This Week. Professor Christopher Hoffman Math 124
|
|
- Warren Moore
- 5 years ago
- Views:
Transcription
1 This Week Sections ,2.5,2.6 First homework due Tuesday night at 11:30 p.m. Average and instantaneous velocity worksheet Tuesday available at m124/ (under week 2) print it out before coming to class
2 Velocity If an arrow is shot upward on the moon with a velocity of 58 meters per second. It height in meters is given by p(t) = 58t 0.83t 2. 1 Find the average velocity over the time intervals [1, 2], [1, 1.5], [1, 1.1], [1, 1.01], and [1, 1.001]. 2 Find the instantaneous velocity at t = 1.
3 Average Velocity between 1 and t Average velocity is change in position divided by change in time. Average velocity between time 1 and t is p(t) p(1). t 1 p(t) h(1) t 1 t
4 Instantaneous Velocity To see what happens as h approaches 0 we do a little algebra. p(1 + h) p(1) (1 + h) 1 = 58(1 + h) 0.83(1 + h)2 ( (1) 2 ) h = h 0.83(1 + 2h + h2 ) ( ) h = 58h 1.66h 0.83h2 ) h = h As h approaches 0 the average velocity approaches We say the instantaneous velocity at t = 1 is and write p(1 + h) p(1) Instantaneous velocity = lim = h 0 h
5 Geometric Interpretation p(1+h) p(1) h is the slope of the secant line between (1, p(1)) and (1 + h, p(1 + h)). As h approaches 0 the secant lines approach the tangent line at (1, p(1)). The instantaneous velocity of is the slope of the tangent line at (1, p(1)).
6 Two Interpretations All of our work over the next few weeks can be interpreted as finding methods to determine the instantaneous velocity for an arbitrary position function. Geometrically, our work can be interpreted as determining the slope of a tangent line for an arbitrary point on the graph of an arbitrary function. In order to do this we first must discuss what we mean by the limit of a function as we approach a point.
7 Limits lim x a f (x) = L means that we can make the value of f (x) arbitrarily close to L by taking x sufficiently close to a but not equal to a. In this picture lim x 2 f (x) = 4.
8 lim x a f (x) does not depend on the value of f (a). In all of these pictures lim x 0 f (x) = 1. For most of the limits lim x a f (x) = L that we take in this course f (a) will not be defined.
9 Naive Idea We will plug in small values of x to try to find lim x 0 sin(x). x sin(x) x x ± ±1/ ± ± ± Based on this we might be tempted to say lim x 0 sin(x)/x = 1.
10 Graphical Interpretation Looking at the graph of y = sin x/x we can see that lim sin(x)/x = 1. x 0
11
12 Precise Definition of a Limit The precise definition of a limit is contained in Section 2.4. We won t cover this section in class. We will be content to know limits when we see them.
13 Problematic Example Find lim sin(π/x). x 0 sin(π) x x ±1 0 ±1/2 0 ±1/3 0 ±1/10 0 ±1/100 0 ±1/n 0 Based on this we might be tempted to say lim sin(π/x) = 0. x 0
14 But if we would have chosen x = n then we would have seen a different behavior as the points approached 0. For these values ( π ) sin(π/x) = sin 2 + 2nπ = 1. Also if we would have chosen x = n then we would have sin(π/x) = sin ( 3π 2 + 2nπ) = 1.
15 sin(π/x) oscillates between 1 and 1 infinitely often as x 0. Thus lim x 0 sin(π/x) does not exist.
16 Check your work On every limit calculation where you want to find lim f (x) x a you should plug in values of x near a. If the numbers you get are not close to your answer then you did something wrong.
17 Calculating limits graphically
18 Calculating limits graphically For the function h whose graph is given, state the value of each quantity if it exists, If it does not exist, explain why 1 lim x 3 h(x) 2 lim x 3 + h(x) 3 lim x 3 h(x) 4 h( 3) 5 lim x 0 h(x) 6 h(0) 7 lim x 2 h(x) 8 h(2) 9 lim x 5 + h(x) 10 lim x 5 h(x)
19 Calculating limits graphically For the function h whose graph is given, state the value of each quantity if it exists, If it does not exist, explain why 1 lim x 3 h(x) = 4 2 lim x 3 + h(x) = 4 3 lim x 3 h(x) = 4 4 h( 3) does not exist 5 lim x 0 h(x) does not exist 6 h(0) = 1 7 lim x 2 h(x) = 2 8 h(2) does not exist 9 lim x 5 + h(x) = 3 10 lim x 5 h(x) does not exist.
20 Infinite limits Look at a graph of the function f (x) = ln(x). The graph has a vertical asymptote at x = 0. As x 0 + the function ln(x) gets more and more negative. We say lim ln(x) =. x 0 +
21 For every n the line x = π/2 + nπ is a vertical asymptote. We can see that the one sided limits are lim tan(x) = and lim tan(x) =. x π/2 x π/2 +
22 The line x = a is a vertical asymptote of y = f (x) if at least one of the following is true lim x a f (x) = or lim x a + f (x) = or lim x a f (x) = or
23 Calculating limits graphically For the function R whose graph is given, state the following 1 lim x 2 R(x) 2 lim x 5 R(x) 3 lim x 3 R(x) 4 lim x 3 + R(x) 5 The equations of the vertical asymptotes.
24 Calculating limits graphically 1 lim x 2 R(x) = 2 lim x 5 R(x) = 3 lim x 3 R(x) does not exist because lim x 3 R(x) = while lim x 3 + R(x) = 4 The equations of the vertical asymptotes are x = 3, x = 2 and x = 5.
25 A function f is continuous at a if lim f (x) = f (a). x a This definition requires three things to happen. 1 f (a) exists 2 lim x a + f (x) = f (a) 3 lim x a f (x) = f (a)
26 Graphs of Continuous Functions This function is continuous at every point except x = 2, 2, 4 and 6.
27 Continuous Functions Any function defined by the following functions is continuous at every point in its domain. polynomials root functions trigonometric functions inverse trigonometric functions exponential functions and logarithmic functions
28 Examples of Continuous Functions All of the following functions are continuous wherever they are defined. ( ) f (x) = sin 3+x x 2 +1 g(t) = e 3t/4+ln(2t) r(x) = x x 5 m(t) = 3t + π tan 1 (4 cos(t π))
29 Where are the following functions continuous 1 x 2 sin(x) (x 3)(x+4) is continuous except when x = 3 and x = 4 2 ln(x 2 1) is continuous except when 1 x 1 3 e x/(x 1) is continuous except when x = 1.
30 Evaluating Limits of Continuous Functions This is the easiest thing we will do all class all quarter. If f (x) is continuous at a then lim f (x) = f (a). x a
31 Evaluating Limits of Continuous Functions Find the following limits. 1 lim x 3 3πx+1 x 2 6 = 2 lim y 2 sin 3y+10 5y lim x 2 3cx+12 7cx+1 = 9π+1 3 = sin(2) cx 1 14c for any c < 2 and c 1/14. When evaluating limits in this course evaluating the function at the point should always be your first approach. If this fails then try pugging in numbers close to a. After you make a guess you must justify your answer.
32 Find lim h 0 (2 + h) 3 8. h Plugging in h = 0 we get 0 0. (2 + h) 3 8 lim h 0 h = lim h 0 (2 + h) 3 8 h h + 6h 2 + h 3 8 = lim h 0 h 12h + 6h 2 + h 3 = lim h 0 h = lim h + h 2 h 0 = 12. If two functions agree everywhere except at one point a then the limits approaching a are the same.
33 Find lim t 0 1/(7 + 3t) 1/7. t Plugging in t = 0 we get /(7 + 3t) 1/7 lim t 0 t 7 (7 + 3t) = lim t 0 t(7 + 3t)(7) 3t = lim t 0 t(7 + 3t)7 3 = lim t 0 (7 + 3t)7 = 3 49.
34 Rationalize the denominator Find lim t 16 Plugging in t = 16 we get t 4 t. lim t t 4 t (16 t)(4 + t) = lim t 16 (4 t)(4 + t) (16 t)(4 + t) = lim t 16 (16 t) = lim t t = 8
35 Find ( 3 lim x 0 x 3 ) x 3. + x Plugging in x = 0 we get. ( 3 lim x 0 x 3 ) x 3 + x = lim x 0 3(x 2 + 1) 3 x 3 + x 3x 2 = lim x 0 x 3 + x = lim x 0 = = 0. 3x x 2 + 1
36 Find Plugging in x = 3 we get 0 0. x 3 27 lim x 3 x 3 x 3 27 lim x 3 x 3 = lim x 0 (x 3)(x 2 + 3x + 9) x 3 = lim x 0 x 2 + 3x + 9 = 27.
37 Limit Laws Suppose c is a constant and the limits exist. Then lim f (x) and lim g(x) x a x a 1 lim x a [ f (x) + g(x) ] = limx a f (x) + lim x a g(x) 2 lim x a [ f (x) g(x) ] = limx a f (x) lim x a g(x) 3 lim x a [ cf (x) ] = c limx a f (x) 4 lim x a [ f (x) g(x) ] = limx a f (x) lim x a g(x) 5 if lim x a g(x) 0 f (x) lim x a g(x) = lim x a f (x) lim x a g(x)
38 Find ( lim cos( 3π ) t 1 t + 1 ) 3 (3(t + 1) 2 4t). This looks like a mess. Let s look at the graph.
39 Find ( lim cos( 3π ) t 1 t + 1 ) 3 (3(t + 1) 2 4t). This looks like a mess. Let s look at the graph.
40 Here is the graph of the function ( g(t) = cos( 3π ) t + 1 ) 3 (3(t + 1) 2 4t) along with f (t) = 2(3(t + 1) 2 4t) and h(t) = 4(3(t + 1) 2 4t). Notice that as t 1 all three functions are approaching 0.
41 As these are continuous functions and defined at t = 1 we get and lim t 1 2(3(t + 1)2 4t) = 2(3( 1 + 1) 4( 1) = 0 lim 4(3(t + 1)2 4t) = 4(3( 1 + 1) 4( 1) = 0. t 1 The function that we are interested in ( g(t) = cos( 3π ) t + 1 ) 3 (3(t + 1) 2 4t) is sandwiched in between so lim t 1 g(t) must be zero as well.
42 Sandwich Theorem This theorem makes the previous slide precise. Theorem If f (x) g(x) h(x) in some interval around a and lim (f (x)) = lim (h(x)) = L x a x a then lim (g(x)) = L x a
43 Find lim x 14 x x. First we plug in x = 14 and get = 0 28 = 0. As this is a continuous function defined at 14 we have lim x 14 x x = 0.
44 Limit Laws and Graphs Here is the graph of a function f (x) Find the following limits. 1 lim x 2 (f (x) 2 ) 2 lim x 2 (f (x)f (x + 3)) 3 lim x 5 f (x)(2 f (x 3)) 2
45 Let f (x) = x 1 (x+2)x 2. 1 Find all vertical asymptotes of f (x). 2 Find the one sided limits of f (x) as x approaches the asymptotes. The function is a rational function so it is continuous at every point where it is defined. It is defined everywhere the denominator is not zero, which is when x = 0 and x = 2. It is important to keep signs right. We must be careful about the terms close to zero.
46 As x 0 +, x 2 approaches 0 from the right. Also when x 0, x 2 approaches 0 from the right. As x 0 the numerator x 1 approaches 1 and x + 2 approaches 2. Thus as x 0 from either side the numerator is negative and the denominator is small and positive. The fraction is thus large and negative. Because of this lim x 0 x 1 (x + 2)x 2 =.
47 As x 2 + the numerator x 1 approaches 3 and the denominator approaches 0 from the positive side. As the numerator is negative and the denominator is small and positive the fraction is large and negative. Because of this x 1 lim x 2 + (x + 2)x 2 =. As x 2 the numerator x 1 approaches 3 and the denominator approaches 0 from the negative side. As the numerator is negative and the denominator is small and negative the fraction is large and positive. Thus x 1 lim x 2 (x + 2)x 2 =.
48 When we graph the function we can see that there are vertical asymptotes at x = 2 and x = 0.
49 1 Find all vertical asymptotes of x 1 (x+2)x 2. 2 Find the one sided limits of f (x) as x approaches the asymptotes. The function is a rational function so it is continuous at every point where it is defined. It is defined everywhere the denominator is not zero, which is when x = 0 and x = 2. It is important to keep signs right. We must be careful about the terms close to zero.
50 As x 0 +, x 2 approaches 0 from the right. Also when x 0, x 2 approaches 0 from the right. As x 0 the numerator x 1 approaches 1 and x + 2 approaches 2. Thus as x 0 from either side the numerator is negative and the denominator is small and positive. The fraction is thus large and negative. Because of this lim x 0 x 1 (x + 2)x 2 =.
51 As x 2 + the numerator x 1 approaches 3 and the denominator approaches 0 from the positive side. As the numerator is negative and the denominator is small and positive the fraction is large and negative. Because of this x 1 lim x 2 + (x + 2)x 2 =. As x 2 the numerator x 1 approaches 3 and the denominator approaches 0 from the negative side. As the numerator is negative and the denominator is small and negative the fraction is large and positive. Thus x 1 lim x 2 (x + 2)x 2 =.
52 Limit Laws Suppose c is a constant and the limits exist. Then lim f (x) and lim g(x) x a x a 1 lim x a [ f (x) + g(x) ] = limx a f (x) + lim x a g(x) 2 lim x a [ f (x) g(x) ] = limx a f (x) lim x a g(x) 3 lim x a [ cf (x) ] = c limx a f (x) 4 lim x a [ f (x) g(x) ] = limx a f (x) lim x a g(x) f (x) limx a f (x) 5 lim x a g(x) = lim x a g(x) if lim x a g(x) 0
53 To infinity and beyond
54 Definition of Limits at We write lim f (x) = L x when the values of f (x) can be made arbitrarily close to L by taking x sufficiently large. Similarly we write lim f (x) = L x when the values of f (x) can be made arbitrarily close to L by taking sufficiently large negative values for x.
55 Limits at Graphically Consider the function f (x) = 3 2e x As x gets large f (x) gets closer to 3. The graph of f (x) gets closer to the graph of y = 3 and we write lim 3 x 2e x = 3.
56 Limits at Graphically As x gets more and more negative 3 2e x gets more and more negative as well. We write lim 3 x 2e x =.
57 Horizontal Asymptotes The line y = L is a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x x Thus y = π/2 and y = π/2 are horizontal asymptotes and lim x tan 1 (x) = π/2 and lim tan 1 (x) = π/2 x
58 Limits of polynomials at If r > 0 then lim x x r = If r < 0 then lim x x r = 0 If r is even then lim x x r = If r is odd then lim x x r =
59 Limits Laws at Suppose c is a constant and the limits exist. Then lim f (x) and lim g(x) x x 1 lim x [ f (x) + g(x) ] = limx f (x) + lim x g(x) 2 lim x [ f (x) g(x) ] = limx f (x) lim x g(x) 3 lim x [ cf (x) ] = c limx f (x) 4 lim x [ f (x) g(x) ] = limx f (x) lim x g(x) f (x) limx f (x) 5 lim x g(x) = lim x g(x) if lim x g(x) 0
60 Extended Real Numbers When working with limits it is useful to use the extended real numbers. = + = + a = a = if a > 0 a = if a < 0 =???
61 Rational Functions Find the following limits. lim x ( 5x 3 + 2(r 1)x) x lim 3/2 +2πx 7 2x x 1 5x lim 3 +2(r 1)x x x 4 +1
62 Polynomials With polynomials the largest term dominates. To see this we factor out the largest power of x. lim ( 5x 3 + 2(r 1)x) = lim x 3 ( 5 + 2(r 1)/x 2 ) x x = lim lim ( 5 + 2(r 1)/x 2 ) x = ( )( 5) = x x 3
63 Divide by the largest power of the denominator For functions like this we divide top and bottom by the largest power of the denominator. x 3/2 + 2πx 7 2x lim x x 2 1 = lim x (1/x 2 )(x 3/2 + 2πx 7 2x) (1/x 2 )(x 2 1) = lim x x 1/2 + 2πx 5 2/x) 1 1/x 2 = lim x (x 1/2 + 2πx 5 2/x) lim x (1 1/x 2 ) = =
64 Divide by the largest power of the denominator For this problem we want to divide by x 2 = x 4. 5x 3 + 2(r 1)x lim x x (1/x 2 )( 5x 3 + 2(r 1)x) = lim x (1/x 2 ) x = lim x 5x + 2(r 1)/x 1 + 1/x 4 = lim x ( 5x + 2(r 1)/x) lim x 1 + 1/x 4 = 5( ) =
65 ( ) lim x 2 + 3x + 2 x x We rationalize the numerator by multiplying the expression by x 2 + 3x x x 2 + 3x x ( ) lim x 2 + 3x + 2 x x ( = lim x 2 + 3x + 2 x x x 2 + 3x + 2 x 2 = lim x x 2 + 3x x 3x + 2 = lim x x 2 + 3x x ) x 2 + 3x x x 2 + 3x x
66 Then we multiply numerator and denominator by 1/x. ( ) lim x 2 + 3x + 2 x x = lim x (3x + 2)(1/x) ( x 2 + 3x x)(1/x) 3 + 2/x = lim x ( 1 + 3/x + 2/x 2 + 1) lim x (3 + 2/x) = lim x ( 1 + 3/x + 2/x 2 + 1) 3 = = 3 2
67 For all real numbers a > 0, b and c find ( ) ax 2 + bx + c 5x and lim x lim x ( ) ax 2 + bx + c 5x As in the previous problem we will multiply the expression times ax 2 + bx + c + 5x ax 2 + bx + c + 5x and then factor out the largest power of the denominator.
68 ( ) lim ax 2 + bx + c 5x x ax = lim ( ax 2 + bx + c 5x) 2 + bx + c + 5x x ax 2 + bx + c + 5x ax 2 + bx + c 25x 2 = lim x ax 2 + bx + c + 5x = lim x (a 25)x 2 + bx + c ax 2 + bx + c + 5x (1/x)(a 25)x 2 + bx + c) = lim x (1/x)( ax 2 + bx + c + 5x) (a 25)x + b + c/x = lim x a + b/x + c/x = lim x ((a 25)x + b + c/x) lim x ( a + b/x + c/x 2 + 5) = lim x ((a 25)x + b + 0) a
69 ( ) lim ax 2 + bx + c 5x = lim x ((a 25)x + b + 0) x a If a > 25 then the leading term has a positive coefficient and the limit approaches. If a < 25 then the leading term has a negative coefficient and the limit approaches. If a = 25 then the limit becomes b/5. The limits as x are found the the same way.
70 Find lim x ( ) lim 4x 4 + 5x (2x 2 + x). x Again we will multiply by the conjugate and divide by the highest power of x in the denominator. ( ) 4x 4 + 5x (2x 2 + x) ( = lim 4x 4 + 5x (2x 2 + x) x = lim x (4x 4 + 5x ) (2x 2 + x) 2 4x 4 + 5x (2x 2 + x) = lim x (4x 4 + 5x ) (4x 4 + 4x 3 + x 2 ) 4x 4 + 5x (2x 2 + x) = lim x x 3 x x 4 + 5x (2x 2 + x) ) 4x 4 + 5x (2x 2 + 4x 4 + 5x (2x 2 +
71 x /x 2 = lim 4 + 5/x + 10/x x /x 2 =
72 1 Find all vertical asymptotes of x 1 (x+2)x 2. 2 Find all horizontal asymptotes of x 1 (x+2)x 2. 3 Find the one sided limits of f (x) as x approaches the asymptotes. The function is a rational function so it is continuous at every point where it is defined. It is defined everywhere the denominator is not zero, which is when x = 0 and x = 2. It is important to keep signs right. We must be careful about the terms close to zero.
73 As x 0 +, x 2 approaches 0 from the right. Also when x 0, x 2 approaches 0 from the right. As x 0 the numerator x 1 approaches 1 and x + 2 approaches 2. Thus as x 0 from either side the numerator is negative and the denominator is small and positive. The fraction is thus large and negative. Because of this lim x 0 x 1 (x + 2)x 2 =.
74 As x 2 + the numerator x 1 approaches 3 and the denominator approaches 0 from the positive side. As the numerator is negative and the denominator is small and positive the fraction is large and negative. Because of this x 1 lim x 2 + (x + 2)x 2 =. As x 2 the numerator x 1 approaches 3 and the denominator approaches 0 from the negative side. As the numerator is negative and the denominator is small and negative the fraction is large and positive. Thus x 1 lim x 2 (x + 2)x 2 =.
75 Sketch a graph of a function f (x) with the following properties. 1 has horizontal asymptotes y = 0 and y = 4 2 has vertical asymptotes x = 3 and x = 4 3 lim x 4 f (x) = 4 lim x 3 f (x) does not exist.
76 Graphical limits Composition of functions. Change of variables.
Calculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
More informationUnit 1 PreCalculus Review & Limits
1 Unit 1 PreCalculus Review & Limits Factoring: Remove common factors first Terms - Difference of Squares a b a b a b - Sum of Cubes ( )( ) a b a b a ab b 3 3 - Difference of Cubes a b a b a ab b 3 3 3
More information2.1 Limits, Rates of Change and Slopes of Tangent Lines
2.1 Limits, Rates of Change and Slopes of Tangent Lines (1) Average rate of change of y f x over an interval x 0,x 1 : f x 1 f x 0 x 1 x 0 Instantaneous rate of change of f x at x x 0 : f x lim 1 f x 0
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
More information1.5 Inverse Trigonometric Functions
1.5 Inverse Trigonometric Functions Remember that only one-to-one functions have inverses. So, in order to find the inverse functions for sine, cosine, and tangent, we must restrict their domains to intervals
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationCalculus I Sample Exam #01
Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6
More information2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Tangents What is a tangent? Tangent lines and Secant lines Estimating slopes from discrete data: Example: 1. A tank holds 1000 gallons of water, which drains from
More informationAdvanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More informationAdvanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More informationCH 2: Limits and Derivatives
2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent
More informationEQ: What are limits, and how do we find them? Finite limits as x ± Horizontal Asymptote. Example Horizontal Asymptote
Finite limits as x ± The symbol for infinity ( ) does not represent a real number. We use to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example,
More informationCalculus (Math 1A) Lecture 5
Calculus (Math 1A) Lecture 5 Vivek Shende September 5, 2017 Hello and welcome to class! Hello and welcome to class! Last time Hello and welcome to class! Last time We discussed composition, inverses, exponentials,
More information(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
More informationAP Calculus Summer Prep
AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have
More information2.2 The Limit of a Function
2.2 The Limit of a Function Introductory Example: Consider the function f(x) = x is near 0. x f(x) x f(x) 1 3.7320508 1 4.236068 0.5 3.8708287 0.5 4.1213203 0.1 3.9748418 0.1 4.0248457 0.05 3.9874607 0.05
More informationLIMITS AND DERIVATIVES
2 LIMITS AND DERIVATIVES LIMITS AND DERIVATIVES 2.2 The Limit of a Function In this section, we will learn: About limits in general and about numerical and graphical methods for computing them. THE LIMIT
More informationMath 106 Calculus 1 Topics for first exam
Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the
More informationLimits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes
Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition
More informationMATH 1902: Mathematics for the Physical Sciences I
MATH 1902: Mathematics for the Physical Sciences I Dr Dana Mackey School of Mathematical Sciences Room A305 A Email: Dana.Mackey@dit.ie Dana Mackey (DIT) MATH 1902 1 / 46 Module content/assessment Functions
More informationCalculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science
Calculus I George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 151 George Voutsadakis (LSSU) Calculus I November 2014 1 / 67 Outline 1 Limits Limits, Rates
More informationMath 150 Midterm 1 Review Midterm 1 - Monday February 28
Math 50 Midterm Review Midterm - Monday February 28 The midterm will cover up through section 2.2 as well as the little bit on inverse functions, exponents, and logarithms we included from chapter 5. Notes
More informationCalculus (Math 1A) Lecture 6
Calculus (Math 1A) Lecture 6 Vivek Shende September 5, 2017 Hello and welcome to class! Hello and welcome to class! Last time Hello and welcome to class! Last time We introduced limits, and discussed slopes
More information2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Ex: When you jump off a swing, where do you go? Ex: Can you approximate this line with another nearby? How would you get a better approximation? Ex: A cardiac monitor
More informationChapter 1 Functions and Limits
Contents Chapter 1 Functions and Limits Motivation to Chapter 1 2 4 Tangent and Velocity Problems 3 4.1 VIDEO - Secant Lines, Average Rate of Change, and Applications......................... 3 4.2 VIDEO
More informationLimits and Continuity
Limits and Continuity Philippe B. Laval Kennesaw State University January 2, 2005 Contents Abstract Notes and practice problems on its and continuity. Limits 2. Introduction... 2.2 Theory:... 2.2. GraphicalMethod...
More informationter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the
Area and Tangent Problem Calculus is motivated by two main problems. The first is the area problem. It is a well known result that the area of a rectangle with length l and width w is given by A = wl.
More informationAnnouncements. Topics: Homework:
Topics: Announcements - section 2.6 (limits at infinity [skip Precise Definitions (middle of pg. 134 end of section)]) - sections 2.1 and 2.7 (rates of change, the derivative) - section 2.8 (the derivative
More informationTopic 3 Outline. What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity. 1 Limits and Continuity
Topic 3 Outline 1 Limits and Continuity What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity D. Kalajdzievska (University of Manitoba) Math 1520 Fall 2015 1 / 27 Topic 3 Learning
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2018, WEEK 3 JoungDong Kim Week 3 Section 2.5, 2.6, 2.7, Continuity, Limits at Infinity; Horizontal Asymptotes, Derivatives and Rates of Change. Section 2.5 Continuity
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall, 2016, WEEK 4 JoungDong Kim Week4 Section 2.6, 2.7, 3.1 Limits at infinity, Velocity, Differentiation Section 2.6 Limits at Infinity; Horizontal Asymptotes Definition.
More informationArnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment LimitsRates0Theory due 01/01/2006 at 02:00am EST.
Arnie Pizer Rochester Problem Library Fall 005 WeBWorK assignment LimitsRates0Theory due 0/0/006 at 0:00am EST.. ( pt) rochesterlibrary/setlimitsrates0theory/c3sp.pg Enter a T or an F in each answer space
More informationMA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
More informationFormulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
More informationChapter 2: Functions, Limits and Continuity
Chapter 2: Functions, Limits and Continuity Functions Limits Continuity Chapter 2: Functions, Limits and Continuity 1 Functions Functions are the major tools for describing the real world in mathematical
More informationMath 115 Spring 11 Written Homework 10 Solutions
Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,
More informationChapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the
Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition 2.1.1 : interval [x 1, x 2 ] is The average Rate of change of y = f(x) with respect to x over the y x = f(x 2) f(x
More informationMath 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS
Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f
More informationMATH 113: ELEMENTARY CALCULUS
MATH 3: ELEMENTARY CALCULUS Please check www.ualberta.ca/ zhiyongz for notes updation! 6. Rates of Change and Limits A fundamental philosophical truth is that everything changes. In physics, the change
More information6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationMATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart)
Still under construction. MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart) As seen in A Preview of Calculus, the concept of it underlies the various branches of calculus. Hence we
More informationSection 1.4 Tangents and Velocity
Math 132 Tangents and Velocity Section 1.4 Section 1.4 Tangents and Velocity Tangent Lines A tangent line to a curve is a line that just touches the curve. In terms of a circle, the definition is very
More informationAnnouncements. Topics: Homework: - sections 2.2, 2.3, 4.1, and 4.2 * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 2.2, 2.3, 4.1, and 4.2 * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the
More informationLecture 3 (Limits and Derivatives)
Lecture 3 (Limits and Derivatives) Continuity In the previous lecture we saw that very often the limit of a function as is just. When this is the case we say that is continuous at a. Definition: A function
More informationThe function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and
Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous
More informationA Library of Functions
LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us
More informationBlue Pelican Calculus First Semester
Blue Pelican Calculus First Semester Student Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function
More informationChapter 4: More Applications of Differentiation
Chapter 4: More Applications of Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 61 In the fall of 1972, President Nixon announced that, the rate of increase of inflation
More informationDr. Sophie Marques. MAM1020S Tutorial 8 August Divide. 1. 6x 2 + x 15 by 3x + 5. Solution: Do a long division show your work.
Dr. Sophie Marques MAM100S Tutorial 8 August 017 1. Divide 1. 6x + x 15 by 3x + 5. 6x + x 15 = (x 3)(3x + 5) + 0. 1a 4 17a 3 + 9a + 7a 6 by 3a 1a 4 17a 3 + 9a + 7a 6 = (4a 3 3a + a + 3)(3a ) + 0 3. 1a
More informationMathematic 108, Fall 2015: Solutions to assignment #7
Mathematic 08, Fall 05: Solutions to assignment #7 Problem # Suppose f is a function with f continuous on the open interval I and so that f has a local maximum at both x = a and x = b for a, b I with a
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationCalculus II Lecture Notes
Calculus II Lecture Notes David M. McClendon Department of Mathematics Ferris State University 206 edition Contents Contents 2 Review of Calculus I 5. Limits..................................... 7.2 Derivatives...................................3
More informationSec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes
Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite
More informationSec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes
Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite
More informationCalculus I. 1. Limits and Continuity
2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity
More informationChapter 1 Limits and Their Properties
Chapter 1 Limits and Their Properties Calculus: Chapter P Section P.2, P.3 Chapter P (briefly) WARM-UP 1. Evaluate: cot 6 2. Find the domain of the function: f( x) 3x 3 2 x 4 g f ( x) f ( x) x 5 3. Find
More information2.4 The Precise Definition of a Limit
2.4 The Precise Definition of a Limit Reminders/Remarks: x 4 < 3 means that the distance between x and 4 is less than 3. In other words, x lies strictly between 1 and 7. So, x a < δ means that the distance
More informationMATH 1910 Limits Numerically and Graphically Introduction to Limits does not exist DNE DOES does not Finding Limits Numerically
MATH 90 - Limits Numerically and Graphically Introduction to Limits The concept of a limit is our doorway to calculus. This lecture will explain what the limit of a function is and how we can find such
More informationReview Guideline for Final
Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice.
More informationMath Fall 08 Final Exam Review
Math 173.7 Fall 08 Final Exam Review 1. Graph the function f(x) = x 2 3x by applying a transformation to the graph of a standard function. 2.a. Express the function F(x) = 3 ln(x + 2) in the form F = f
More informationWEEK 7 NOTES AND EXERCISES
WEEK 7 NOTES AND EXERCISES RATES OF CHANGE (STRAIGHT LINES) Rates of change are very important in mathematics. Take for example the speed of a car. It is a measure of how far the car travels over a certain
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationReview Problems for Test 1
Review Problems for Test Math 6-03/06 9 9/0 007 These problems are provided to help you study The presence of a problem on this handout does not imply that there will be a similar problem on the test And
More informationInfinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.
Infinite Limits Return to Table of Contents Infinite Limits Infinite Limits Previously, we discussed the limits of rational functions with the indeterminate form 0/0. Now we will consider rational functions
More informationTangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationChapter 4: More Applications of Differentiation
Chapter 4: More Applications of Differentiation Autumn 2017 Department of Mathematics Hong Kong Baptist University 1 / 68 In the fall of 1972, President Nixon announced that, the rate of increase of inflation
More information1. Introduction. 2. Outlines
1. Introduction Graphs are beneficial because they summarize and display information in a manner that is easy for most people to comprehend. Graphs are used in many academic disciplines, including math,
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2018, WEEK 3 JoungDong Kim Week 3 Section 2.3, 2.5, 2.6, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity; Horizontal Asymptotes. Section
More information3 Polynomial and Rational Functions
3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,
More informationChapter 1: Limits and Continuity
Chapter 1: Limits and Continuity Winter 2015 Department of Mathematics Hong Kong Baptist University 1/69 1.1 Examples where limits arise Calculus has two basic procedures: differentiation and integration.
More informationMATH 1040 Objectives List
MATH 1040 Objectives List Textbook: Calculus, Early Transcendentals, 7th edition, James Stewart Students should expect test questions that require synthesis of these objectives. Unit 1 WebAssign problems
More informationSection 2.5. Evaluating Limits Algebraically
Section 2.5 Evaluating Limits Algebraically (1) Determinate and Indeterminate Forms (2) Limit Calculation Techniques (A) Direct Substitution (B) Simplification (C) Conjugation (D) The Squeeze Theorem (3)
More informationDuVal High School Summer Review Packet AP Calculus
DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and
More informationInfinite Limits. By Tuesday J. Johnson
Infinite Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Graphing functions Working with inequalities Working with absolute values Trigonometric
More information106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM Fermat s Theorem f is differentiable at a, then f (a) = 0.
5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function
More informationAP Calculus Chapter 3 Testbank (Mr. Surowski)
AP Calculus Chapter 3 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.). If f(x) = 0x 4 3 + x, then f (8) = (A) (B) 4 3 (C) 83 3 (D) 2 3 (E) 2
More information1.10 Continuity Brian E. Veitch
1.10 Continuity Definition 1.5. A function is continuous at x = a if 1. f(a) exists 2. lim x a f(x) exists 3. lim x a f(x) = f(a) If any of these conditions fail, f is discontinuous. Note: From algebra
More informationLimits, Continuity, and the Derivative
Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change
More informationFunction Terminology and Types of Functions
1.2: Rate of Change by Equation, Graph, or Table [AP Calculus AB] Objective: Given a function y = f(x) specified by a graph, a table of values, or an equation, describe whether the y-value is increasing
More informationChapter 2: Limits & Continuity
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 2: Limits & Continuity Sections: v 2.1 Rates of Change of Limits v 2.2 Limits Involving Infinity v 2.3 Continuity v 2.4 Rates of Change and Tangent Lines
More informationDecember Exam Summary
December Exam Summary 1 Lines and Distances 1.1 List of Concepts Distance between two numbers on the real number line or on the Cartesian Plane. Increments. If A = (a 1, a 2 ) and B = (b 1, b 2 ), then
More informationLIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or
More informationCalculus & Analytic Geometry I
Functions Form the Foundation What is a function? A function is a rule that assigns to each element x (called the input or independent variable) in a set D exactly one element f(x) (called the ouput or
More informationUnit #3 : Differentiability, Computing Derivatives
Unit #3 : Differentiability, Computing Derivatives Goals: Determine when a function is differentiable at a point Relate the derivative graph to the the graph of an original function Compute derivative
More informationEvaluating Limits Analytically. By Tuesday J. Johnson
Evaluating Limits Analytically By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews
More informationSolutions to Math 41 First Exam October 15, 2013
Solutions to Math 41 First Exam October 15, 2013 1. (16 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether
More informationMath 131. The Derivative and the Tangent Line Problem Larson Section 2.1
Math 131. The Derivative and the Tangent Line Problem Larson Section.1 From precalculus, the secant line through the two points (c, f(c)) and (c +, f(c + )) is given by m sec = rise f(c + ) f(c) f(c +
More informationCalculus: Early Transcendental Functions Lecture Notes for Calculus 101. Feras Awad Mahmoud
Calculus: Early Transcendental Functions Lecture Notes for Calculus 101 Feras Awad Mahmoud Last Updated: August 2, 2012 1 2 Feras Awad Mahmoud Department of Basic Sciences Philadelphia University JORDAN
More information80 Wyner PreCalculus Spring 2017
80 Wyner PreCalculus Spring 2017 CHAPTER NINE: DERIVATIVES Review May 16 Test May 23 Calculus begins with the study of rates of change, called derivatives. For example, the derivative of velocity is acceleration
More informationTRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)
TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents
More informationWelcome to AP Calculus!!!
Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you
More informationUnit #3 : Differentiability, Computing Derivatives, Trig Review
Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is differentiable at a point Relate the derivative graph to the the graph of an original function Compute
More informationAP Calculus Summer Homework
Class: Date: AP Calculus Summer Homework Show your work. Place a circle around your final answer. 1. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES We have: Seen how to interpret derivatives as slopes and rates of change Seen how to estimate derivatives of functions given by tables of values Learned how
More informationfunction independent dependent domain range graph of the function The Vertical Line Test
Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding
More informationObjectives List. Important Students should expect test questions that require a synthesis of these objectives.
MATH 1040 - of One Variable, Part I Textbook 1: : Algebra and Trigonometry for ET. 4 th edition by Brent, Muller Textbook 2:. Early Transcendentals, 3 rd edition by Briggs, Cochran, Gillett, Schulz s List
More informationNAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018
NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 A] Refer to your pre-calculus notebook, the internet, or the sheets/links provided for assistance. B] Do not wait until the last minute to complete this
More informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationMath Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8
Name: Math Academy I Fall Study Guide CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 1-A Terminology natural integer rational real complex irrational imaginary term expression argument monomial degree
More information