4) Set derivative of the function equal to zero and solve. Maximum And Minimum Quadratic Word Problems Worksheet. (solution by Calculus). His method was essentially geometrical, and this made the. Find the length of the shortest ladder that will reach over an 8-ft. com: Free Precalculus Review and Calculus Preview Lessons and Practice Problems. Define calculus. Provide a Testimonial Introduction. Thus there is only one relative minimum in this function, and it occurs at x=0. While a fair number of the exercises involve only routine computations, many of the exercises and most of the problems are meant to illuminate points that in my experience students have found confusing. S Publisher: McGraw-Hill, Year: 2001 ISBN: 0071358978 Search in Amazon. We are trying to do things like maximise the profit in a company, or minimise the costs, or find the least amount of material to make a particular object. It also has its application to commercial problems, such as finding the least dimensions of a carton that is to contain a given volume. Finding maximum and minimum values of polynomial functions help us solve these types of problems. We introduce an explicit Polya approach to a maximizing an area problem, with an. Pre-Calculus Assignment Sheet Unit 4 - Graphing & Writing Sine & Cosine Functions; Application Problems October 21 to November 5th, 2013 Date Topic Assignment Monday Gr 10/21 changes. The process of finding maximum or minimum values is called optimisation. Read the problem- write the knowns, unknowns and draw a diagram if applicable L y 8 3 x-3 x 2. What is the maximum volume for such a box? Let xand ybe as is shown in the gure above. PRACTICE PROBLEMS: For problems 1-10, identify all critical points of the given function. 5 Optimization Problems Practice Solve each optimization problem. " That x-coordinate of a relative maximum or minimum value of the function. What are the dimensions if the printed area is to be a maximum? 2) A cylindrical container with circular base is to hold 64 cubic centimeters. The minimum value of the function f (x) = x 2 + 1 is y = 1. Find the maximum and minimum on the domain [-10, 10], and graph the function as well as its derivative using Wolfram|Alpha. Find the local maximum and minimum values and saddle point(s) of the function. You can find the maximum or minimum if your original function is written in general form, f(x)=ax2+bx+c{\displaystyle f(x)=ax^{2}+bx+c}, or in standard form, f(x)=a(x−h)2+k{\displaystyle f(x)=a(x-h)^{2}+k}. Read each problem slowly and carefully. And I can tell you right away. Every word is important and must be clearly understood if. » Questions » Science/Math » Math » Calculus » Find the global minimum and maximum values Questions Courses Find the global minimum and maximum values of f(x,y)=x^2+4x-7y+y^2, Where D is the square whose vert. $$h=-5t^2+20t+1$$ a) Find the maximum height of the ball and the time when it occurs. If relevant, make a drawing and label the measurements on your drawing. When do the minimum speeds occur? What are they? Answer: the speed is zero at b and d; Students often benefit from a verbal explanation of all this. We use the derivative to determine the maximum and minimum values of particular functions (e. Now look at the same places and think about what the slope is at those two locations. 1 Problem 47E. Louis University) Fundamentals of Engineering Calculus, Differential Equations & Transforms, and Numerical Analysis1 / 30. These constraints are usually very helpful to solve optimization problems. • Find absolute ex. Check out my Calculus limits playlist if you want some harder examples. Math 1A: Calculus Worksheets 7th Edition Department of Mathematics, University of California at Berkeley and Problems. Give all decimal answers correct to three decimal places. In this section we work on word problems where we try to find the maximum or minimum value of a variable. 1) The problem may state in words what you're supposed to find, in which case you have to translate those words into symbols. These slides act like unfinished lecture notes. Login to reply the answers Post; Ray. (d) A function continuous on an open interval may not have an absolute minimum or absolute maximum on that interval. Understanding Calculus: Problems, Solutions, and Tips covers all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college. For example, if the length of the base is x = 5 cm then the surface area is850cm2. Problems (1) A man has 1200 feet of fence with which to enclose a rectangular area. However, before we differentiate the right-hand side, we will write it as a function of x only. Written Response: 1. F INDING a maximum or a minimum has its application in pure mathematics, where for example we could find the largest rectangle that has a given perimeter. We de ned a critical number of f(x) on domain D to be any number x = c on the domain D such that. It is through word problems that we find value in math. Give all decimal answers correct to three decimal places. Determine a feasible domain • 4. Calculus Word Problems Mathematical Analysis Math Word Work. Read the problem carefully. superprof resources. Each problem was designed to show a reasonable portrayal of actual scenarios faced. As of September 3, 2019, all WORD Problem tutorials have been reprogrammed as lessons with answers. Again, we can use the vertex to find the maximum or the minimum values, and roots to find solutions to quadratics. Finding the maximum of a parabola can tell you the maximum height of a ball thrown into the air, the maximum area of a rectangle, the minimum value of a company's profit, and so on. Maxima and minima mc-TY-maxmin-2009-1 In this unit we show how diﬀerentiation can be used to ﬁnd the maximum and minimum values of a function. You run away at a speed of 6 meters per second. Steps in Solving Maxima and Minima Problems. Graphing with Calculus. -2- ©u u210 R143j hK Eu Ht4as nSwo1f2t Vwlagr7eE ELyL VCE. Many problems in the math section will be presented as word problems. Its maximum cruising speed is 110 mi/h. However, we are today equipped with graphing calculators and computers to find the maximum and minimum values of functions. Every word is important and must be clearly understood if. When a function has a maximum or minimum on an infinite domain, the derivative is _____. At that point, they'll want you to differentiate to find the maximums and minimums; at this point, you'll find the vertex, since the vertex will be the maximum or minimum of the related graphed. Learning Calculus limits is a breeze with this problem and introduction. 5, which should be followed by the multivariable max-min problems in section 6. These are sometimes called quadratic word problems. aphing Sine and Cosine functions: amplitude, phase shift, and vertical slide Cofunctions (Notes pp. Find the global maximum value and the global minimum value of the function f (x) = x 4-2 x 2 + 3 on [-1, 2]. For success with such word problems, Professor Edwards stresses the importance of first framing the problem with precalculus, reducing the equation to one independent variable, and then using calculus to find and verify the answer. Shormann provides 21st Century math and science homeschool curriculum with a Christian foundation featuring expert video instruction, automated grading, and email support. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Pre -Calculus 11 Extra Practice - Word Problems & Max/Min Name _____ Block _____ 1. What are the values of t at which the speed obtains its (local) maximum values? Answer: x = a, c, and e. To find this value, we set dA/dx = 0. This new title. Check that this value is a minimum or maximum and read exactly what form the answer should be. The constraints may be equalities or inequalities. Minimum And Maximum Some of the worksheets for this concept are Assignment date period, For each problem find all points of absolute minima and, Name, Work 17 maxima and minima, Quadratic work name maximums and minimums, Bwc payroll reporting guidelines, Ira required minimum distribution work, Calculus work problems. Maximum And Minumum. MAXIMUM AND MINIMUM IN APPLICATIONS To find a maximum or minimum, solve f'(x) = 0. Calculus: Derivatives Maximum/Minimum Word Problems Topics include cost function, ellipse, distance, volume, surface area, and more. Question: How to solve if instead, the problem asked for the minimum profit? In some problems, the minimum is the value of x (example: the 0. Now you have an equation of one variable. Optimization problems (also called maximum-minimum problems) occur in many fields and contexts in which it is necessary to find the maximum or minimum of a function to solve a problem. We don't go higher than that in many problems, but the second derivative is an important--the derivative of the derivative is an important thing to know, especially in problems with maximum and minimum, which is the big application of derivatives, to locate a maximum or a minimum, and to decide which one it is. Max & Min applications. A complete set of Class Notes, Handouts, Worksheets, PowerPoint Presentations, and Practice Tests. MAXIMUM AND MINIMUM VALUES. Such problems, involving the determination of the form of a curve having a certain maximum or minimum property, were quite different from the ordinary maximum and minimum problems of the differential calculus. There is another way to measure an angle, which involves arc length. Justify your answer. This worksheet generates AB Calculus Topics/Questions: To keep server load down, there is a maximum of 100 questions per worksheet. Behind every calculus problem, neatly packaged and ready to solve, is a word problem. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum. Area and perimeter worksheets. Free Maximum Calculator - find the Maximum of a data set step-by-step This website uses cookies to ensure you get the best experience. asked by Marissa on August 16, 2007; Algebra2. One that is very useful is to use the derivative of a function (and set it to 0) to find a minimum or maximum to find either the smallest something can be, or the largest it can be. Hairy inflection point problem. Maximum and Minimum of. Find the dimensions of the rectangle with the maximum area that can be in-scribed in a circle of radius 10. Improve your math knowledge with free questions in "Find the maximum or minimum value of a quadratic function" and thousands of other math skills. Solution of exercise 3. Maximum And Minumum - Displaying top 8 worksheets found for this concept. exponential functions 25. Think about the English meaning of the word constraint, and remember that the constraint equation will have an equals sign. 2 The chapter also has maximum-minimum and optimization word problemsin 2. com Description: Considered to be the hardest mathematical problems to solve, word problems continue to terrify students across all math disciplines. How to Find Maximum and Minimum Points Using Differentiation ? In this section, we will see some example problems of finding maximum and minimum values of the function. For example, in any manufacturing business it is usually possible to express profit as function of the number of units sold. Topic: Applications of Derivatives • Find the maximum or minimum value of a function in an optimization problem by finding its critical points and applying the second derivative test. The tip of the corner is no more than 4 inches above the bottom edge of the paper. Basic calculus with applications to problems in the life and social sciences. Three major examples are geometry, number theory, and functional equations. ©a 62C0z1 b3e DKGuStwa S rS9odf 6tVwVaHr Ve f 7LBLhC U. Check that this value is a minimum or maximum and read exactly what form the answer should be. Find the Local Maxima and Minima. As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. feet and then drops the same distance. More information. The largest is the absolute maximum whereas the smallest is the absolute minimum. 512 The only critical point of A is r =1. Pre-Calculus Online. The best ticket prices to maximize the revenue is then: $10−0. 75 cm and at the sides 0. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal. Linear Approximation and Differentials- Finding approximate values of a function using linearizations, definition of the differential and how it relates to linear approximations. Some have short videos. Determine a feasible domain • 4. These are often referred to as word problems. For example, companies often want to minimize production costs or maximize revenue. In the optimization sections (2. word problem involving the maximum or minimum of a quadratic function. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the. 512 is a global maximum. 371 in the problem above) after differentiating the given equation and equating it to 0. Absolute maximum. Such areas must be learned outside class. By finding the value of x where the derivative is 0, then, we have discovered that the vertex of the parabola is at (3, −4). Piecewise Functions Worksheet. Maximum Minimum. Mathematics has its own “grammar”. Listed below are word lessons that focus on giving students instruction on how to solve most types of word problems commonly found in algebra, geometry, and trigonometry. 18 for each thing-a-ma-bob. Rather, the purpose is to show them real world situations in which calculus is helpful. 3 Increasing & Decreasing Functions and the 1st Derivative Test F: intervals where f is incr, decr relative extrema step­by­step: 1st Deriv Test Calculus Home Page Problems for 3. If θ is the angle between the ground and her line of sight to the balloon, at what rate is this angle changing at the instant the balloon hits the ground?. ) > I'm really keen to understand if it is possible to calculate maximum and > minimum points of cubic graphs WITHOUT the use of calculus. To get started a sketch may help you get a sense of what the curve looks like. We’ll break these two big Stages into smaller steps below. All students take calculus. Louis University Brody Dylan Johnson (St. Maximum and Minimum Value Word Problems - Quadratic Equations This algebra video tutorial explains how to solve word problems that asks you to calculate the maximum value of a function or How to solve word problems with quadratic equations Geometry Teachers Never Spend Time Trying to Find. You'll find a variety of solved word problems on this site, with step by step examples. 10) Give an example function f (x) where f '' (0) = 0 and there is a relative maximum at x = 0. Students must know how to perform the first and second derivative tests and how to identify maximum, minimum and inflection points from those tests. Chapter 6: Word Problems - Rate And Distance; Chapter 7: Word Problems - Percentages; Chapter 8: Word Problems - Investments; Chapter 9: Word Problems - Ratios; Chapter 10: Word Problems - Age; Chapter 11: Word Problems - Rates Of Performing Work; Chapter 12: Word Problems - Slope-Intercept Equations; Chapter 13: Word Problems With 2 Variables. Reduce the primary equation to one variable by substitutions from other equations/information given • 3. Page Navigation. Maximum And Minumum. Because the derivative provides information about the gradient or slope of the graph of a function we can use it to locate points on a graph where the gradient is zero. Pre-Calculus 11- Course Outline: File Size: Maximum and Minimum Problems. 79 to make each thing-a-ma-bob and the company charges$ 5. 1 decade ago. In the optimization sections (2. Here is a set of practice problems to accompany the Finding Absolute Extrema section of the Applications of Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. By using this website, you agree to our Cookie Policy. Linear Approximation and Differentials- Finding approximate values of a function using linearizations, definition of the differential and how it relates to linear approximations. 1 through 3. Worksheet # 17: Maxima and Minima 1. use differentiation to solve related rate problems in a variety of pure and applied contexts. Proof for all n is delayed until we discuss numerical sequences and series later on. Examples train understanding, translation into the mathematical language (eg, equations), solve it, check the accuracy and solution discussion. Nov 24, 2009 Equation Calculator - Quadratic Solver. Word problems involving integrals usually fall into one of two general categories: alien related and non-alien related. Complete each question on lined paper. They illustrate one of the most important applications of the first derivative. Sample Minimum/Maximum Problem (05:27) Steps for finding the minimum of a function are demonstrated. 265 Clove Road, New Rochelle, New York 10801 Pre-Calculus. " That x-coordinate of a relative maximum or minimum value of the function. We we've seen, there are many useful applications of differential calculus. Finding a maximum for this function represents a straightforward way of maximizing profits. She is watching the balloon as it travels at a steady rate of 20 feet per second towards the ground. Calculus Optimization Problems/Related Rates Problems Solutions 1) A farmer has 400 yards of fencing and wishes to fence three sides of a rectangular field (the fourth side is along an existing stone wall, and needs no additional fencing). l'hopital's rule 26. A local extrema is the point where the function takes on the largest or smallest value in a small region around the point. The best ticket prices to maximize the revenue is then: $10−0. Write a function for each problem, and justify your answers. Reduce the primary equation to one variable by substitutions from other equations/information given • 3. cost, strength, amount of material used in a building, profit, loss, etc. Know how to compute absolute maxima and minima on closed regions. •• This might not be the case in applied “word problems. This is what Wolfram|Alpha does with elementary word problems: it not only gives you the answers, but it also helpfully translates the information in the problems into mathematical symbols, showing you the first (and most important) steps toward finding a solution. In the pdf version of the full text, clicking. However after t months the value of the investment, in dollars, is. The expert examines calculus and inequalities in a word problem. It is through word problems that we find value in math. Folsom Lake College's mathematics program provides students with the ability to think logically and abstractly and develop the problem-solving and computational skills necessary for success in any field of study. ©a 62C0z1 b3e DKGuStwa S rS9odf 6tVwVaHr Ve f 7LBLhC U. There are three problems, each of which has a background discussion, an illustrative example, and an exercise for you to do. Get an answer for 'Maximum profit, given revenue and cost equations. (1 pt) A company that makes thing-a-ma-bobs has a start up cost of$ 47060. If each rented room costs $10. 3 - Radians So far we have measured angles in degrees. What is a Limit? Local Maximum and Minimum Values/ Function of Two Variables The Simplex Method – Finding a Maximum / Word Problem Example, Part 5. 01 Minimum length of cables linking to one point;. MAT 111 - Pre-Calculus Chapter 6 – Trigonometric Functions 6 6. Trakimas Math WHS. Determine a feasible domain • 4. Be able to solve word problems involving maxima and minima. Read the problem carefully. Using Derivatives to Find the Absolute Maximum and Minimum Values of f(x)OBJECTIVES• Find absolute extrema using Maximum-Minimum Principle I. Optimization Problems; Introduction to Optimization. 3 Increasing & Decreasing Functions and the 1st Derivative Test F: intervals where f is incr, decr relative extrema step­by­step: 1st Deriv Test Calculus Home Page Problems for 3. (b) Find the velocity and acceleration vectors of the particle. The functions that maximize or minimize the functionals are can be found using the Euler - Lagrange of the calculus of variations. For each$1. Word problems with max/min Example: Optimization 1 A rancher wants to build a rectangular pen, using one side of her barn for one side of the pen, and using 100m of fencing for the other three sides. Can anyone help me to solve these derivative word problems pls. Find the global maximum value and the global minimum value of the function f (x) = x 4-2 x 2 + 3 on [-1, 2]. Rational Equations Word Problems. Max Min Word Problems Our approach to max min word problems is modeled after our approach to related rates word problems. Now look at the same places and think about what the slope is at those two locations. Pre-Calculus Assignment Sheet Unit 4 - Graphing & Writing Sine & Cosine Functions; Application Problems October 21 to November 5th, 2013 Date Topic Assignment Monday Gr 10/21 changes. Pre-Calculus 11- Course Outline: File Size: Maximum and Minimum Problems. We don't go higher than that in many problems, but the second derivative is an important--the derivative of the derivative is an important thing to know, especially in problems with maximum and minimum, which is the big application of derivatives, to locate a maximum or a minimum, and to decide which one it is. Can You Show Me Examples Similar to My Problem? Optimization is a tool with applications across many industries and functional areas. Applications of Extrema of Functions of Two Variables. 2017 AP® CALCULUS AB FREE-RESPONSE QUESTIONS 2. Linear Programming A linear programming problem may be defined as the problem of maximizing or minimizing a linear function subject to system of linear constraints. Graphing with Calculus. In the process of solving these problems we will develop a procedure in which all “word problems” can be effectively solved. The analytical tutorials may be used to further develop your skills in solving problems in calculus. The course concentrates on the various functions that are important to the study of the calculus. Proof for all n is delayed until we discuss numerical sequences and series later on. A high point is called a maximum (plural maxima). OPTIMATIZATION - MAXIMUM/MINIMUM PROBLEMS - BC CALCULUS. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. Maxima and minima mc-TY-maxmin-2009-1 In this unit we show how diﬀerentiation can be used to ﬁnd the maximum and minimum values of a function. 3 FINDING THE MAXIMUM AND MINIMUM OF WORD PROBLEMS. Optimization Problems (Calculus Fun) Many application problems in calculus involve functions for which you want to find maximum or minimum values. Some worksheets contain more problems than can be done during one discussion section. There is some vital stuff in this unit. While they might not actually work out the quadratic function to come up with a precise number, managers at movie theaters,. Read the problem- write the knowns, unknowns and draw a diagram if applicable L y 8 3 x-3 x 2. 2) There are three levels of information. Below is the graph of some function, $$f\left( x \right)$$. There was not a good enough understanding of how the Earth, stars and planets moved with respect to each other. Listed below are word lessons that focus on giving students instruction on how to solve most types of word problems commonly found in algebra, geometry, and trigonometry. Identify all of the relative extrema and absolute extrema of the function. Get an answer for 'Maximum profit, given revenue and cost equations. Give all decimal answers correct to three decimal places. One-dimensional and two-dimensional gravity problems, range, vector components of velocity, etc. Clearly, negative values are not allowed by our problem, so we are left with only two cut points and the following line graph: Therefore the minimum occurs for x 3. Free Maximum Calculator - find the Maximum of a data set step-by-step This website uses cookies to ensure you get the best experience. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. In this maximum and minimum worksheet, students solve and complete 10 various types of word problems. Max and Min's. We shall see that such. We don't go higher than that in many problems, but the second derivative is an important--the derivative of the derivative is an important thing to know, especially in problems with maximum and minimum, which is the big application of derivatives, to locate a maximum or a minimum, and to decide which one it is. Topic: Applications of Derivatives • Find the maximum or minimum value of a function in an optimization problem by finding its critical points and applying the second derivative test. (4 Credits) Differential calculus for engineers and scientists. Worksheets are Assignment date period, For each problem find all points of absolute minima and, Work 17 maxima and minima, Work 5, Lessonunit plan name key features of graphs swbat, Fha maximum mortgage work, Name, Quadratic work name maximums and minimums. If a baseball is projected upward from ground level with an initial velocity of 64 feet per second, then its height is a function of time, given by s(t) = -16t^2 + 54t. opens downward and thus has a maximum point. A ball is thrown vertically upward. Review: Review the definitions and first derivative test from previous lesson. for which the stated problem makes sense. The concavity of functions is discussed. mapped lines to other lines. Hey thanks for all your help but you kinda confused me on a few: 1)Determine whether f(x)=-5x^2-10x+6 has a maximum or minimum value and find that value A. MAT 111 - Pre-Calculus Chapter 6 – Trigonometric Functions 6 6. Constrained Optimization: Step by Step Most (if not all) economic decisions are the result of an optimization problem subject to one or a series of constraints: • Consumers make decisions on what to buy constrained by the fact that their choice must be affordable. And the absolute minimum point for the interval happens at the other endpoint. After the ball has hit the floor for the first time it rises 10. 5 applied maximum and minimum problems We have used derivatives to help find the maximums and minimums of some functions given by equations, but it is very unlikely that someone will simply hand you a function and ask you to find its extreme values. 2 show that a square has the maximum area inscribed in a circle. The following problems range in difficulty from average to challenging. We de ned a critical number of f(x) on domain D to be any number x = c on the domain D such that. New Calculus 5. Some of rst year calculus repeats senior school calculus: including \Word problems" which have always been di cult for students. Lesson 10: Max and Min Problems 3. (1 pt) A company that makes thing-a-ma-bobs has a start up cost of $47060. Each question is accompanied by a table containing the main learning objective(s), essential knowledge statement(s), and Mathematical Practices for AP Calculus that the question addresses. Such problems, involving the determination of the form of a curve having a certain maximum or minimum property, were quite different from the ordinary maximum and minimum problems of the differential calculus. How to Solve World Problems in Calculus reviews important concepts in calculus and provides solved problems and step-by-step solutions. K Worksheet by Kuta Software LLC. Optimization problems (also called maximum-minimum problems) occur in many fields and contexts in which it is necessary to find the maximum or minimum of a function to solve a problem. (solution by Calculus). What are the dimensions if the printed area is to be a maximum? 2) A cylindrical container with circular base is to hold 64 cubic centimeters. I plan on working through them in class. AP Calculus Sec 3. 265 Clove Road, New Rochelle, New York 10801 Pre-Calculus. Maximum Minimum. It also has its application to commercial problems, such as finding the least dimensions of a carton that is to contain a given volume. Login to reply the answers Post; Ray. So the function has a relative maximum at x=-5. So let's look. One example is the path of an airplane. 50$ , with 27,000+300(5) = 28,500 spectators and a revenue of \$ R(5) = 270,750. While they might not actually work out the quadratic function to come up with a precise number, managers at movie theaters,. Maximizing/Minimizing word problem: Minimum and Maximum Values. Area and perimeter worksheets. Maximum and Minimum of. To get started a sketch may help you get a sense of what the curve looks like. Displaying all worksheets related to - Maximum Minimum. Quadratic applications are very helpful in solving several types of word problems (other than the bouquet throwing problem), especially where optimization is involved. Read the problem carefully. To find the local maximum and minimum values of the. Next, the first- and second-derivative tests for local extrema are studied. What calculus is useful for is science, economics, engineering, industrial operations, finance, and. But we will not always be able to look at the graph. However once you know these 6 steps, then you should be able to solve any Calculus related rates problems you like. Nov 24, 2009 Equation Calculator - Quadratic Solver. superprof resources. of cardboard, what are the dimensions of the biggest box that can be made? 2. OPTIMATIZATION - MAXIMUM/MINIMUM PROBLEMS - BC CALCULUS. Maximum and Minimum Value Word Problems - Quadratic Equations This algebra video tutorial explains how to solve word problems that asks you to calculate the maximum value of a function or How to solve word problems with quadratic equations Geometry Teachers Never Spend Time Trying to Find. Limit of Sequence Problems. Rates of change: the derivative, velocity, and acceleration. More Lessons for Calculus Math Worksheets A series of free Calculus Video Lessons. Precalculus review and Calculus preview - Shows Precalculus math in the exact way you'll use it for Calculus - Also gives a preview to many Calculus concepts. Using calculus you can calculate its average cruising altitude, velocity and acceleration.