Fluid Pressure and Fluid Force


 Sybil Wilcox
 2 years ago
 Views:
Transcription
1 0_0707.q //0 : PM Page 07 SECTION 7.7 Section 7.7 Flui Pressure an Flui Force 07 Flui Pressure an Flui Force Fin flui pressure an flui force. Flui Pressure an Flui Force Swimmers know that the eeper an object is submerge in a flui, the greater the pressure on the object. Pressure is efine as the force per unit of area over the surface of a bo. For eample, because a column of water that is 0 feet in height an inch square weighs. pouns, the flui pressure at a epth of 0 feet of water is. pouns per square inch.* At 0 feet, this woul increase to 8. pouns per square inch, an in general the pressure is proportional to the epth of the object in the flui. Definition of Flui Pressure The pressure on an object at epth h in a liqui is Pressure P wh where w is the weightensit of the liqui per unit of volume. The Granger Collection Below are some common weightensities of fluis in pouns per cubic foot. BLAISE PASCAL ( ) Pascal is well known for his work in man areas of mathematics an phsics, an also for his influence on Leibniz. Although much of Pascal s work in calculus was intuitive an lacke the rigor of moern mathematics, he nevertheless anticipate man important results. Ethl alcohol Gasoline Glcerin Kerosene Mercur Seawater Water When calculating flui pressure, ou can use an important (an rather surprising) phsical law calle Pascal s Principle, name after the French mathematician Blaise Pascal. Pascal s Principle states that the pressure eerte b a flui at a epth h is transmitte equall in all irections. For eample, in Figure 7.8, the pressure at the inicate epth is the same for all three objects. Because flui pressure is given in terms of force per unit area P F A, the flui force on a submerge horizontal surface of area A is Flui force F PA (pressure)(area). h The pressure at h is the same for all three objects. Figure 7.8 * The total pressure on an object in 0 feet of water woul also inclue the pressure ue to Earth s atmosphere. At sea level, atmospheric pressure is approimatel.7 pouns per square inch.
2 0_0707.q //0 : PM Page CHAPTER 7 Applications of Integration EXAMPLE Flui Force on a Submerge Sheet Fin the flui force on a rectangular metal sheet measuring feet b feet that is submerge in feet of water, as shown in Figure 7.9. The flui force on a horizontal metal sheet is equal to the flui pressure times the area. Figure 7.9 c L( i ) h( i ) Calculus methos must be use to fin the flui force on a vertical metal plate. Figure 7.70 Solution Because the weightensit of water is. pouns per cubic foot an the sheet is submerge in feet of water, the flui pressure is P. 7. pouns per square foot. Because the total area of the sheet is A square feet, the flui force is F PA pouns. This result is inepenent of the size of the bo of water. The flui force woul be the same in a swimming pool or lake. In Eample, the fact that the sheet is rectangular an horizontal means that ou o not nee the methos of calculus to solve the problem. Consier a surface that is submerge verticall in a flui. This problem is more ifficult because the pressure is not constant over the surface. Suppose a vertical plate is submerge in a flui of weightensit w (per unit of volume), as shown in Figure To etermine the total force against one sie of the region from epth c to epth, ou can subivie the interval c, into n subintervals, each of with. Net, consier the representative rectangle of with an length L i, where i is in the ith subinterval. The force against this representative rectangle is F i w epth area wh i L i. P wh The force against n such rectangles is n F i w n h i L i. i i pouns square feet square foot Note that w is consiere to be constant an is factore out of the summation. Therefore, taking the limit as 0 n suggests the following efinition. Definition of Force Eerte b a Flui The force F eerte b a flui of constant weightensit w (per unit of volume) against a submerge vertical plane region from c to is F w lim 0 n h i L i i w c h L where h is the epth of the flui at an L is the horizontal length of the region at.
3 0_0707.q //0 : PM Page 09 SECTION 7.7 Flui Pressure an Flui Force 09 EXAMPLE Flui Force on a Vertical Surface ft 8 ft ft (a) Water gate in a am ft h() = (, ) 0 (, 9) (b) The flui force against the gate Figure 7.7 A vertical gate in a am has the shape of an isosceles trapezoi 8 feet across the top an feet across the bottom, with a height of feet, as shown in Figure 7.7(a). What is the flui force on the gate when the top of the gate is feet below the surface of the water? Solution In setting up a mathematical moel for this problem, ou are at libert to locate the  an aes in several ifferent was. A convenient approach is to let the ais bisect the gate an place the ais at the surface of the water, as shown in Figure 7.7(b). So, the epth of the water at in feet is Depth h. To fin the length L of the region at, fin the equation of the line forming the right sie of the gate. Because this line passes through the points, 9 an,, its equation is In Figure 7.7(b) ou can see that the length of the region at is Length L. Finall, b integrating from 9 to, ou can calculate the flui force to be F w h L c ,9 pouns. NOTE In Eample, the ais coincie with the surface of the water. This was convenient, but arbitrar. In choosing a coorinate sstem to represent a phsical situation, ou shoul consier various possibilities. Often ou can simplif the calculations in a problem b locating the coorinate sstem to take avantage of special characteristics of the problem, such as smmetr.
4 0_0707.q //0 : PM Page 0 0 CHAPTER 7 Applications of Integration EXAMPLE Flui Force on a Vertical Surface 8 Observation winow The flui force on the winow Figure A circular observation winow on a marine science ship has a raius of foot, an the center of the winow is 8 feet below water level, as shown in Figure 7.7. What is the flui force on the winow? Solution To take avantage of smmetr, locate a coorinate sstem such that the origin coincies with the center of the winow, as shown in Figure 7.7. The epth at is then Depth h 8. The horizontal length of the winow is, an ou can use the equation for the circle,, to solve for as follows. Length Finall, because ranges from to, an using pouns per cubic foot as the weightensit of seawater, ou have F w L h L c 8. Initiall it looks as if this integral woul be ifficult to solve. However, if ou break the integral into two parts an appl smmetr, the solution is simple. F The secon integral is 0 (because the integran is o an the limits of integration are smmetric to the origin). Moreover, b recognizing that the first integral represents the area of a semicircle of raius, ou obtain F pouns. So, the flui force on the winow is 08. pouns. 0. f is not ifferentiable at ±. Figure 7.7. TECHNOLOGY To confirm the result obtaine in Eample, ou might have consiere using Simpson s Rule to approimate the value of 8 8. From the graph of f 8 however, ou can see that f is not ifferentiable when ± (see Figure 7.7). This means that ou cannot appl Theorem.9 from Section. to etermine the potential error in Simpson s Rule. Without knowing the potential error, the approimation is of little value. Use a graphing utilit to approimate the integral.
5 0_0707.q //0 : PM Page SECTION 7.7 Flui Pressure an Flui Force Eercises for Section 7.7 Force on a Submerge Sheet In Eercises an, the area of the top sie of a piece of sheet metal is given. The sheet metal is submerge horizontall in feet of water. Fin the flui force on the top sie.. square feet. square feet See for workeout solutions to onumbere eercises. Flui Force of Water In Eercises, fin the flui force on the vertical plate submerge in water, where the imensions are given in meters an the weightensit of water is 9800 newtons per cubic meter.. Square. Square Buoant Force In Eercises an, fin the buoant force of a rectangular soli of the given imensions submerge in water so that the top sie is parallel to the surface of the water. The buoant force is the ifference between the flui forces on the top an bottom sies of the soli... h h ft ft ft ft ft 8 ft. Triangle. Rectangle Flui Force on a Tank Wall In Eercises 0, fin the flui force on the vertical sie of the tank, where the imensions are given in feet. Assume that the tank is full of water. 9. Rectangle. Triangle 7. Trapezoi 8. Semicircle 9. Parabola, 0. Semiellipse, 9 Force on a Concrete Form In Eercises 8, the figure is the vertical sie of a form for poure concrete that weighs 0.7 pouns per cubic foot. Determine the force on this part of the concrete form.. Rectangle. Semiellipse, 0 ft ft 7. Rectangle 8. Triangle ft ft ft ft ft ft 9. Flui Force of Gasoline A clinrical gasoline tank is place so that the ais of the cliner is horizontal. Fin the flui force on a circular en of the tank if the tank is half full, assuming that the iameter is feet an the gasoline weighs pouns per cubic foot.
6 0_0707.q //0 : PM Page CHAPTER 7 Applications of Integration 0. Flui Force of Gasoline Repeat Eercise 9 for a tank that is full. (Evaluate one integral b a geometric formula an the other b observing that the integran is an o function.). Flui Force on a Circular Plate A circular plate of raius r feet is submerge verticall in a tank of flui that weighs w pouns per cubic foot. The center of the circle is k k > r feet below the surface of the flui. Show that the flui force on the surface of the plate is F wk r. (Evaluate one integral b a geometric formula an the other b observing that the integran is an o function.). Flui Force on a Circular Plate Use the result of Eercise to fin the flui force on the circular plate shown in each figure. Assume the plates are in the wall of a tank fille with water an the measurements are given in feet. (a) (b). Flui Force on a Rectangular Plate A rectangular plate of height h feet an base b feet is submerge verticall in a tank of flui that weighs w pouns per cubic foot. The center is k feet below the surface of the flui, where h k. Show that the flui force on the surface of the plate is F wkhb.. Flui Force on a Rectangular Plate Use the result of Eercise to fin the flui force on the rectangular plate shown in each figure. Assume the plates are in the wall of a tank fille with water an the measurements are given in feet. (a) (b). Submarine Porthole A porthole on a vertical sie of a submarine (submerge in seawater) is square foot. Fin the flui force on the porthole, assuming that the center of the square is feet below the surface.. Submarine Porthole Repeat Eercise for a circular porthole that has a iameter of foot. The center is feet below the surface Moeling Data The vertical stern of a boat with a superimpose coorinate sstem is shown in the figure. The table shows the with w of the stern at inicate values of. Fin the flui force against the stern if the measurements are given in feet. w Water level 8. Irrigation Canal Gate The vertical cross section of an irrigation canal is moele b f Stern where is measure in feet an 0 correspons to the center of the canal. Use the integration capabilities of a graphing utilit to approimate the flui force against a vertical gate use to stop the flow of water if the water is feet eep. In Eercises 9 an 0, use the integration capabilities of a graphing utilit to approimate the flui force on the vertical plate boune b the ais an the top half of the graph of the equation. Assume that the base of the plate is feet beneath the surface of the water Think About It (a) Approimate the epth of the water in the tank in Eercise if the flui force is onehalf as great as when the tank is full. (b) Eplain wh the answer in part (a) is not. Writing About Concepts. Define flui pressure.. Define flui force against a submerge vertical plane region.. Two ientical semicircular winows are place at the same epth in the vertical wall of an aquarium (see figure). Which has the greater flui force? Eplain. w 7
Exponential Functions: Differentiation and Integration. The Natural Exponential Function
46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential
More informationSolutions to Examples from Related Rates Notes. ds 2 mm/s. da when s 100 mm
Solutions to Examples from Relate Rates Notes 1. A square metal plate is place in a furnace. The quick temperature change causes the metal plate to expan so that its surface area increases an its thickness
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationWork Done by a Constant Force
6_75.qd // :9 PM Page 87 SECTION 7.5 Work 87 Section 7.5 Work Find the work done b a constant force. Find the work done b a variable force. Work Done b a Constant Force The concept of work is important
More informationAnswers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationExample Optimization Problems selected from Section 4.7
Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point
More informationCalculating Viscous Flow: Velocity Profiles in Rivers and Pipes
previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar
More informationSECTION 91 Conic Sections; Parabola
66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships
More informationSection 72 Ellipse. Definition of an Ellipse The following is a coordinatefree definition of an ellipse: DEFINITION
7 Ellipse 3. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a (  h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b (  h) 3 ) + k Resource Locker Eplore 1
More information6.1 Area between curves
6. Area between curves 6.. Area between the curve and the ais Definition 6.. Let f() be continuous on [a, b]. The area of the region between the graph of f() and the ais is A = b a f()d Let f() be continuous
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
More informationChapter 6 Momentum Analysis of Flow System
57:00 Mechanics of Fluis an Transport Processes Chapter 6 Professor Fre Stern Tpe b Stephanie Schraer Fall 005 Chapter 6 Momentum nalsis of Flow Sstem 6. Momentum Equation RTT with B = M an β = [ F S +
More informationThe Rectangular Coordinate System
3.2 The Rectangular Coordinate Sstem 3.2 OBJECTIVES 1. Graph a set of ordered pairs 2. Identif plotted points 3. Scale the aes NOTE In the eighteenth centur, René Descartes, a French philosopher and mathematician,
More informationThere are good hints and notes in the answers to the following, but struggle first before peeking at those!
Integration Worksheet  Using the Definite Integral Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the various types of problems. Important:
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationSection 105 Parametric Equations
88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative
More informationA Summary of Curve Sketching. Analyzing the Graph of a Function
0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph
More informationy or f (x) to determine their nature.
Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic
More informationTHE PARABOLA section. Developing the Equation
80 (0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying
More informationCoordinate Geometry. Positive gradients: Negative gradients:
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
More information3 Rectangular Coordinate System and Graphs
060_CH03_13154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
More informationMathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400
hsn.uk.net Higher Mathematics UNIT OUTCOME 4 Circles Contents Circles 119 1 Representing a Circle 119 Testing a Point 10 3 The General Equation of a Circle 10 4 Intersection of a Line an a Circle 1 5 Tangents
More informationSTRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs
6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous
More informationYears t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304
Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More informationAttributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest yvalue of a function. Minimum value the least
 Attributes and Transformations of Quadratic Functions TEKS FCUS TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening.
More information20122013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 08
01013 Enhance Instructional Transition Guie Unit 08: Exponents an Polynomial Operations (18 ays) Possible Lesson 01 (4 ays) Possible Lesson 0 (7 ays) Possible Lesson 03 (7 ays) POSSIBLE LESSON 0 (7 ays)
More informationAre You Ready? Simplify Radical Expressions
SKILL Are You Read? Simplif Radical Epressions Teaching Skill Objective Simplif radical epressions. Review with students the definition of simplest form. Ask: Is written in simplest form? (No) Wh or wh
More informationAnalyzing the Graph of a Function
SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the
More informationRectangle Square Triangle
HFCC Math Lab Beginning Algebra  15 PERIMETER WORD PROBLEMS The perimeter of a plane geometric figure is the sum of the lengths of its sides. In this handout, we will deal with perimeter problems involving
More informationTranslating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More information25. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math
Chapter 2 Lesson 25 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular
More informationVOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.
Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:
More informationSECTION 22 Straight Lines
 Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationPhysics 213 Problem Set 8 Solutions Fall 1997
Phsics 213 Problem Set 8 Solutions Fall 1997 1. Reaing Assignment a) This problem set covers Serwa 25.325.8, 26.126.2 b) For next week, we will cover material in chapters 26.326.7, 28.428.6. 2. In
More informationnparameter families of curves
1 nparameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,
More informationHere the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and
Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric
More informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More informationM147 Practice Problems for Exam 2
M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work
More informationCOORDINATE PLANES, LINES, AND LINEAR FUNCTIONS
G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs
More informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationSOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH
SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH Basic Definitions In homework set # one of the questions involves basic unerstaning of the hyperbolic functions sinh an cosh. We will use this
More informationPerimeter, Area, and Volume
Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all
More informationApplications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
More informationManure Spreader Calibration
Agronomy Facts 68 Manure Spreaer Calibration Manure spreaer calibration is an essential an valuable nutrient management tool for maximizing the efficient use of available manure nutrients. Planne manure
More information11 CHAPTER 11: FOOTINGS
CHAPTER ELEVEN FOOTINGS 1 11 CHAPTER 11: FOOTINGS 11.1 Introuction Footings are structural elements that transmit column or wall loas to the unerlying soil below the structure. Footings are esigne to transmit
More informationHIRES STILL TO BE SUPPLIED
1 MRE GRAPHS AND EQUATINS HIRES STILL T BE SUPPLIED Differentshaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More informationCOORDINATE PLANES, LINES, AND LINEAR FUNCTIONS
a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated
More informationPurpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of
Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More information2.7 Mathematical Models: Constructing Functions
CHAPTER Functions and Their Graphs.7 Mathematical Models: Constructing Functions OBJECTIVE Construct and Analyze Functions Construct and Analyze Functions Realworld problems often result in mathematical
More information88 Volume and Surface Area of Composite Figures. Find the volume of the composite figure. Round to the nearest tenth if necessary.
Find the volume of the composite figure. Round to the nearest tenth if necessary. The figure is made up of a triangular prism and a rectangular prism. Volume of triangular prism The figure is made up of
More informationAP Calculus Testbank (Chapter 7) (Mr. Surowski)
AP Calculus Testbank (Chapter 7) (Mr. Surowski) Part I. MultipleChoice Questions. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved
More informationThe Natural Logarithmic Function: Integration. Log Rule for Integration
6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. EXPLORATION Integrating Rational Functions Earl in Chapter, ou learned rules that allowed ou to integrate
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More information6.3 Parametric Equations and Motion
SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More informationExploration 72a: Differential Equation for Compound Interest
Eploration 72a: Differential Equation for Compoun Interest Objective: Write an solve a ifferential equation for the amount of mone in a savings account as a function of time. When mone is left in a savings
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationContents. How You May Use This Resource Guide
Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More informationSLOPES AND EQUATIONS OF LINES CHAPTER
CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8 The Slope of a Line 8 The Equation of a Line 83 Midpoint of a Line Segment 84 The Slopes of Perpendicular Lines 85 Coordinate Proof 86 Concurrence of the
More informationThe Standard Form of a Differential Equation
Fall 6 The Stanar Form of a Differential Equation Format require to solve a ifferential equation or a system of ifferential equations using one of the commanline ifferential equation solvers such as rkfie,
More informationMath 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas
Math 0 Chapter Lecture Notes Professor Miguel Ornelas M. Ornelas Math 0 Lecture Notes Section. Section. The Rectangular Coordinate Sstem Plot each ordered pair on a Rectangular Coordinate Sstem and name
More informationTransformations of Function Graphs
   0        Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),
More informationMathematical Modeling and Optimization Problems Answers
MATH& 141 Mathematical Modeling and Optimization Problems Answers 1. You are designing a rectangular poster which is to have 150 square inches of tet with inch margins at the top and bottom of the poster
More information121 Representations of ThreeDimensional Figures
Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 121 Representations of ThreeDimensional Figures Use isometric dot paper to sketch each prism. 1. triangular
More information15.2. FirstOrder Linear Differential Equations. FirstOrder Linear Differential Equations Bernoulli Equations Applications
00 CHAPTER 5 Differential Equations SECTION 5. FirstOrer Linear Differential Equations FirstOrer Linear Differential Equations Bernoulli Equations Applications FirstOrer Linear Differential Equations
More information2.8 FUNCTIONS AND MATHEMATICAL MODELS
2.8 Functions and Mathematical Models 131 2.8 FUNCTIONS AND MATHEMATICAL MODELS At one time Conway would be making constant appeals to give him a year, and he would immediately respond with the date of
More informationSection 14.5 Directional derivatives and gradient vectors
Section 4.5 Directional derivatives and gradient vectors (3/3/08) Overview: The partial derivatives f ( 0, 0 ) and f ( 0, 0 ) are the rates of change of z = f(,) at ( 0, 0 ) in the positive  and directions.
More informationMathematics. Total marks 100. Section I Pages marks Attempt Questions 1 10 Allow about 15 minutes for this section
04 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Boardapproved calculators ma be
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
More informationNotes on tangents to parabolas
Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,
More information2 Analysis of Graphs of
ch.pgs116 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More information64 : Learn to find the area and circumference of circles. Area and Circumference of Circles (including word problems)
Circles 64 : Learn to fin the area an circumference of circles. Area an Circumference of Circles (incluing wor problems) 83 Learn to fin the Circumference of a circle. 86 Learn to fin the area of circles.
More informationPerimeter. 14ft. 5ft. 11ft.
Perimeter The perimeter of a geometric figure is the distance around the figure. The perimeter could be thought of as walking around the figure while keeping track of the distance traveled. To determine
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationAppendix D: Variation
Appendi D: Variation Direct Variation There are two basic types of linear models. The more general model has a yintercept that is nonzero. y = m + b, b 0 The simpler model y = k has a yintercept that
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More information