f ( ), 5 Eample 6 Show that the it does not eist, 0 Eample 7 1 Discuss the eistence of the it 0 4 Eample 8 Discuss the eistence of the it 1 cos 0 Limits That Fail to Eist

3 4/7/015 Properties of Limits and Direct Substitution Properties of Limits and Direct Substitution Eample 9 Properties of Limits and Direct Substitution Find each it. 3 a) b) c) d) tan Find each it. Eample 10 a) 5 4 b) Techniques for Evaluating Limits Precalculus 1. 3

4 4/7/015 Eample 1 Eample Find the it: 8 Find the it: 3 4 Eample 3 Eample 4 Find the it: Approimate the it: f()? Eample 5 Approimate the it graphically: 1 cos 0 One-Sided Limits You saw that one way in which a it can fail to eist is when a function approaches a different value from the left side of c than it approaches from the right side of c. This type of behavior can be described more concisely with the concept of a one-sided it. 4

5 4/7/015 Eample 6 Find the it as 0 from the le and the it as 0 from the right for the func on. f ( ) 4 One-Sided Limits For the it of a function to eist as c, it must be true that both one-sided its eist and are equal. Eample 7 Find the it of f() as approaches. 1, f ( ) 1 6, Eample 8 To ship a package overnight, a delivery service charges $9 for the first pound and $1 for each additional pound or portion of a pound. Let represent the weight of the package and let f() represent the shipping cost. Show that the it of f() as approaches 3 does not eist. A Limit from Calculus Eample 9 You will study an important type of it from calculus the it of a difference quotient. f ( h) f ( ) h0 h For the function given by f ( h) f (). h0 h f ( ) 1, find 5

6 4/7/015 Tangent Line to a Graph The Tangent Line Problem Precalculus 1.3 Calculus is a branch of mathematics that studies rates of change of functions. If you go on to take a course in calculus, you will learn that rates of change have many applications in real life. Earlier in the tet, you learned how the slope of a line indicates the rate at which a line rises or falls. For a line, this rate (or slope) is the same at every point on the line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point. Tangent Line to a Graph For instance, in Figure 1.0, the parabola is rising more quickly at the point ( 1, y 1 ) than it is at the point (, y ). At the verte ( 3, y 3 ), the graph levels off, and at the point ( 4, y 4 ), the graph is falling. Tangent Line to a Graph To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at that point. In simple terms, the tangent line to the graph of a function f at a point P( 1, y 1 ) is the line that best approimates the slope of the graph at the point. Figure 1.0 Tangent Line to a Graph Figure 1.1 shows other eamples of tangent lines. Eample 1 Visually Approimating the Slope of a Graph Use the graph in Figure 1. to approimate the slope of the graph of f () = at the point (1, 1). Figure 1.1 Figure 1. 6

7 4/7/015 Slope and the Limit Process In Eample 1, you approimated the slope of a graph at a point by creating a graph and then eyeballing the tangent line at the point of tangency. A more precise method of approimating tangent lines makes use of a secant line through the point of tangency and a second point on the graph, as shown in Figure 1.4. Slope and the Limit Process If (, f()) is the point of tangency and ( + h, f( + h)) is a second point on the graph of f, the slope of the secant line through the two points is given by Slope of secant line The right side of this equation is called the difference quotient. The denominator h is the change in, and the numerator is the change in y. Figure 1.4 Slope and the Limit Process The beauty of this procedure is that you obtain more and more accurate approimations of the slope of the tangent line by choosing points closer and closer to the point of tangency, as shown in Figure 1.5. Slope and the Limit Process Using the it process, you can find the eact slope of the tangent line at (, f()). As h approaches 0, the secant line approaches the tangent line. From the definition above, you can see that the difference quotient is used frequently in calculus. Using the difference quotient to find the slope of a tangent line to a graph is a major concept of calculus. Figure 1.5 Eample 3 Eample 4 Find the slope of the graph of point (,8). f ( ) 3 at the Find the slope of f ( ) 3 5 when =4. 7

8 4/7/015 Eample 5 Find the formula for the slope of the graph of f ( ). What are the slopes at (-3,7) and (1,-1)? The Derivative of a Function You started with the function f() = + 1 and used the it process to derive another function, m =, that represents the slope of the graph of f at the point (, f()). This derived function is called the derivative of f at. It is denoted by f(), which is read as f prime of. The Derivative of a Function Eample 6 Find the derivative of f ( ) 4 5 Remember that the derivative f() is a formula for the slope of the tangent line to the graph of f at the point (, f()). Eample 7 Find the equation of the tangent line of f ( ) 4 5 at =1. Find f () for f ( ) 1. Then find the slopes of the graph of f at the points (4,3) and (9,4). Where does f ( ) 4 5 tangent line? have a horizontal 8

9 4/7/015 Derivative Shortcuts! Limit Process / Definition of Derivative Vs Shortcut Method If f() = c, then f () = If f() = a + b then f () = If f ( ) a b c then f () = Find the derivative of: f() = 4 f() = f ( ) 3 6 f() = 1/ f ( ) Limits at Infinity and Horizontal Asymptotes Limits at Infinity & Limits of Sequences The graph of f is shown in Figure Precalculus 1.4 Figure 1.30 Limits at Infinity and Horizontal Asymptotes Limits at Infinity and Horizontal Asymptotes From earlier work, you know that is a horizontal asymptote of the graph of this function. Using it notation, this can be written as follows. Horizontal asymptote to the left Horizontal asymptote to the right These its mean that the value of f() gets arbitrarily close to as decreases or increases without bound. 9

10 4/7/015 Limits at Infinity and Horizontal Asymptotes To help evaluate its at infinity, you can use the following definition. Find the it: Eample Eample Limits at Infinity and Horizontal Asymptotes Find the it as approaches for each function. 4 4 a) b) c) Eample 3 You are manufacturing greeting cards that cost $0.65 per card to produce. Your initial investment is $4500, which implies that the total cost of producing cards is given by C( ) Eample 3 cont d The average cost per card is given by C(). Find the average cost per card when a)=5000, b) =50,000, c) =500,000. d) What is the it of C() as approaches? C( ) 10

11 4/7/015 Limits of Sequences The following relationship shows how its of functions of can be used to evaluate the it of a sequence. Eample 4 Find the it of each sequence (assume n begins with 1) a) a 4n 5 4n 5 b) c) n n 1 n 1 4n 5 1n a n a n A sequence that does not converge is said to diverge. Eample 5 Find the it of the sequence whose nth term is a n 5 n( n 1)(n 1) 3 n 6 The Area Problem Precalculus 1.5 Limits of Summations We have used the concept of a it to obtain a formula for the sum S of an infinite geometric series Limits of Summations The following summation formulas and properties are used to evaluate finite and infinite summations. Using it notation, this sum can be written as 11

12 4/7/015 Eample 1 Evaluate the summation. 50 i1 i Eample Simplify the summation. n i n 3 S... n n n n n i1 Eample 3 Find the it of S(n) as n. n i 1 S( n) i1 n n The Area Problem You now have the tools needed to solve the second basic problem of calculus: the area problem. The problem is to find the area of the region R bounded by the graph of a nonnegative, continuous function f, the -ais, and the vertical lines = a and = b, as shown in Figure Figure 1.33 The Area Problem If the region R is a square, a triangle, a trapezoid, or a semicircle, you can find its area by using a geometric formula. For more general regions, however, you must use a different approach one that involves the it of a summation. Eample 4 Use five rectangles of equal width to approimate the area of the region bounded by f ( ) 8, the -ais, and the line =0 and =. The basic strategy is to use a collection of rectangles of equal width that approimates the region R, as illustrated in Eample 4. 1

13 4/7/015 The Area Problem Based on the procedure illustrated in Eample 4, the eact area of a plane region R is given by the it of the sum of n rectangles as n approaches. Eample 5 Find the area of the region bounded by the graph of f ( ), and the -ais between =0 and =1. Eample 6 Find the area of the region bounded by the graph of f ( ) 4, and the -ais between =1 and =3. 13