#### project-22

University of Florida , MAC 2311
Excerpt: ... MAC 2311 PROJECT 22 LECTURE 22 NAME SSN SECTION 1. Use the First Derivative Test to find all relative extreme values (if any) of . relative max: relative min: 2. Use the First Derivative Test to find all relative extreme values of . relative max: relative min: MAC 2311 PROJECT 22 LECTURE 22 3. Use the Second Derivative Test to determine the relative extreme values (if any) of the function . relative max: relative min: 4. Use the Second Derivative Test to determine the relative extreme values (if any) of the function . relative max: relative min: 5. Use the Second Derivative Test to determine where has relative extreme values on . relative max: relative min: ...

#### 1009

University of Texas, M 1009
Excerpt: ... Contents 2 3 4 5 6 7 Lecture topics 10 - 09 - 98 The Second Derivative Test Why I Dont Like The Second Derivative Test Why I Still Dont Like . . . Let The Computer Do The Work Maybe the Second Derivative Test Isnt That Bad Lecture topics 10 - 09 - 98 1) I said I was going to look for extremes of functions, rst on open intervals, like (0, 1). Noted that the whole real number line is an open interval: (, ). 1 2) Worked examples of functions with extremes: f (x) = x on (0, 1), f (x) = x2x on (, ), f (x) = cos x on (, ). Used these to +1 dene absolute maximum, absolute minimum and absolute extremes. 3) Showed example: f (x) = x + sin x on (, ), to dene local 2 maximum, local minimum and local extremes. 4) Worked examples: f (x) = 1 |x| on (, ), and f (x) = x 3 on (, ) to to see that extremes can happen where f (c) = 0 or f (c) dne. 5) Dened critical point to be those places in the domain where 1 f (c) = 0 or f (c) dne. Remarked tha ...

#### q16pracs9

George Mason, MATH 108
Excerpt: ... April 3, 2009 Math 108 - Practice Quiz 16 This is more practice for part of section 3.1 and part of section 3.2. This practice quiz gives practice in applying the first and second derivative test s for classifying relative extrema. In Question 1 the second derivative test is easier to use than the first derivative test because it is difficult to factor f (x). On the other hand in Question 2 the first derivative test is easier to use than the second derivative test because it is hard to calculate f (x). In Question 3 either method is equally easy to use. 1. Let f (x) = x4 + 6x2 - 56x + 60. (a) Verify that x = 2 is a critical number of f . (b) Use the second derivative test in order to classify x = 2 as a relative maximum or relative minimum. 2. Let f (x) = (x - 7)3 . x2 (a) Find all critical numbers of f . (b) Use the first derivative test to classify each of the critical numbers as a relative maximum or relative minimum. (Note: Even though f (0) does not exist, x = 0 is not considered to be a critical number ...

#### Math112Lecture018BHandouts

UMBC, MATH 112
Excerpt: ... if f (x ) < 0 on I. Clint Lee Math 112 Lecture 18: Mean Value Theorem 10/13 Inflection Points Inflection Points Suppose that f is continuous at c. The point with x-coordinate c is an inflection point of the graph of f if the concavity of the graph of f changes at c or equivalently the sign of f (x ) changes at c It is not necessary that f (c ) or, for that matter, f (c ) be defined for there to be an inflection point at c. The only candidates for inflection points are x-values c for which f (c ) = 0 or f (c ) does not exist. Clint Lee Math 112 Lecture 18: Mean Value Theorem 11/13 The Second Derivative Test The Second Derivative Test Suppose that the function f is twice differentiable at c and that c is a critical number for which f (c ) = 0. f has a local maximum at c if f (c ) < 0, that is, the graph of f is concave down at c. f has a local minimum at c if f (c ) > 0, that is, the graph of f is concave up at c. If f (c ) = 0, the Second Derivative Test fails to determine whether t ...

#### Math112Lecture018BSlides

UMBC, MATH 112
Excerpt: ... The Mean Value Theorem The First Derivative and the Shape of the Graph Concavity and the Second Derivative Lecture 18: Mean Value Theorem The Mean Value Theorem The Secant Line over an Interval The Tangent Line Parallel to the Secant Line The Mean Value Theorem The First Derivative and the Shape of the Graph Increasing and Decreasing Functions Testing for Increase and Decrease The First Derivative Test Concavity and the Second Derivative Concavity Testing for Concavity Inection Points The Second Derivative Test Clint Lee Math 112 Lecture 18: Mean Value Theorem 1/13 The Mean Value Theorem The Secant Line over an Interval The First Derivative and the Shape of the Graph Concavity and the Second Derivative The Secant Line over an Interval Consider the graph y = f (x). y y = f (x) a b x Clint Lee Math 112 Lecture 18: Mean Value Theorem 2/13 The Mean Value Theorem The Secant Line over an Interval The First Derivative and the Shape of the Graph Concavity and the Second Derivative The Secant ...

#### Guide3

University of Illinois, Urbana Champaign, MATH 220
Excerpt: ... ection 3.2 Indeterminate Forms and L'H^pital's Rule. The Indeterminate forms that work o for L'H^pital's Rule. What needs to be true in order to use L'H^pital. Manipo o ulating other indeterminate forms into forms on which L'H^pital can be used. o Why indeterminate forms are indeterminate. Section 3.3 Maximum and Minimum Values. Definitions of absolute and local extrema. The difference between absolute and local extrema. The Extreme Value Theo- 1 rem, critical numbers and Fermat's Theorem. The relationship between critical numbers and extrema. Section 3.4 Increasing and Decreasing Functions. Definition of strictly increasing and decreasing functions, relation to the derivative. Why this relationship is true (MVT). The First Derivative Test. Section 3.5 Concavity and the Second Derivative Test . Concave up and concave down, what this means for the function and its derivatives. Inflection points. The Second Derivative Test . Section 3.6 Overview of Curve Sketching. Using the results of sections 3.3-3.5 (an ...

#### wk11calclect1

Allan Hancock College, MAT 1102
Excerpt: ... MAT1102 Algebra&Calculus 1, S1,2004, Week 11 First Lecture in Calc ' 1 \$ Calculus Week 11 - Lecture 1 - Outline Look Back - Chain Rule -One More Example Study Book 5.1 Using 1st & 2nd Derivatives Slide 1 Curve sketching Local Maxima and Minima (Extrema) Other Critical Points First Derivative Test Second Derivative Test Concavity & Inflection Points & % \$ ' One More Chain Rule Example Example: y = esin x cos x, find y (/2). Slide 2 & % MAT1102 Algebra&Calculus 1, S1,2004, Week 11 First Lecture in Calc ' 2 \$ Study book 5.1 Using 1st & 2nd Derivatives Given y = f (x), why should we find where f (x) = 0? Coming up Slide 3 Curve sketching Local Maxima and Minima (Extrema) Other Critical Points First Derivative Test Second Derivative Test Concavity & Inflection Points & ' % \$ Curve Sketching Slide 4 Curve sketching - Read Text 2nd Edition p 240, 241, or 3rd edition p 166, 167. f (x) > 0 means curve increasing f (x) < 0 means curve decreasing f (x) = 0 means curve flat ...

#### act23_S08

Oakland University, M 10250
Excerpt: ... Name Date Math 10250 Activity 23: Second Derivative Test s (Section 4.2) GOAL: To study how the graph of a given f (x) bends, and how these features of the graph are described by f (x) and f (x). The second derivative test for concavity Example 1 Water is lling up each of the following vessels at a constant rate of 1 cm3 /sec. B A C Let h be the height of the water level in the vessel at time t. h a. Sketch the graphs of h verses t for Vessels A and B in the axes for Figure 1. Indicate which graph belong to A and which to B. Figure 1 Figure 2 b. Sketch the graph of h verses time t for Vessel C in the axes for Figure 2. c. Comment on how the bending (up or down) of the graph changes with h (t). Mark on the graph where the bending changes. We now introduce terminologies that describe the bending of a graph. Case 1: For a < x < b, slope of the graph f (x) is increasing as x increases i.e. f (x) is increasing. So for a < x < b. (Portions of u-shape) f (x) is y y = f (x) y = f ...

#### 3_4

Morningside, MATH 205
Excerpt: ... 3.4 Lecture Notes Math 205 02/20/09 Mammenga CONCAVITY AND THE SECOND DERIVATIVE TEST A. Denition of Concavity: Let f be dierentiable on an open interval I. The graph of f is concave upward on I if f is increasing on the interval and concave downward on I if f is decreasing on the interval. B. Test for Concavity: Let f be a function whose second derivative exists on an open interval I. If f (x) > 0 for all x in I, then the graph of f is concave upward in I. If f (x) < 0 for all x in I, then the graph of f is concave downward in I. C. Denition: Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c, f (c), then this point is a point of inection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point. If (c, f (c) is a point of inection of the graph of f , then either f (c) = 0 or f does not exist at x = c. D. Second Derivative Test : Le ...

#### lec17

Minnesota, MATH 1371
Excerpt: ... Math 1371 Lecture 17 Bryan Mosher Wednesday, October 31, 2007 1 Nuts and bolts 1. Oce hours this week: MW 11-12, and anytime Thursday by appointment. 2. The second exam is Thursday, either 5-6 or 6-7. The room for this lecture is once again 250 Anderson on the West Bank. Note: other lectures of Math 1371 have dierent rooms. 2 Whats happening today 1. Graph sketching based on rst and second derivatives 2. Optimization 3. Related rates 4. LHpitals rule? Anything else? o Review for exam 3 Graph sketching Based on the sign of the rst and second deriviative, Here are all the possible shapes near x = a. For each shape, is there a critical point or inection point? Is there a local extremum? How do you know? (rst or second derivative test ?) Putting it all together: Example 1. Suppose that f (x) is a continuous function with the following information given: f (x) < 0 for < x < 2 and 2 < x < , f (x) > 0 for 2 < x < 0 and 0 < x < 2, and f is undened only ...

#### P24

University of Florida , MAC 2311
Excerpt: ... curve on the interval. ( There are four possibilities: each would be the left or right half of the shapes and .) Now try to piece your shapes together to make a rough free-hand sketch of the function f (x) below. Label your cusps, horizontal tangents, and inflection points. As one last experiment, use the second derivative test to verify the local extrema that you found. Do you have any critical numbers at which second derivative test cannot be used? 2. Suppose a particle moves in a straight line so that its position in feet at time t seconds is given by s(t) = t2 e-t . How would you find where the particle has its greatest speed (absolute value of velocity)? Find the greatest speed on [ 0, 2 ]; [ 0, 4 ]. To what special type of point on the graph of s(t) do these correspond? Explain. . . ...

#### wksht27

University of Illinois, Urbana Champaign, MATH 220
Excerpt: ... Merit Worksheet #27, 10/29/08 1. With your group, make a detailed, clearly-written, absolutely correct answer key for the quiz that was given Friday. 2. For each of the curves shown, state whether the curve is increasing or decreasing, and whether it is concave up or concave down. 3. Using your textbook and notes, state the denition of an inection point. True or false: The place where a graph is steepest is an inection point, and vice versa. 4. Find the intervals on which the graph of y = x4 ex is concave up/down. Also identify any points of inection (give both coordinates). 5. Find all critical points of the function from the previous problem, and use the Second Derivative Test , if possible, to determine whether each is a local maximum or a local minimum. (If its not possible to get an answer with the Second Derivative Test , explain why not and use another method to answer the question.) 6. For a function f , which of f (x), f (x), or f (x) would you need to nd out the follow ...

#### 110StudyGuideforExamIII

Occidental, MATH 110
Excerpt: ... extremum, global extremum, concavity, root First Derivative Test and Second Derivative Test to classify local extrema Extreme Value Theorem, its Conditions (continuous function on a closed finite interval) and Conclusion (function must achieve one global maximum value and one global minimum value ) Determine graphical behavior of function analytically (,), (,U), (,+,-) and curve sketching Optimization o Find critical points. (In this class, always do that by solving f = 0 , by hand when possible and otherwise with a calculator; I will not accept any argument that doesn't use calculus, for example just graphing f .) o Classify them using the first derivative test o Classify them using the second derivative test o Applied max-min problems (if there is only one critical point that makes sense, you aren't required to show that it's the requested global max or global min) Check whether a given function satisfies an IVP (initial value problem) which consists of a rate equation and an initial condition, i ...

#### Section 3.2LN

Kennesaw, MATH 1106
Excerpt: ... ond derivative negative - a maximum p.213 second derivative test 1 find critical points (first derivative 0 or doesn't exist) evaluate second derivative at critical point if second derivative is 0 (negative), critical point is at a maximum if second derivative is 0 (positive), critical point is at a minimum if second derivative is 0, test is inconclusive; use first derivative test p.214, Ex.3.2.5 fx f f 2x 3 3x 2 2 x 6x 6x x 12x 6 12x 12 7 find critical points using first derivative 6x 2 6x 12 0 6 x2 x 2 0 x 2 x 2 divide both sides by 6 0 x2 x 1 x-2 x1 critical points use second derivative test to evaluate critical points f f 1 2 0 12 12 1 2 6 6 negative positive relative maximum relative minimum find y coordinates for relative maximum, relative minimum use ORIGINAL function f 2 f1 2 21 2 3 3 3 2 2 2 12 2 7 7 13 relative maximum occurs at (-2, 13) relative minimum occurs at (1, -14) 3 1 12 1 14 Class Exercise p.217, #28 extrema fx f f use second derivative test to ...

#### 301-05-3-tut1

Sveriges lantbruksuniversitet, ECON 301
Excerpt: ... ECON 301 Tutorial #1 Homework The following are 2-variable optimization problems, both unconstrained and constrained. Prepare them for your first tutorial. If you have difficulty, review the notes "Optimizaion and Economics" found on the website SH ...

#### lecture8

Dartmouth, OPENCALC 2
Excerpt: ... ve down to concave up. So, to summarize: if if if d2 f dx2 (p) d2 f dx2 (p) d f dx2 (p) 2 > 0 at x = p, then f (x) is concave up at x = p. < 0 at x = p, then f (x) is concave down at x = p. = 0 at x = p, then we do not know anything new about the behavior of f (x) at x = p. 1 For an example of nding and using the second derivative of a function, take f (x) = 3x3 6x2 + 2x 1 as above. Then f (x) = 9x2 12x + 2, and f (x) = 18x 12. So at x = 0, the second derivative of f (x) is 12, so we know that the graph of f (x) is concave down at x = 0. Likewise, at x = 1, the second derivative of f (x) is f (1) = 18 1 12 = 18 12 = 6, so the graph of f (x) is concave up at x = 1. Critical Points and the Second Derivative Test We learned before that, when x is a critical point of the function f (x), we do not learn anything new about the function at that point: it could increasing, decreasing, a local maximum, or a local minimum. We can often use the second derivative of the f ...

#### MAT1102_calc_l14_wk10_2solutions

Allan Hancock College, MAT 1102
Excerpt: ... MAT1102 Algebra & Calculus I Review question Find local extrema of f(x)=x3 Calculus MAT1102 Algebra & Calculus I Calculus Lecture 14 (week 10 part 2) Slide 1 Special presentation During enrichment tutorial time, TOMORROW, Friday 8-10am in T122 Dr Ashley Plank (Statistics, Maths and Computing) will give a presentation on recursion, iteration and applications All welcome! Morning tea will be provided Please let me know if you are coming (need numbers for catering), lochb@usq.edu.au MAT1102 Algebra & Calculus I Calculus Lecture 14 (week 10 part 2) Slide 2 Outline Second derivative test Necessary and sufficient conditions for local extrema Concavity and Inflection Points Modelling & Optimisation Antiderivatives Readings and Activities for next week MAT1102 Algebra & Calculus I Calculus Lecture 14 (week 10 part 2) Slide 3 Concavity f'(x) > 0, concave up. f'(x) < 0, concave down. f'(x) = 0, don't know! MAT1102 Algebra & Calculus I Calculus Lecture 14 (week 10 part 2) Slide 4 ...

#### wk11calclect2

Allan Hancock College, MAT 1102
Excerpt: ... MAT1102 Algebra&Calculus 1, S1 2004, Week 11 Second Lecture in Calc 1 ' \$ Calculus Week 11 - Lecture 2 - Outline Study Book 5.1 Using 1st & 2nd Derivatives - continued SB 5.2 Families of Curves Slide 1 SB 5.3 Optimization SB 5.4 Marginality SB 5.5 Modelling & Optimisation Homework ' & % \$ Look back: First & Second Derivative Test for Local Max or Min Example Revisited: y = x3 - 3x + 1 y (x) = 3x2 - 3 = 3(x - 1)(x + 1) Slide 2 y = 0 at x = 1 and x = -1, thus critical points at (1, -1) and (-1, 3). Case p = 1. y < 0 for x < 1 and y > 0 for x > 1 Hence slope goes from negative to zero to positive, ie a minimum at x = 1. Case p = -1. y > 0 for x > -1 and y < 0 for x < -1. Hence slope goes from positive to zero to negative, ie a maximum at x = 1. Sketch. & % MAT1102 Algebra&Calculus 1, S1 2004, Week 11 Second Lecture in Calc 2 ' \$ Look back: Second Derivative Test for Local Max or Min Given y = f (x) Slide 3 Find critical point x = p where f (x) = 0 (ie curve flat). Find f (p) ...

#### exam2rev

Johns Hopkins, MTS 251
Excerpt: ... stive enumeration of paths from START to FINISH in the PERT/CPM network. 5. Be able to formulate a linear program to determine a project's duration. 6. Be able to formulate a linear program to determine how much the activities of a project should be crashed to meet a given project duration or project cost. Nonlinear Optimization Models (Chapter 13) General: The student should be able to formulate a problem with a nonlinear objective function and/or constraints. 1. Single variable unconstrained problem: be able to solve by hand using 1st derivative and second derivative test for concavity (max)/convexity (min). 2. Single variable constrained problem: be able to solve by hand using 1st derivative and second derivative test for concavity (max)/convexity (min) and end point check. 3. Key terms: (ordinary) derivative (e.g., y , f (x), df /dx, d2 f /dx2 , etc.); critical point (where first derivatives are zero), global optimum, local (relative) optimum 4. Key mathematical concepts: y (x) < 0 maximizer; y (x) > 0 ...

#### finalstudyguide

Rutgers, MATH 151
Excerpt: ... of a given function, which will mean being able to identify the pattern established by taking repeated derivatives and substituting that into the general formula for the coefficents. Good practice problems are Section 8.8 #17, 21, 25, 28, 29. The remaining twenty or so percent of the exam will cover functions of multiple variables. Topics to review include the following: Level Curves: Discussion on pp. 974-975 Section G1.1 #15, 17, 23, 24, 25, 26 Partial Differentiation: Discussion on pp. 978-985 Section G1.2 #1, 3, 5, 7, 9, 11, 19 Maxima and Minima of Functions of Two Variables: Discussion on pp. 987-992, especially First Derivative Test on p. 989 and the Second Derivative Test on p. 992 Example 4 on p. 992 Section G1.3 #1, 3, 5, 11, 13, 17, 19, 21 You'll be given the formula sheet from the first exam plus the formula for D(x, y) in the statement of the Second Derivative Test for functions of two variables. ...

#### calc notes lecture 16

Union, MAT 111
Excerpt: ... Lecture Section 16 3.3 Objectives Concavity and The Second-derivative Test Assignment 3.3: 1, 7, 9-17 odd, 23-33 odd, 47, 65 Understanding Goals: 1. Understand the definition of concavity and points of inflection. 2. Understand how to determine th ...

#### math120_chapters45_material_outline

E. Michigan, MATH 120
Excerpt: ... Math 120 Outline of material in Chapters 4 & 5 for Final Exam 4.1 Maximum and Minimum Values Functions Global (a.k.a. absolute) max & min Local (a.k.a. relative) max & min Extreme Value Theorem Critical number, point Local max & min at critical number Closed Interval Method to find global max & min 4.3 How Derivatives Affect the Shape of a Graph What does f ' say about f ? Increasing/Decreasing Test First Derivative Test for local max & min What does f ' say about f ? Concave upward & downward Concavity test Inflection point Second Derivative Test for local max & min Inconclusive when f '(c)=0 or does not exist 4.7 Optimization Problems To solve an optimization problem, follow the 6-step procedure outlined on the first page of the section. Steps 15: Set up max or minimization problem, making sure to specify domain Step 6 : Find global max or min using methods of Section 4.1 or 4.3: A finite non-closed domain interval arising from the physical problem can usually be extended to the correspond ...