# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

8 Pages

### lesson11notesforclassfiillpart

Course: M 1314, Fall 2009
School: U. Houston
Rating:

Word Count: 975

#### Document Preview

11 1 Math M1314 lesson 1314 Lesson 11 Applications of the Second Derivative Concavity Earlier in the course, we saw that the second derivative is the rate of change of the first derivative. The second derivative can tell us if the rate of change of the function is increasing or decreasing. In business, for example, the first derivative might tell us that our profits are increasing, but the second derivative...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Texas >> U. Houston >> M 1314

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
11 1 Math M1314 lesson 1314 Lesson 11 Applications of the Second Derivative Concavity Earlier in the course, we saw that the second derivative is the rate of change of the first derivative. The second derivative can tell us if the rate of change of the function is increasing or decreasing. In business, for example, the first derivative might tell us that our profits are increasing, but the second derivative will tell us if the pace of the increase is increasing or decreasing. Example 1: From these graphs, you can see that the shape of the curve change differs depending on whether the slopes of tangent lines are increasing or decreasing. This is the idea of concavity. Definition: Let the function f be differentiable on an interval (a, b). Then f is concave upward on (a, b) if f ' is increasing on (a, b) and f is concave downward on (a, b) if f ' is decreasing on (a, b). Determining Where a Function is Concave Upward and Where it is Concave Downward By Analyzing the Sign of the Second Derivative Algebraically We can also determine concavity algebraically. The procedure for doing this should look pretty familiar: M1314 lesson 11 2 1. Find the second derivative of the function. 2. Determine all values of x for which f ( x ) = 0 or is undefined. 3. Use the values found in step 2 to divide the number line into open intervals. 4. Choose a test value, c, in each open interval. 5. Substitute each test value, c, into the second derivative to determine the sign of f (c) . 6. Apply the following theorem: Theorem: If f ' ' ( x ) > 0 for each value of x in (a, b), then f is concave upward on (a, b). If f ' ' ( x ) < 0 for each value of x in (a, b), then f is concave downward on (a, b). Example 1: Determine where the function f ( x ) = x 3 - 3x 2 - 24 x + 32 is concave upward and where it is concave downward. M1314 lesson 11 3 Example 2: Determine where the function f ( x ) = xe 2 x is concave upward and where it is concave downward. Example 3: Determine where the function f ( x ) = x 4 + 6 x 3 is concave upward and where it is concave downward. M1314 lesson 11 4 Inflection Points The point where a function changes from being concave upward to concave downward (or from being concave downward to concave upward) is called a point of inflection or an inflection point. We'll show the significance of this point by an example. Example 4: Note that, at the beginning, the slopes of the lines tangent to the graph are increasing slowly. Then the slopes of the tangent lines increase rapidly. But after a certain point, the rate of increase slows down. Suppose you are in business and this is a graph of your total sales, where x represents the amount spent (in millions of dollars) on advertising and y represents total receipts (in millions of dollars). When your company an begins advertising campaign, the rate of increase in sales speeds up. But this can't go on forever. There comes a point where the advertising campaign won't have as much of an effect on sales. This is the point of inflection. You may think of it as the point of diminishing returns. In business, it might be the time to start a new ad campaign. M1314 Definition: A point on the graph for a differentiable function f at which the concavity changes is called an inflection point. Finding Inflection Points To find the inflection point(s) of a function, lesson 11 5 1. Compute f ' ' ( x ) . 2. Find all points in the domain of f for which f ' ' ( x ) = 0 . 3. Determine the sign of f ' ' ( x ) to the left and to the right of each point x = c found in step 2. If there is a change in the sign of f ' ' ( x ) as we move across the point x = c from left to right, then the point (c , f (c)) is an inflection point of f. Note: to find the y coordinate of an inflection point, substitute the x coordinate into the original function. Example 5: Determine any points of inflection if f (x) = x 4 - 2 x 3 + 6 . M1314 lesson 11 6 Example 6: Determine any points of inflection if f ( x ) = xe 2 x . Example 7: Determine any points of inflection if f (x) = x 4 - 5x 3 + 2 x + 3 . M1314 lesson 11 7 The Second Derivative Test Recall that we use the first derivative test to determine if a critical point is a relative extremum. There is also a second derivative test to find relative extrema. It is sometimes convenient to use; however, it can be inconclusive. The Second Derivative Test: 1. Find f ' ( x ) and f ' ' ( x ) . 2. Find all critical points. 3. Compute f ' ' (c) fo...

Textbooks related to the document above:
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

U. Houston - M - 1314
Section 19304Assignment 008 Grade ID Form A3Use for popper 1 and 2: Given the position function: s(t ) = -.1t+ 2 t 2 + 24 t1. Find the velocity at 2 seconds. A. 30.8 ftsec2. Find the position at 2 seconds. A. 55.2 Ft at 2 seconds3. Sup
U. Houston - M - 1314
Review over chain Rule:1. f(x) = x 2 - 22. f(x) = ( 5x - 2 )43. f(x) = 4e2 x 2x - 1 4. f(x) = ln 4x + 3
U. Houston - M - 1314
Review of Derivatives: Find the derivatives of the following:1 1. f ( x ) = x 2 - x2 2. f (x ) = x 2 6 ln x3. f ( x ) =2x2 4x - 54. f ( x ) = x 2 - 1This we are covering today . Same with this problem.5. f ( x ) = x - 2e 3 xSolutions: 1
U. Houston - M - 1314
M1314lesson 51Math 1314 Lesson 5 The Product and Quotient Rules In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [f (x)g(x)] = f (x)g' (x) + g(x)f ' (x) dxExamp
U. Houston - M - 1314
Review of derivates (lesson 4): Find the derivative:1. f ( x ) = 23 x2. f ( x ) = 4 x - 2 x + 435x 4 - 2 x2 - 4 x 3. f ( x ) = x24. f ( x ) = -3e + 6 ln xx5. f ( x ) = 2 x e2x6. f ( x ) =2x - 1 4x + 2
U. Houston - M - 1314
M1314Lesson 41Math 1314 Lesson 4 Basic Rules of Differentiation We can use the limit definition of the derivative to find the derivative of every function, but it isnt always convenient. Fortunately, there are some rules for finding derivatives
U. Houston - M - 1314
Section 19304Assignment 001 Grade ID Form Alim x 31. Evaluate3x 2 - 2 D. Limit does not existA.10B. 5C. 5,-52. Evaluatex2 + 2 x - 3 lim x 1 x2 - 1 B. 4 C. 1 D. 2A. Limit does not exist 3. Evaluatex -1lim f ( x ) if - 2 x +
U. Houston - M - 1314
M1314Lesson 31Math 1314 Lesson 3 The Derivative The Limit Definition of the Derivative We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specifi
U. Houston - M - 1314
M1314Lesson 11Math 1314 Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18th century. 1. How can we find the line tangent to a curve
U. Houston - M - 1314
U. Houston - M - 1314
Review for test 2: Find the limit. 1.- x 3 - 2 x 2 + 2x - 2x + 3 x -1lim2.x4limx 2 - x - 12 x-4x+1 x2 - 13.x -1lim4.xlimx 2 - 2x + 2 x3 - 15x - 2 6x + 15.xlim6. lim1 - x2 x x - 1x&lt;2 x 7. f ( x ) = x - 1 x
U. Houston - M - 1314
M1314lesson 3 Math 1314 Lesson 3 The Derivative The Limit Definition of the Derivative1We now address the first of the two questions of calculus, the tangent line question. We are interested in finding the slope of the tangent line at a specifi
U. Houston - M - 1314
Math 1314 Lesson 5 The Product and Quotient Rules In this lesson, we continue with more rules for finding derivatives. These are a bit more complicated. Rule 8: The Product Ruled [ f ( x ) g ( x )] = f ( x ) g ' ( x ) + g ( x) f ' ( x ) dxExample
U. Houston - M - 1314
Math 1314 Lesson 7 Higher Order Derivatives Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second derivative, and is
U. Houston - M - 1314
Math 1314 Lesson 10 Applications of the First Derivative Determining the Intervals on Which a Function is Increasing or Decreasing From the Graph of f Definition: A function is increasing on an interval (a, b) if, for any two numbers x1 and x2 in (a,
U. Houston - M - 1314
Math 1314 Lesson 13 Absolute Extrema In earlier sections, you learned how to find relative (local) extrema. These points were the high points and low points relative to the other points around them. In this section, you will learn how to find absolut
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 1 1. Find lim f ( x) given the graph of f (x). (10.4.3)x32. Evaluate lim 3 x 2 - 2 (10.4.35)x 3x 2 + 4x + 3 (10.4.57) x 1 x2 -1 2 x - 1, x &lt; -1 4. Evaluate lim f ( x) if f ( x) = 2 (10.4.18) x -1 x + 2, x -1 5. Eval
U. Houston - MATH - 1314
Math 1314 Daily Poppers Lesson 3f ( x + h) - f ( x ) . h 2. True or False: If you need to find an instantaneous rate of change, you will find the derivative.1. True or False: The formula for the average rate of change is3. Find ( x + h ) . (10.6
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 4 1. Find the derivative: f ( x) = 12 . (11.1.2)2. Find the derivative: f ( x) = x 6 . (11.1.4) 3. Find the derivative: f ( x) = 5 x . (11.1.13)4. Find the derivative: f ( x) = x 4 - 7 x 2 + 5. (11.1.17) 5. Find the deriva
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 5 1. Find the derivative: f ( x) = 5 x( x 2 + 4) . (11.2.1) 2. Find the derivative: f ( x) = (6 x - 3)( x + 4) . (11.2.3)3. Find the derivative: f ( x) = xe x (13.3.7) 4. Find the derivative: f ( x) = x ln x . (13.4.15) 5. S
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 6 1. If f ( x) = 3 x + 4 and g ( x) = x 2 , then find g o f .2. Find the derivative: f ( x) = (4 x 5) . (11.3.1)63. Find the derivative: f ( x) = x 2 + 2 . (11.3.12) 4. Find the derivative: f ( x) =( x 5 )26. (11.
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 7 1. Find the second derivative: f ( x) = 6 x 2 - 9 x + 3 . (11.5.1) 2. Find the second derivative: f ( x) = 2e x . 1 . (11.5.15) 3. Find the second derivative: f ( x) = ( x - 4 )24. Find the second derivative: f ( x) = ln(
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 8 1. Find the slope of the tangent line to f ( x) = 3 x 2 - 4 when x = 1. (10.6.20) 2. Write an equation of the tangent line to the graph of f ( x) = -2 x 2 + 5 at the point (-2, -3). (10.6.23) 3. Let f ( x) = x 3 - 8 x 2 . F
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 9 Use this information for questions 1 9. Suppose a company's weekly cost incurred in producing x of its products is given by the function C ( x) = .000002 x 3 - .02 x 2 + 150 x + 80,000 and the weekly demand for the product
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 10 1. Suppose f ( x) = x 2 5 x . State the interval(s) where the function is increasing and the interval(s) where it is decreasing. (12.1.13) 2. Suppose f ( x) = x 4 5 x 3 + 6 . State the interval(s) where the function is i
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 11 1. Suppose f ' ' ( x) &gt; 0 for all x in the domain of f. Then which of these statements is true? (12.2.16 21) o f is concave upward for all x in the domain of f o f is concave downward for all x in the domain of f o f has
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 13 1. Suppose f ( x) = x - 2 x . Find any critical points. (12.4.28)2. Suppose f ( x) = x - 2 x . Find the absolute maximum of f on the interval [0, 9]. (12.4.28) 3. Suppose f ( x) = x - 2 x . Find the absolute minimum of f
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 14 1. By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 25 inches long and 10 inches wide, (12.5.4) a. writ
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 15 1. Write the general form of a function expressing exponential growth. 2. Write the general form of a function expressing exponential decay. 3. Find the rate of change of the exponential growth function. 4. At the beginnin
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 16 1. Find the antiderivative: (x2- 4 x + 6 dx)(14.1.24)2. Find the antiderivative: e x1 + dx x(14.1.44) x2 - 7x + 4 dx 3. Find the antiderivative: x 4. Find the antiderivative: x dx 3 6 x 2
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 17 1. Rewrite the problem (4 x - 7) dx using a substitution.5(14.2.1)2. Rewrite the problem 5 5 x 2 + 6 dx using substitution. (14.2.7) 3. Find the indefinite integral: 4. Find the indefinite integral:55 x 2 + 6 d
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 18 1. Suppose f ( x) = 2 x 2 + 6 and you are asked to use Riemann sums to approximate the area under the curve on the interval [0, 8]. (14.3.13) a. if you are asked to use four subintervals of equal length, find x . b. if you
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 20 1. Evaluate: x(x1 02+ 6 dx)2(14.5.1)2. Evaluate:31x 3 x 2 2 dx6(14.5.4)3. Evaluate: (5 x 3) dx1 0(14.5.9)4. Evaluate:41e 3 x dx(14.5.16)5. Evaluate:1 0x 2x + 12dx
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 21 1. Write the integral(s) needed to find the area between the x axis and f ( x) = x 3 3 x 2 . (14.6.1)2. Write the integral(s) needed to find the area between the x axis and f ( x) = x 3 x . (14.6.5)3. Write the integ
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 22 1. Suppose f ( x, y ) = x 2 y - 4 xy + 6 y 2 . Find f (-1, 2) .(16.1.3)2. Suppose f ( x, y ) = 3 x - 4 y x + 5 xy 2 . Find f (4, - 2 ). (16.1.5)3. Suppose f ( x, y, z ) = 2 xy + 3 xz - 4 yz. Find f (0,1,2). 4. Suppose
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 23 1. Find f x : f ( x, y ) = x 2 + 3 xy + 2 y 2 2. Find f y : f ( x, y ) = x 2 + 3 xy + 2 y 2 (16.2.23) (16.2.23) (16.2.4)3. Find the first partial derivatives: f ( x, y ) = 3 - 5 x 2 + 7 y 2 4. Findf : f ( x, y ) = x 2 e
U. Houston - MATH - 1314
Math 1314 Poppers Lesson 24 1. Suppose f ( x, y ) = x 2 - 4 xy + 2 y 2 + 4 x + 8 y - 1 . a. find the first partial derivatives b. find the critical points c. find the second order partial derivatives d. find f xx (a, b) for each critical point e. fin
U. Houston - M - 1300
Math 1300Section 1.2 NotesMake each sentence true by using &lt; or &gt;. 1) -9 _ -8 4) 4 _ 5 2) 0 _-7 5) -5 _11 3) -3 _ -23 6) _ 2/3Adding Integers: o Same signs add and keep the sign o Different signs subtract and take the sign of the number with
U. Houston - M - 1300
Math 1300Section 1.3 NotesGCF (Greatest Common Factor) 1) Write each of the given numbers as a product of prime factors. 2) The GCF of two or more numbers is the product of all prime factors common to every number. Examples: 1. Find the GCF of 24
U. Houston - M - 1300
Math 1300 Rules for Exponents: Multiplying Powers: Section 1.4 Notesa m a n = a m+n am Dividing Powers: = a m-n n a 1 1 Negative Powers: a - m = m and = a n (Note: a 0 ) a a-n Zero Power Rule: a0 = 1 If no power is shown, then the exponent
U. Houston - M - 1300
Math 1300 Rules for Exponents: Multiplying Powers: Section 1.4 Notesa m a n = a m+n am Dividing Powers: = a m-n n a 1 1 Negative Powers: a - m = m and = a n (Note: a 0 ) a a-n Zero Power Rule: a0 = 1 If no power is shown, then the exponent
U. Houston - M - 1300
Math 1300Section 1.5 NotesOrder of Operations: P E M D A SExamples: 1.(2 - 4)3 - (-1 + 6) 2 22.3(-6) + 0.5(-60) (32 - 2) + (-7 - 2)3.-7 + 24.-6 25. 1 - 226. 2 + 3(4 - 5) 2 4 1Math 13001 , 2Section 1.5 NotesE
U. Houston - M - 1300
Math 1300Section 1.6 NotesSolving Linear Equations Steps: 1) Distribute if the equation has parentheses 2) Combine any like terms 3) Isolate the variable by doing addition/subtraction before multiplication/division Examples: 1. x + 2 =82. 4
U. Houston - M - 1300
Math 1300Section 1.7 NotesSolving Linear Inequalities An inequality is similar to an equation except instead of an equal sign &quot;=&quot; you find one of the following signs: &lt;, , &gt;, or . Now &gt; and &lt; are strict inequalities, and and are inequalities th
U. Houston - M - 1300
Math 1300Section 1.8 NotesSolving Absolute Value Equations: To solve and equation involving absolute values, use the following property: If C is positive, then |x| = C is equivalent to x = C. Special cases for |x| = C: Case 1: If C is negative
U. Houston - M - 1300
Math 1300Section 2.1 NotesThe Coordinate Plane:3 2 1 X -3 -2 -1 0 -1 -2 -3 YPlotting: When plotting (a, b) on a graph, go in `alphabetical' order. Go a units in the xdirection (horizontally) first, then in go b units the y-direction (verticall
U. Houston - M - 1300
Math 1300Section 2.1 NotesThe Coordinate Plane:3 2 1 X -3 -2 -1 0 -1 -2 -3 YPlotting: When plotting (a, b) on a graph, go in alphabetical order. Go a units in the xdirection (horizontally) first, then in go b units the y-direction (vertically)
U. Houston - M - 1300
Math 1300Section 2.2 NotesThe Distance and Midpoint Formulas: The distance between two points A = (x1 , y1 ) and B = (x 2 , y 2 ) is given by the distance formula d ( A, B) = (x 2 - x1 )2 + ( y 2 - y1 )2 . 1. Find the distance between the two poi
U. Houston - M - 1300
Math 1300Section 2.2 NotesThe Distance and Midpoint Formulas: The distance between two points A = (x1 , y1 ) and B = (x 2 , y 2 ) is given by the distance formula d ( A, B) = (x 2 - x1 )2 + ( y 2 - y1 )2 . 1. Find the distance between the two poi
U. Houston - M - 1300
Math 1300Section 2.3 NotesSlope and Intercepts of a Line: The slope, m, of a line that passes through the points A = (x1 , y1 ) and B = (x 2 , y 2 ) is y 2 - y1 rise = given by the slope formula m = . x 2 - x1 runPositive SlopeNegative Slope
U. Houston - M - 1300
Math 1300Section 2.3 NotesSlope and Intercepts of a Line: The slope, m, of a line that passes through the points A = (x1 , y1 ) and B = (x 2 , y 2 ) is y 2 - y1 rise = given by the slope formula m = . x 2 - x1 runPositive SlopeNegative Slope
U. Houston - M - 1300
Math 1300Section 2.4 NotesEquations of a Line: Forms of Lines: 1. The standard form of a linear equation is given by Ax + By = C where A and B cannot both be equal to zero. 2. The point-slope form of a linear equation is given by y y1 = m ( x x
U. Houston - M - 1300
Math 1300Section 2.5 NotesParallel and Perpendicular Lines: Two lines with slopes m1 and m2 are parallel if and only if m1 = m2.3 2 1 X -3 -2 -1 0 -1 -2 -3 YTwo lines with slopes m1 and m2 are perpendicular if and only if m1m2 = -1.012
U. Houston - M - 1300
Math 1300Section 2.5 NotesParallel and Perpendicular Lines: Two lines with slopes m1 and m2 are parallel if and only if m1 = m2.3 2 1 X -3 -2 -1 0 -1 -2 -3 YTwo lines with slopes m1 and m2 are perpendicular if and only if m1m2 = -1.012
U. Houston - M - 1300
Math 1300Section 2.6 NotesFunctions and Domains: Definition: A relation in mathematics is a set of one or more ordered pairs. It can be described by: 1. A set of ordered pairs: {(-3, -1), (-2, 1), (-1, 1), (1, 3), (3, 1), (3, 2), (0, 3)} 2. Graph
U. Houston - M - 1300
Math 1300Section 2.7 NotesFunctions and Domains: Definition: The graph of a function f is the set of all points (x, y) in the coordinate plane where the x-coordinates are the elements of the domain of f and where the ycoordinates are given by y =
U. Houston - M - 1300
Math 1300Section 2.7 NotesFunctions and Domains: Definition: The graph of a function f is the set of all points (x, y) in the coordinate plane where the x-coordinates are the elements of the domain of f and where the ycoordinates are given by y =
U. Houston - M - 1300
Math 1300Section 3.1 NotesPolynomials A polynomial is an expression made up of terms called monomials. A monomial is an expression made up of one real-number coefficient and (at least) one variable to some whole-number power. Examples are -3; 7x;
U. Houston - M - 1300
Math 1300 Operations with PolynomialsSection 3.2 NotesAddition of Polynomials The sum of two polynomials is found by combining like terms. To add like terms, add the coefficients and do not change the variable and exponents in common. Example: Pe
U. Houston - M - 1300
Math 1300 Section 2.2 Note: For the final result, use a reduced radical form (i.e., write 8 as 2 2 ).Test 2 ReviewUse the Pythagorean Theorem to find the missing side of each of the following right triangles.1. 2. 3.If a = 5, c =7, find b. If
U. Houston - M - 1300
Section 2.2 1. b = 24 = 2 6 2. c = 125 = 5 5 3. a = 57 5 241 1 = 241 5. d = 100 10 17 6. d = 5 5 7. d = 8 641 1 8. d = = 641 100 10 9. d = 109 3 10. ,4 10 1 5 11. , 5 4 1 1 12. , 2 5 4. d =Section 2.3 13. m = undefined15 2 32 15. m