Integrals as Areas - Review. Integration to Find Areas and Volumes, Volumes of. Revolution

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1 Integrals as Areas - Review - Unit #3 : Revolution Goals: Integration to Find Areas and Volumes, Volumes of Be able to apply a slicing approach to construct integrals for areas and volumes. Be able to visualize surfaces generated by rotating functions around different axes. Integrals as Areas - Review Reading: Section 8. Example: Write the integral that represents the area underneath the graph y = x 2 between x = 0 and x = 2. Reading: Sections 8., 8.2 Evaluate the integral to find the area. Integrals as Areas - Review - 2 Illustrate how this shape can be constructed by accumulating small rectangles of varying heights. Integrals as Areas - Review - 3 Now show how the exact same area can be constructed by using horizontal rectangles. Write the analogous intervals, widths, etc. on the diagram.

2 Integrals as Areas - Review - 4 Write out first a sum, then a new integral, that would represent the exact area in the sketch. Evaluate the integral. Integrals as Sums of Slices Integrals as Sums of Slices - Most people visualize this approach as slicing the shape into thin pieces. To find the total area, the process is: decide along which axis you want to slice (say slices perpendicular to x) find the size of a generic slice, as a function of the position x write out the sum that represents to the total you want transform the sum into an integral evaluate the integral Integrals as Sums of Slices - 2 Integrals as Sums of Slices - 3 Example: Find the area between x = y 2 and y = x using horizontal slices

3 Pyramid Volume - Pyramid Volume - 2 Pyramid Volume A pyramid with its base being an equilateral triangle with sides 3 units long, is 5 units high. What is its volume? Helpful fact: the area of an equilateral 3 triangle with all sides length a is 4 a2 square units. Cone Volume - Cone Volume - 2 Cone Volume Example: Use a similar slicing strategy to find the volume of a cone of height h and bottom radius r.

4 Volumes of Revolution - Introduction - Volumes of Revolution - Introduction - 2 Volumes of Revolution - Introduction Reading: Section 8.2 We can extend the cone example to find the volume of more complex spun shapes. These are often called volumes of revolution. Example: Consider the graph of y = e x shown below, and the solid we would build if we spun this shape around the x axis, cutting it off at x = 0 and x = 3. Express the volume of a cut, x thick, in terms of the location of the cut, x Write down an integral that represents the total volume of the shape What is the shape of any cut made perpendicular to the x axis? Volumes of Revolution - Introduction - 3 Volumes of Revolution - Changing Rotation Axis - Evaluate the integral for the volume of the shape. Volumes of Revolution - Changing Rotation Axis Example: Now consider the shape we would get if we spun y = e x around the y axis. Suppose we cut the shape off between y = 0. and y = What is the shape of any cut made perpendicular to the x axis? Would this make a good way to cut up the shape? What is the shape of any cut made perpendicular to the y axis? Would this make a good way to cut up the shape?

5 Volumes of Revolution - Changing Rotation Axis - 2 Volumes of Revolution - Changing Rotation Axis - 3 Express the volume of a cut, y thick, in terms of the location of the cut, y. Evaluate the integral from the previous page Write down an integral that represents the total volume of the shape. Volumes of Revolution - Changing Rotation Axis - 4 Volumes of Revolution - Slices as Rings - Volumes of Revolution - Slices As Rings Example: The region bounded by y = x, y = 0 and x = 4 is rotated around the y-axis. Find the volume.

6 Volumes of Revolution - Slices as Rings - 2 Volumes of Revolution - Displaced Rotation Axis - Volumes of Revolution - Displaced Rotation Axis Example: The region bounded by y = x, y = 0 and x = 4 is rotated around the line y = 3. Find the volume. Volumes of Revolution - Displaced Rotation Axis - 2 Volumes of Revolution - Displaced Rotation Axis 2 - Volumes of Revolution - Displaced Rotation Axis 2 Example: The region bounded by y = x, y = 0 and x = 4 is rotated around the line x = 6. Find the volume.

7 Volumes of Revolution - Displaced Rotation Axis 2-2

8 Mass with Varying Density - Linear Example - Unit #4 : Goals: Center of Mass, Improper Integrals Apply the slicing integral approach to computing more complex totals calculations, including center of mass. Learn how to evaluate integrals involving infinite quantities. Reading: Sections 8.4 and 7.7 In earlier units we reviewed how to compute the total area or volume of complex shapes by taking slices of the shape. This week we will study more abstract applications of the same idea. Example: A metal rod has length two meters. At a distance of x meters from its left end, the density of the rod is given by δ(x) = x kg per meter. Why can t we simply compute the total mass using mass (kg) = [lineal density (kg/m)] [ length (m)] Mass with Varying Density - Linear Example - 2 Breaking the rod down into small segments x long, construct a Riemann sum for the total mass of the metal rod. Mass with Varying Density - Linear Example - 3 Find the exact mass of the rod by converting the Riemann sum to an integral and evaluating it.

9 Mass with Varying Density - Linear Example - 4 Can you verify that the answer you obtained is reasonable with a quick alternative calculation? Do so. Mass with Varying Density - Circular Example - Example: The density of oil in a circular oil slick on the surface of the ocean, at a distance of r meters from the center of the slick, is given by 00 r2 δ(r) = kg/m 2. We want to find the total mass of the oil in the slick. 0 This problem is similar to Example 4 on page 47 in the textbook. Consider a ring of radius r and thickness dr concentric with the oil slick. Write down the expression for the mass of the oil contained within that ring. Mass with Varying Density - Circular Example - 2 Write down and evaluate the integral that represents the total mass of the oil slick. (Think about what outer radius we should use.) Center of Mass Center of Mass - A property of an object related to density and mass is the center of mass. It can be thought of as the balance point of the object or system. In general, when dealing with a set of point masses m i at locations x i, their center of mass is given by xi m i x = mi Example: A mother and daughter sit on opposite ends of a 3 m long seesaw; the mother has a mass of 60 kg, while her daughter weighs 20 kg. How close to the mother s end would the support need to be put so that the two of them would be in perfect balance?

10 Center of Mass - 2 Unlike the last question, many problems in physics and chemistry involve mass spread out in a more continuous way. Extend the formula for center of mass using a slicing approach to find the center of mass for the metal rod we studied earlier: two meters long, with density δ(x) = x kg/m. Center of Mass - Example - Example: Find the center of mass for a triangular piece of sheet metal shown with dimensions shown below. Improper Integrals Improper Integrals - Introduction - A common type of integral that appears naturally in many sciences and economics is an integral with an infinite limit. This can force us to face some slippery ideas about infinity... Example: A chemical reaction produces a desired chemical at a rate of r(t) = e t g/s. What amount is produced between t = 0 and some unspecified later time, t = T? Improper Integrals - Introduction - 2 How would you answer the question What is the total amount of chemical that this reaction could produce, if it were run forever? We can write the answer to this question as an improper integral.

11 Improper Integrals - Introduction - 3 Improper Integrals Improper integrals are integrals which somehow involve an infinite quantity. Both are computed using limits. Infinite limit of integration: Infinity in the integrand: f(t) dt = lim f(t) dt T } a{{} usual form exact definition } a {{} if f(a) is undefined, b a T f(t) dt = lim T a b T f(t) dt If we can evaluate the limit and obtain a finite number, we say the integral converges. If the limit does not exist, we say the integral diverges. Example: Use the definition of improper integrals to evaluate Improper Integrals - Examples - t 2 dt Improper Integrals - Examples - 2 Improper Integrals - Examples - 3 Example: Evaluate t dt Sketch both graphs and indicate what area is being evaluated in each case. How much does graph sketching help in analyzing these types of integrals?

12 Families of functions Show that all the integrals e kt dt converge for k < 0. Improper Integrals - Common Families - Show that all the integrals dt converge for k >. tk Improper Integrals - Common Families - 2 Reasonableness of Infinite Integrals Reasonableness of Infinite Integrals - Reasonableness of Infinite Integrals - 2 Can we ever have an infinitely long object that only has finite area? Students often approach infinite integrals initially from one of two stances: (b) If the function we re integrating is going to zero, how (a) If we integrate out to infinity, how can the area ever be finite? can the area ever be infinite? Both of these ideas reflect the struggle about the role of infinity and zero in the integral.

13 Reasonableness of Infinite Integrals - 3 Can we ever have a function that goes to zero, but has infinite area underneath it? Common Family Concept Check Common Family Concept Check - From our earlier analysis of common families of functions, we can say that for an integral up to infinity to converge, the integrand must to go zero and it must go to zero faster than t. Question:. Yes 2. No Does e 0.0t dt converge / is the area it represents finite? Common Family Concept Check - 2 Common Family Concept Check - 3 Question:. Yes Does dt converge / is the area it represents finite? t2 Question:. Yes Does dt converge / is the area it represents finite? t 2. No 2. No

14 Infinite Integrands - Infinite Integrands - 2 Infinite Integrands Sometimes, an integral involves an implicit infinity e.g. when the integrand (function) goes to infinity at the edge of our interval. Fortunately, we can use a similar limit approach to evaluate this case as well. Again, these integrals may end up being infinite or not, depending on how quickly the function approaches infinity. Example: Evaluate the integral 0 dx x Infinite Integrands - 3 Escape Velocity as an Improper Integral - Example: Evaluate the integral 0 x 2 dx Applications of Improper Integrals: Universal Law of Gravitation The Universal Law of Gravitation gives the force of attraction between two masses m and m 2 (in kilograms) which are a distance of r meters apart by the formula F = Gm m 2 r 2, where G = is the gravitational constant and F is measured in newtons. This formula is especially interesting when m 2 is the mass of the Earth and m is the mass of an object in its gravitational field. For objects close to the Earth s surface, r r 0 = meters, the radius of the Earth.

15 Escape Velocity as an Improper Integral - 2 When an object of mass m is lifted away from the Earth s surface by a small distance d meters (say a math prof going up two stories in an elevator), then the amount of work done by the elevator is equal to force times distance; that is, Escape Velocity as an Improper Integral - 3 Write down the integral that represents the work required to move a mass of m kg from the surface of the earth, r 0 from the center, to a point completely out of the reach of Earth s gravity. Work = F d = Gm m 2 r0 2 d, with the result measured in joules (J). On the other hand, to put a satellite in an orbit at a height r meters from the Earth s center, we have to account for the fact that force decreases as we move away from the Earth. This means that the total work done will be represented by an integral rather than simply the product F (r r 0 ). Escape Velocity as an Improper Integral - 4 Evaluate the integral you found, leaving the constants in as letters. Escape Velocity as an Improper Integral - 5 Sub in the following values based on lifting an object from the surface of the earth: m = 70 kg, the rough mass of a person m 2 = kg as the mass of the earth G = N m 2 /kg 2 is the universal gravitation constant r 0 = m is the distance from the center to the Earth s surface

16 Escape Velocity as an Improper Integral - 6 To escape the Earth s pull without further assistance, a rocket must be moving fast enough so that its kinetic energy while moving at the Earth s surface is equal to the amount of energy we just found. If kinetic energy of an object moving at velocity v is given by E = 2 mv2, how quickly must an object at the surface of the earth be moving escape the Earth s gravitational pull completely? (This speed is know as the escape velocity for the Earth.)

17 Differential Equations Intro - Unit #5 : Goals: Differential Equations To introduce the concept of a differential equation. Discuss the relationship between differential equations and slope fields. Discuss Euler s method for solving a differential equation numerically. Discuss the method of separation of variables to solve a differential equation exactly. Reading: Sections 6.3,..4. Differential Equations Reading: Section 6.3 and. A differential equation (DE) is an equation involving the derivative(s) of an unknown function. Many of the laws of nature are easily expressed as differential equations. For example, here is one way to define the exponential function: dy dt = y Write this mathematical formula as a sentence, and then find a solution to the equation. Differential Equations Intro - 2 Differential Equations Intro - 3 How can the solution you found be altered and still satisfy the DE dy dt = y? Making a further alteration to the function y = e t, find a family of functions all of which satisfy the DE dy dt = ky.

18 Differential Equations Intro - 4 The differential equation dy = ky, when expressed as an English sentence, says dt that the rate at which y changes is proportional to the magnitude of y. If k > 0, this is one way of characterizing exponential growth. If k < 0, the rate of change becomes negative and we are dealing with exponential decay. Second-Order Differential Equations - If a DE involves the second derivative of a function, it is called a second order differential equation. Try to think of two functions that satisfy the differential equation d 2 y dt 2 = y. Then, try to combine these two functions to get even more solutions for this DE. Sources of Differential Equations - Sources of Differential Equations We study differential equations primarily because many natural laws and theories are best expressed in this format. Translate the following sentence into an equation: The rate at which the potato cools off is proportional to the difference between the temperature of the potato and the temperature of the air around the potato. Translate the following sentence into an equation: Sources of Differential Equations - 2 The rate at which a rumour spreads is proportional to the product of the people who have heard it and those who have not.

19 Translate the following sentence into an equation: Sources of Differential Equations - 3 The rate at which water is leaking from the tank is proportional to the square root of the volume of water in the tank. Translate the following sentence into an equation: Sources of Differential Equations - 4 As the meteorite plummets toward the Earth, its acceleration is inversely proportional to the square of its distance from the centre of the Earth. The previous examples indicate how easily differential equations can be constructed. Unfortunately, starting with those equations, we have a lot of work to do before we can predict will happen given the equation. Slope Fields Reading: Section.2 in the textbook. Consider the differential equation dy dx = cos x. Recall how we would use this derivative information to sketch y: Slope Fields - dy values give the slopes of the graph of y dx Said another way, we are looking for a function y(x) which has, at each point, a slope given by cos(x). Give the most general function y that satisfies dy dx = cos(x). Slope Fields - 2 We are now going to introduce an alternate way to get to this solution through graphical techniques. These are an extension of our slope interpretation. Example: Below is a slope field graph for the DE dy = cos(x). How was dx it constructed? Sketch the functions y found earlier on the slope field.

20 Slope Fields - 3 Slope fields are especially useful when we study more general differential equations, which can be written in the form d y = f(x, y). dx Examples are dy dx = x y, dy dx = xy, and dy dx = y x. Note: These forms cannot be directly integrated to find y(x). Try to solve for y given dy dx = (x y) 2 Slope Fields - 4 Sketch the slope field for the differential equation dy dx = (x y), and sketch 2 two solutions This sketch gives an idea of the form of y which satisfies dy dx = (x y), without 2 needing to solve for y as a function. Slope Fields - Logistic Growth - Example: (Logistic Growth) The growth of a population is often modeled by the logistic differential equation. For example, if bacteria are grown on a petri dish which really cannot support a bacterial culture larger than L, then a useful differential equation model for the population is dp dt = k P (L P ), where P (t) is the size of the culture at time t. For what values of P is the function k P (L P ) zero? largest? smallest? Sketch the slope field associated with the differential equation dp dt = k P (L P ). Slope Fields - Logistic Growth - 2 On the slope field, draw several solutions using different initial conditions.

21 Euler s Method - Euler s Method - 2 Euler s Method Reading: Section.3 in the textbook. We can extend the idea of a slope field (a visual technique) to Euler s method (a numerical technique). Euler s method can be used to produce approximations of the curve y(x) that satisfy a particular differential equation. Here is the idea: Knowing where you are in x and y, you look at the slope field at your location, set off in that direction for a small distance, look again and adjust your direction, set off in that direction for a small distance, etc. Algorithmically, Start at a point (x i, y i ) Compute the slope there, using the DE dy dx = f(x i, y i ) Follow the slope for a step of x: x i+ = x i + x y i+ = y i + dy dx x }{{} y Euler s Method - 3 Follow this procedure for the differential equation dy = x + y with initial condition y(0) = 0.. Use x = 0.. dx x y slope y = slope x (0.)(0.) = Euler s Method - 4 Here is a picture of the slope field for dy = x + y. On this slope field, sketch dx what you have done in creating the table of values. From the picture, would you say the values for y(x) in your table are over-estimates or under-estimates of the real y values?

22 Separation of Variables - Separation of Variables - 2 Separation of Variables Reading: Section.4 in the textbook. We have now considered both visual and approximate techniques for solving differential equations, which can be obtained with no calculus. The problem with those approaches is that they do not result in formulas for the function y that we want to identify. We (at last!) proceed to calculus-based techniques for finding a formula for y. Consider the differential equation dy dx = k y. Treating dx and dy as separable units, transform the equation so that only terms with y are on the left, and only terms with x are on the right. Evaluate the integrals. Solve for y. Place an integral sign in front of each side. The solution gives a family of functions, one for each value of the integration constant. k is also a parameter, of course, but it is presumed to be specified in the differential equation. Separation of Variables - 3 As soon as we are given an initial value, say y(0) = 0, the solution becomes unique. The solution with initial value y(0) = 0 is given by Separation of Variables - 4 Use the method of separation of variables to solve the differential equation dr = 2R + 3, dx and find the particular solution for which R(0) = 0. If y 0 > 0, this function describes exponential growth (k > 0) or decay (k < 0).

23 Classifying Differential Equations - Classifying Differential Equations - 2 Classifying Differential Equations For any differential equation which is separable, we can at least attempt to find a solution using anti-derivatives. For equations which are not separable, we ll need other techniques. It is important, as a result, to be able to tell the difference! Indicate which of the following differential equations are separable. For those which are separable, set up the appropriate integrals to start solving for y. dy dx = x2 dy dx = x + y dy dx = cos(x) cos(y) dy dx = ey x dy dx Classifying Differential Equations - 3 = cos(xy) A. Separable B. Not separable dy dx = ex + e y A. Separable B. Not separable Classifying Differential Equations - 4 Note: all the original anti-derivatives we studied in first term are of the form dy dx = f(x) and so y = F (x) = f(x) dx. These are all immediately separable. The challenge is that most interesting scientific laws expressed in differential equation form aren t that easy to work with. dy dx = e(x+y) A. Separable B. Not separable

24 Differential Equation Modelling - Unit #6 : Goals: Differential Equations To develop skills needed to find the appropriate differential equations to use as mathematical models. Reading: Sections In working with the models in Sections.5.7, it should not be your goal to memorize particular differential equations and their solutions as if they were the correct formulas for the problems under discussion. Instead, you should concentrate on the process that leads to the differential equations. You should try to become comfortable with the various parts of this process, which involves translating relationships derived from common sense or elementary science, expressed in ordinary English, into mathematical equations, judgments (regarding which effects are insignificant enough to disregard or which factors vary little enough to be regarded as constant), and recognizing special cases that are amenable to solution. DE Models of Growth and Decay Differential Equation Modelling - 2 Reading: Section.5 in the textbook. We have already seen that exponential growth and decay have simple formulations using differential equations. A summary is shown in the box below. Differential Equation for Exponential Growth/Decay Every solution to the equation dp dt = k P can be written in the form P (t) = P 0 e kt where P 0 is the initial value of P ; Equations in which k > 0 represent growing systems and with k < 0 represents decaying ones. Newton s Law of Heating and Cooling - Example (Newton s Law of Heating and Cooling) Newton s Law of Cooling asserts that the rate at which an object cools off (or heats up) is proportional to the difference between the temperature of the object and that of its environment. where H is the temperature of the object, dh dt = α (H T env) Tenv is the temperature of the environment, and α is a property of the object, involving such things as the ability of the surface of the object to conduct heat. Explain why α must be positive. We now proceed to study some related modeling problems using differential equations.

25 Newton s Law of Heating and Cooling - 2 Example: If we put a potato at room temperature (20 C) in an oven at 200 C, then Tenv = 200. Suppose that α = 0.25 and t is measured in minutes. Write the differential equation describing the rate of temperature change of the potato. Newton s Law of Heating and Cooling - 3 Solve this differential equation to get an equation for the temperature of the potato, and use the equation to predict the potato s temperature after 0 minutes. Newton s Law of Heating and Cooling - 4 Newton s Law of Heating and Cooling - 5 Sketch a slope field for the DE dh = 0.25 (H 200) dt and draw in some solutions, including one for which the initial condition is greater than 200 C H (deg. C) t (mins)

26 Equilibrium Solutions - Equilibrium Solutions It is important to realize that although solving differential equations can be hard (because solving implicitly involves integration), we can learn important characteristics about our model directly from the differential equation itself. One important feature of many models are the equilibrium solution(s). For the potato problem, the DE we found was dh = 0.25 (H 200). By considering only this equation, find the equilibrium temperature(s) of the dt potato. Equilibrium Solutions - 2 For water leaking out of a cylindrical reservoir under the force of gravity, we can use conservation of energy arguments to derive the differential equation dh for the depth of the water, h: dt = k h. Find the equilibrium level of water. Applications and Modeling Reading:.6 and.7 in the textbook. Modelling - Ice Formation - As the textbook indicates, a mathematical model is sometimes obtained by finding a function that fits certain experimental measurements, or by a theoretical approach that leads directly to a certain formula, or by a combination of the two approaches. The models in the book are chosen to illustrate this idea. Example: Thickness of Ice The surface of water freezes first, through contact with cold air. As heat from the water travels up through the ice and is lost to the air, more ice is formed. We want a function describing the thickness of ice as a function of time. Let y(t) = the thickness at time t, measured from the time at which the ice begins to form. The rate at which ice forms is proportional to the rate at which heat escapes from the water to the air. The rate at which heat escapes from the water (through the ice) to the air is inversely proportional to the thickness of the ice. Modelling - Ice Formation - 2 Translate the rate information into a differential equation, and an initial condition.

27 Solve this differential equation, subject to the initial condition. Modelling - Ice Formation - 3 Modelling - Ice Formation - 4 If it takes two hours to form /2 cm of ice, then how long will it take to form 2 cm? Modelling - Drug Dosing - Example: Drug Dosing Consider a patient who is being given morphine at a rate of 2.5 mg/hr. Their body metabolizes the drug at a rate of 30% per hour. Write a differential equation for the amount of morphine in the patient over time. Find the general solution to the differential equation. Modelling - Drug Dosing - 2

28 Modelling - Drug Dosing - 3 In the long run, how much morphine will the patient have in their body? Modelling - Drug Dosing - 4 Based on the earlier analysis, sketch several solutions to morphine DE on the axes below. Modelling - Terminal Velocity - Example: Terminal Velocity In physics, we note that an object near the earth will accelerate at 9.8 m/s 2. This would imply that, if we drop an object from high enough, it will accelerate until it reaches any velocity we want. For example, the main visitor s gallery of the CN tower is 346 m above the ground. Using just the force of gravity, an object dropped from this height should take about 8 seconds to reach the ground, and have the object falling at 78 m/s or 280 km/h, or /4 the speed of sound. If you dropped an object out of a plane at 2,000 m, it would take 50 seconds to reach the ground, and would be falling at 580 m/s or almost twice the speed of sound. Why don t we observe objects falling this fast in practice? Draw a diagram indicating the forces on a falling object. Modelling - Terminal Velocity - 2

29 Modelling - Terminal Velocity - 3 Assuming that the force exerted by air resistance is proportional to velocity, write a differential equation for the velocity of a falling object. Modelling - Terminal Velocity - 4 Explain what happens to an object that is traveling slower than the equilibrium velocity. Find the equilibrium velocity based on the differential equation. Explain what happens to an object that is traveling faster than the equilibrium velocity. Solve the differential equation to obtain an expression for v(t). Modelling - Terminal Velocity - 5 Sketch the shape of the possible graphs of v(t) that satisfy the DE. Modelling - Terminal Velocity - 6

30 Modelling - Logistic Growth - Example: Logistic Growth The logistic model is a model for population growth that takes into account the fact that population growth slows down when the environment occupied by the population is saturated. In many cases, the environment has a certain carrying capacity L which represents a degree of overpopulation that would prevent further growth. In the logistic model, it is assumed that the rate of growth of the population P (t) is proportional to the product of P (t) and L P (t). Write down the differential equation that expresses this relationship. Modelling - Logistic Growth - 2 Recall that when we studied this differential equation in an earlier unit, we found that the growth rate of the population is largest when the size of the population is halfway between 0 and L. When the population size, P, is close to 0 or close to L, we found the rate of growth to be very small. When the population size, P, is greater than L, the rate of growth of P becomes negative. Find the equilibrium levels of the population. Modelling - Logistic Growth - 3 Show that the population will always increase if the initial population is in the range 0 < P < L. Modelling - Compartmental Analysis - Compartmental Analysis An aquarium holding 25 m 3 of water at a zoo is kept clean using a large pipe system equipped with filters. Water is pumped into and out of the aquarium at constant rate of 0 m 3 per day. In the pipe flowing into the aquarium, a filter has broken down, so ammonia (toxic to fish) is flowing into the tank with the water, at a concentration of 50 g / m 3. The water in the aquarium starts out at time t = 0 free of ammonia. Find the rate of ammonia entering the aquarium in grams/day. Show that the population will always decrease if the initial population is above L.

31 Modelling - Compartmental Analysis - 2 If the water in the aquarium is well mixed (same concentration of ammonia everywhere within the tank), and the water is leaving the tank at 0 m 3 /day, find an expression for the rate that ammonia is leaving the tank, given that the amount of ammonia in the tank at time t is S(t) grams. Modelling - Compartmental Analysis - 3 Use your two rates to find an expression for the net rate of change of the ammonia mass, ds dt. Convert your differential equation to show the rate of change of the concentration of ammonia, dc dt. Modelling - Compartmental Analysis - 4 Use the differential equation to find the equilibrium concentration of ammonia in the aquarium. Explain why the equilibrium level makes sense for the situation. Modelling - Compartmental Analysis - 5 Find C(t), the solution to the differential equation, given that the initial ammonia concentration is zero.

32 Modelling - Compartmental Analysis - 6 dc dt = 60 C 2.5 Modelling - Compartmental Analysis - 7 A concentration of 0. mg/l (or 00 g/m 3 ) is considered dangerous for the fish in the tank. How long after the filter breakdown do the fish have for the filter problem to be identified and fixed before the ammonia levels become dangerous?

33 Functions of Two Variables - Unit #7 : Goals: Functions of Two Variables To introduce functions of two variables. To learn what is meant by the graph of a function of two variables. To learn to visualize and even sometimes sketch the graph of a function of two variables. Functions of Two Variables Reading: Section 2.. Example: (Temperature in Ontario) On a particular summer day, the daytime high temperature varied from location to location in Southern Ontario according to the following table: Reading: Sections 2. and 2.2 Functions of Two Variables - 2 longitude in west latitude in north Functions of Two Variables - 3 If you wanted to find out the temperature at a point, what information you would need to specify? In this example, temperature is a function of two variables. We can write it as T (x, y).

34 Functions of Two Variables - 4 Functions of Two Variables - 5 Generally speaking, we define a real-valued function of two variables as follows: A function of two variables is a rule that takes certain pairs of numbers as inputs and assigns to each a definite output number. The set of all input pairs is called the domain of the function and the set of resulting output numbers is called the range of the function. Inputs are pairs of real numbers = Can be represented as points in the xy-plane = Domain can be pictured as a region in the xy-plane Output is a single real number = Can be represented as an interval of the real line= Range can be pictured as an interval of IR Returning to the Temperature in Ontario example, what is the range of T (x, y)? Functions of Two Variables - 6 When you travel from Toronto, at approximate latitude 43.0 and longitude 79.5, to North Bay, at approximate latitude 46.5 and longitude 79.5, the trip is almost exactly due North. latitude in north longitude in west Construct a graph of these temperatures, indicating the high temperature along that route on that July day. Functions of Two Variables - 7 This example shows that when you have a function of two variables, you can use it to construct a function of a single variable by keeping one of the original variables constant. Describe (in words) another way to construct a function of a single variable from T (x, y). (There are several possible answers.)

35 Multivariate Functions - The Ideal Gas Law - Multivariate Functions - The Ideal Gas Law - 2 Example: (Gas Law) A well-known formula in chemistry relates the pressure, temperature, and volume of an ideal gas as P V = nrt, where n and R are constant for a given amount of the gas. Thus, P = P (V, T ) = nrt V If we keep the temperature T constant, say at 300 K, what is the interpretation of the resulting function P (V, 300)? Graph the function P (V, 300). Multivariate Functions - The Ideal Gas Law - 3 Multivariate Functions - The Ideal Gas Law - 4 Sometimes in discussing functions of several variables, we can isolate the general effect one of the input variables has on the output. When we make these statements, we assume that all the other variables are fixed. Question: Is P = nrt an increasing function of T? I.e. if we increase T, holding V the other values constant, does P necessarily increase as a result? Remember that n, R, T and V are all positive values. A. Yes. B. No. C. Depends. Question: How are P = nrt and V related? V A. P is an increasing function of V. B. P is a decreasing function of V. C. Neither consistently/depends on other factors. Now consider the function z = xy 2. Question: How is z related to x? A. z is an increasing function of x. B. z is a decreasing function of x. C. Neither consistently/depends on other factors. Question: How is z related to y? A. z is an increasing function of y. B. z is a decreasing function of y. C. Neither consistently/depends on other factors.

36 Multivariate Functions - Finance - Example: (Finance) The monthly payments, P dollars per month, on a mortgage in which A dollars were borrowed, at an annual interest rate of r%, to be repaid over an amortization period of t years, is given by P = f(a, r, t). Is P an increasing function of A? of r? of t? Interpret the statement f(250000, 4, 0) Give units. Multivariate Functions - Finance - 2 Picturing f(x, y) : Graphs Reading: Sections 2. and 2.2 Graphs as Surfaces - A function of a single variable is usually visualized by means of its graphs. Recall that for a function of one variable, the graph is defined to be the set Real-valued functions of two variables also have graphs. dimensions for the two input variables, and for the output variable. Graphs as Surfaces - 2 As a result, graphs for functions of two variables will be drawn in how many dimensions? { (x, y) y = f(x) } Such a graph can be drawn on a plane, or a space referred to as IR 2. We need two dimensions to draw the graph of a function of one variable: one dimension for the input variable x, and one for the output variable y. The graph of g(x, y) is defined as the set of points { (x, y, z) z = g(x, y) } We refer to the 3-dimensional space as IR 3 since three copies of the real line, IR, are used as axes.

37 Graphs as Surfaces - 3 A typical graph for a function of two variables is a surface in IR 3 and it looks like this: Graphs as Surfaces - 4 To draw this graph, you begin with the x-, y-, and z-axes. Each point on the graph has coordinates (x, y, f(x, y)). Points on the graph lie above the xy-plane if z = f(x, y) is positive and below if z = f(x, y) is negative. Example: Describe and then sketch the surface represented by f(x, y) = 5. Graphs as Surfaces - 5 Graphs as Surfaces - 6 Example: Describe and then sketch the surface represented by f(x, y) = 2. Example: Describe the graph of f(x, y) = { if y > 0 if y 0.

38 Graphs as Surfaces - 7 As we have just seen, sketching graphs of 2 variables can be very difficult to do by hand. It is much more practical to use computer software to generate plots of such functions. Since that is not a tool you all have access to, we are going to focus on looking for patterns in such graphs, so we can predict structure without necessarily being able to draw the graphs by hand. Graphs of Multivariate Functions - Strategies and Examples - Graphs of Functions of 2 Variables To describe or identify graphs of 2 variables, z = f(x, y), some techniques that will help are the following. Determine if z is always positive or negative. When z does take on different signs, find the regions where it is consistently the same sign. Substitute x = 0 or y = 0 into the function. You will get a -variable function representing the cross-section of f(x, y) along one of the axes. Substitute in other values of x or y to get a different cross-section. Experiment with large values of x and y. Does the function approach the same limit everywhere, or does it depend on the values of x and y? Useful patterns in f(x, y) If f(x, y) is linear in x and y, the function represents a plane. If x 2 +y 2 always appear together, the graph will be circularly symmetric around the origin. If only x or y is present in f(x, y), the graph will be a simple -variable function stretched along the missing variable s axis. Graphs of Multivariate Functions - Strategies and Examples - 2 Use the points above to identify the characteristics that tie the following functions with their matching graphs. If possible, indicate clearly the x and y axes. f(x, y) = e (x2 +y 2 ) Graphs of Multivariate Functions - Strategies and Examples - 3 f(x, y) = x 2 + y 2

39 Graphs of Multivariate Functions - Strategies and Examples - 4 Determine what the graph of f(x, y) = x 2 + y 2 looks like based on cross-sections through x = 0 and y = 0. f(x, y) = x y Graphs of Multivariate Functions - Strategies and Examples - 5 f(x, y) = sin x Graphs of Multivariate Functions - Strategies and Examples - 6 Graphs of Multivariate Functions - Example - Example: Suppose we wanted to consider the graph of the function f(x, y) = y 2 x 2 in detail. We would ask some of the following questions. Where is f(x, y) positive? Where is it negative? Where is it zero? To get a more accurate picture of the graph of f(x, y), we can study it by cutting it with planes. We do this by selecting a sample value of either x, y or z, and graphing the resulting -variable function.

40 Graphs of Multivariate Functions - Example - 2 Fix the value of x to a constant. What will the resulting z vs. y graphs look like? Graphs of Multivariate Functions - Example - 3 Fix the value of y to a constant. What will the resulting z vs. x graphs look like? Graphs of Multivariate Functions - Example - 4 Fix the value of z to a constant. What will the resulting y vs. x graphs look like? z = y 2 x 2 Graphs of Multivariate Functions - Example - 5

41 Question: Which of the following graphs represents the function z = 2 x 2 y 2 Identifying Graphs - Practice - Question: Which of the following graphs represents the function z = x 2 + y 2 Identifying Graphs - Practice - 2 Graphs of Multivariate Functions - Further Examples - Even when functions are more complicated, we can still perform some mental experiments that allow us to get some sense of their shape without a full sketch or computer-generated surface. Consider the surface defined by z = cos(x) + y 2. What happens to the function when you set x = 0? Graphs of Multivariate Functions - Further Examples - 2 What does this tell you about the intersection between the surface and the plane x = 0? If you cut the surface at the planes x = π 4 and x = π 2, how is the resulting function different than at x = 0?

42 Graphs of Multivariate Functions - Further Examples - 3 Graphs of Multivariate Functions - Further Examples - 4 Describe the shape of the surface. Indicate the x and y axes on the graph below. Relations in Several Variables - Relations in Several Variables - 2 Relations in several variables The line, plane, and 3D space By analogy, what shape in IR 3 is described by the relation x 2 + y 2 + z 2 = 25? Points on the real line, x, belong to IR Points on the plane, (x, y), belong to IR 2 Points in 3D space, (x, y, z), belong to IR 3 Some shapes in IR 2 and IR 3 are best described by relations, rather than functions. For example: In IR 2, what shape is described by the relation x 2 + y 2 = 25? Give a formula for the distance from any point (x, y, z) to the origin. In IR 2, what shape is described by the relation (x 2) 2 + (y + ) 2 = 25?

43 Relations in Several Variables - 3 What is the formula for a sphere of radius 5 centered at (2, 4, )? Give a formula for the distance between two arbitrary points in IR 3, (a, b, c) and (x, y, z).

44 2 4 Graphs of Surfaces and Contour Diagrams - Unit #8 : Goals: Level Curves, Partial Derivatives To learn how to use and interpret contour diagrams as a way of visualizing functions of two variables. To study linear functions of two variables. To introduce the partial derivative. Picturing f(x, y): Contour Diagrams (Level Curves) Reading: Section 2.3 We saw earlier how to sketch surfaces in three dimensions. However, this is not always easy to do, or to interpret. A contour diagram is a second option for picturing a function of two variables. Suppose a function h(x, y) gives the height above sea level at the point (x, y) on a map. Then, the graph of h would resemble the actual landscape. Reading: Sections 2.3, 2.4, 4. and 4.2. Graphs of Surfaces and Contour Diagrams - 2 Graphs of Surfaces and Contour Diagrams - 3 Suppose the function h looks like this: Together they usually constitute a curve or a set of curves called the contour or level curve for that value. In principle, there is a contour through every point. In practice, just a few of them are shown. The following is the contour diagram for the earlier surface Then, the contour diagram of the function h is a picture in the (x, y) plane showing the contours, or level curves, connecting all the points at which h has the same value. Thus, the equation h(x, y) = 00 gives all the points where the function value is Indicate the location of the peaks and pits/valleys on the contour diagram.

45 Graphs of Surfaces and Contour Diagrams - 4 Graphs of Surfaces and Contour Diagrams - 5 Topographic maps are also contour maps. In principle, the contour diagram and the graph can each be reconstructed from the other. Here is a picture illustrating this: Identify first a steep path, and then a more flat path, from the town up to Signal hill. As shown above, the contour f(x, y) = k is obtained by intersecting the graph of f with the horizontal plane, z = k, and then dropping (or raising) the resulting curve to the (x, y) plane. The graph is obtained by raising (or dropping) the contour f(x, y) = k to the level z = k. Interpreting Contour Diagrams - Interpreting Contour Diagrams - 2 Interpreting Contour Diagrams Match each of the following functions to their corresponding contour diagram. () h(x, y) is the degree of pleasure you get from a cup of coffee when - x is the temperature, and - y is the amount of ground coffee used to brew it. (2) f(x, y) is the number of TV sets sold when - x is the price per TV set, and - y is the amount of money spent weekly on advertising. (3) g(x, y) is the amount of gas per week sold by a gas station when - x is the amount spent on bonus gifts to customers, and - y is the price charged by a nearby competitor.

46 Interpreting Contour Diagrams Useful Properties of Contour Plots If the contour lines are evenly spaced in their z values, Interpreting Contour Plots - Examples - contour lines closer together indicate more rapid change/steeper slopes contour lines further apart indicate flatter regions peaks and valleys look the same; only the values of the contours let you distinguish them Interpreting Contour Plots - Examples - 2 Example Draw the contours of f(x, y) = (x + y) 2 for the values, 2, 3, and 4. Give a verbal description of the surface defined by f(x, y). Interpreting Contour Plots - Examples - 3

47 Interpreting Contour Plots - Examples - 4 Draw the contours of f(x, y) = x 2 + y 2 for the values, 2, 3, and 4. Interpreting Contour Plots - Examples - 5 From the contour diagram, is the value of f at (0, 0) a local minimum or maximum? Is the surface becoming more or less steep as you move away from the origin? Try to associate how the lines on the contour diagram could help you to imagine the actual surface: [See also H-H, Section 2.5, Examples -3.] Linear Functions of Two Variables Linear Functions of Two Variables - A function of two variables is linear if its formula has the form f(x, y) = c+mx+ ny. The textbook shows that m and n can be interpreted as slopes in the x- direction and the y-direction, respectively, and that c is the z-intercept. [H-H beginning of Section 2.4. For similar examples to the following see Examples 2 and 3 in Section 2.4.] Consider the plane z = 2 x y. This plane has slope - in both the x- and the y-directions. Is there any direction in which it has a steeper slope? It may help to experiment by holding up a book or other flat object. Linear Functions of Two Variables - 2 Linear Example Given that the following is a table of values for a linear function, f, complete the table. y x Give a formula for the function f(x, y) in the preceding example.

48 Linear Functions of Two Variables - 3 Partial Derivatives - Sketch contours of the linear function f. Based on this example, hypothesize properties of the contour diagrams for all linear functions. Partial Derivatives Reading: Sections 4. and 4.2 Just as df is the rate of change of f(x) when x is changed, so the derivatives of dx f(x, y) are the rates of change of the function value when one of the variables is changed. Since there are two variables to choose from, there are two derivatives, one to describe what happens when you change x only and one to describe what happens when you change y only. Because either one by itself describes the behaviour of the function only partly, they are called partial derivatives. Partial Derivatives - 2 If we look at a graph of z = f(x, y), partial derivatives tell us how the height of the graph, z, is changing as the point (x, y, 0) moves along a line parallel to the x-axis or parallel to the y-axis. (x 0, y 0, 0) (x 0 + h, y 0, 0) (along solid line parallel to the x-axis). (x 0, y 0, 0) (x 0, y 0 + h, 0) (along dotted line parallel to the y-axis). Definition of Partial Derivatives f x (x f(x 0 + h, y 0 ) f(x 0, y 0 ) 0, y 0 ) = lim h 0 h = f x (x 0, y 0 ) f y (x f(x 0, y 0 + h) f(x 0, y 0 ) 0, y 0 ) = lim h 0 h = f y (x 0, y 0 ) Partial Derivatives - 3

49 Partial Derivatives - 4 Partial Derivatives - 5 To actually calculate f when we have a formula for f(x, y), we imagine that y x is fixed, then we have a function of only one variable and we take its derivative in the usual way. To calculate f, we imagine that x is fixed and y is not. y Example: Consider f(x, y) = x 2 y + sin(xy). Write a new single-variable function: g(x) = f(x, y 0 ) Find dg dx. Partial Derivatives - 6 Partial Derivative Practice - More practically, when we are looking for f (x, y) a formula in terms of x and x y we do not actually replace y by y 0, but simply think of it as a constant. f(x, y) = x 2 y + sin(xy) Partial Derivative Practice Find both partial derivatives for the following functions. f(x, y) = ( + x 3 )y 2 Write f (x, y). x Find f (x, y). y

50 Partial Derivative Practice - 2 Partial Derivative Practice - 3 f(x, y) = e x sin(y) f(x, y) = x2 4y ( Question: x 2 tan(y) + y 2 + x ) is y (a) 2x sec 2 (y) + 2y + (b) x 2 sec 2 (y) + 2y (c) 2x tan(y) + (d) x 2 tan(y) + y 2 + Partial Derivative Practice - 4 Partial Derivative Practice - 5 If f(x, y) = x 2 sin y + e xy2, find f f (, 0) and (, 0). x y What do these values tell you about the graph of f(x, y) near (, 0)?

51 Partial Derivatives - Economics Example: What are the signs of g g (x, y) and (x, y) if x y x is its price, and y is the price charged by a nearby competitor? g(x, y) is the amount of gas sold per week by a gas station, Partial Derivatives - Economics - Example: if Partial Derivatives - Economics - 2 What would you expect the signs of h h (x, y) and (x, y) to be x y x is the number of ski boots sold, and y is the number of tickets from Canada to warmer vacation spots. h(x, y) is the number of pairs of ski lift tickets sold in a year in Canada, Economics key words: substitutes and complements. Partial Derivatives - Ideal Gas Law - Partial Derivatives - Ideal Gas Law - 2 Partial Derivatives - Ideal Gas Law Evaluate both derivatives at T = 300 o K and V = 0 liters. Recall the ideal gas law, written with pressure as a function of temperature and volume: P (V, T ) = nrt V Use n = mol and R = 8.3 J/K mol in the formula and find P T and P V.

52 Partial Derivatives - Ideal Gas Law - 3 Partial Derivatives from Contour Diagrams - Express the meaning of both values in words. Partial Derivatives from Contour Diagrams Example: Consider the contour diagram shown below, representing the function h(x, y). Question: h (, ) is x (a) positive (b) negative (c) zero Question: h (, ) is y (a) positive (b) negative (c) zero Partial Derivatives from Contour Diagrams - 2 Partial Derivatives from Contour Diagrams - 3 Question: h at the point B is x (a) positive (b) negative (c) zero On the same contour diagram, mark a point where h x large. seems particularly Question: h at the point B is y (a) positive (b) negative (c) zero

53 Partial Derivatives from Table Data - Partial Derivatives from Table Data - 2 Partial Derivatives from Table Data What do these values tell you about the graph of f(x, y) near (, )? Given the following table of values for f(x, y), calculate approximate values for f f (, ) and (, ). (Multiple answers are possible, because we are x y estimating.) y x

54 Local Linearity and the Tangent Plane - Unit #9 : Functions of Many Variables, and Vectors in R 2 and R 3 Goals: To introduce tangent planes for functions of two variables. To consider functions of more than two variables and their level surfaces. To study the differential of a function and its interpretation as the linear approximation of measurement error. To learn about vectors. Local Linearity (Tangent Plane) Reading: Section 4.3. Just as a graph of a function of a single variable looks like a straight line when you zoom in to a point, the graph of a function f of two variables looks flat when you zoom in to a point (a, b, f(a, b)) on it. Reading: Sections 2.5, 4.3, 3., and 3.2. Let us write the equation of the tangent plane to z = f(x, y) in point/slope form: z = c + m(x a) + n(y b), What is the slope of f(x, y) in the x direction at (a, b)? Local Linearity and the Tangent Plane - 2 Local Linearity and the Tangent Plane - 3 Based on those calculations, if we are given f(x, y), and want the tangent plane at (a, b), what values should we pick for the slopes m and n in the tangent plane formula? What is the slope of the tangent plane in the x direction at (a, b)? If the tangent plane passes through (a, b, f(a, b)), what value do we need for c in z = c + m(x a) + n(y b)?

55 Local Linearity and the Tangent Plane - 4 Local Linearity and the Tangent Plane - 5 Tangent Plane The plane tangent to z = f(x, y) at the point (x, y) = (a, b) is defined by z = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) This can be used to define the local linear approximation to f(x, y) at points (x, y) near (a, b): An alternate form of this same relationship is f f x (a, b) x + f y (a, b) y We will revisit use these two forms interchangeably, depending on whether we want to calculate local changes in f(x, y), or the estimated values of f(x, y). f(x, y) f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) Note the similarity of the right hand side of a tangent line formula for single variable functions. Tangent Plane Examples - Tangent Plane Examples Example: What is the equation of the tangent plane to the graph of the function f(x, y) = 3xe y/2 at (, 2, 3e)? Tangent Plane Examples - 2 Use your answer to the preceding question to find an approximate value for f(0.95, 2.03) and use a calculator to check the accuracy of your answer.

56 Tangent Plane Examples - 3 Example: Let f(x, y) = 2 + x 2 2x + y2. Find the points on the graph where the tangent plane is parallel to the (x, y)-plane (i.e. horizontal). Sketch what the surface might look like near this/these points. Tangent Plane Examples - 4 [See also examples 3, 4, 5 in Section 4.3.] Functions of More Than Two Variables - Functions of More Than Two Variables - 2 Functions of More Than Two Variables Reading: Section 2.5. Example: (Temperature in a Room) To specify a point in the room, you must specify three coordinates (x, y, z). So, the temperature is a function of three variables: T = T (x, y, z). What is another variable that we could potentially include in this function, that would make temperature a function of four variables? Pictures for a Function of Three Variables To picture f(x, y, z) by means of a graph is not very useful. The graph would have to have the following definition: {(x, y, z, w) : w = f(x, y, z)}. That is, it would be a set in 4 dimensional space, which is not something we can directly visualize. However, we can make use of the analogue of the contour diagram. Suppose we pick a constant C and collect all the points (x, y, z) for which f(x, y, z) = C. In general, this gives a surface called a level surface. By sketching a number of level surfaces, we can get a sense of the function f. [See also H-H, Examples 2-5 in Section 2.5.]

57 Functions of More Than Two Variables - 3 Functions of More Than Two Variables - 4 Example (Steam Heating) Queen s University is heated by steam which is distributed to the buildings via underground pipes. Depict the level surfaces for the temperature function near the steam pipe under Agnes Benidickson Field. Contour of fixed density in a CT scan A Simple and Flexible Volume Rendering Framework for Graphics-Hardware-based Raycasting, Stegmeier et. al. Material contours based on MRI of a head. isocaps from Mathworks/MATLAB. Schlumberger Water Services - Groundwater Modelling Software MT3D99 A program for generating electron density isosurfaces for presentation in protein crystallography. M. C. Lawrence, P. D. Bourke Partial Derivatives in Higher Dimensions - Partial Derivatives in Higher Dimensions If g(x, y, z) = x2y + 3yezy g, find. 2x y In general, for a function of several variables f (x,..., xn), the partial derivative f (x,..., xn) xi is the derivative obtained by holding all other variables fixed. f. z GMU s EastFire Cluster - Real-time wind velocities for fire spread prediction. Xianjun Hao and John J. Qu Partial Derivatives in Higher Dimensions - 2 General Partial Derivatives If f (x, y, z) = xeyz + z 2 cos(x2y), find Visualization of Fluid Turbulence using AVS/Express, CEI/Ensight and Paraview. P.K. Yeung et. al. If h(x, y, z) = sin(xy) + xz 3, find h (0, 2, 3). x