48. Logistic Growth (BC) Classwork

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1 48. Logistic Growth (BC) Classwork Using the exponential growth model, the growth of a population is proportion to its current size. The differential equation for exponential growth is dp = kp leading to dt P = Ce kt. This models growth that continues forever: lim P( t) =. t In real-life situations though, exponential growth does not continue forever. There is a limit to how large the population can be. For instance, in an office building, a flu virus is growing. It may grow exponentially but there is a limit to how large the population can be which is the number of people working in the building. We call this number the Carrying Capacity and use the variable M. So in this model, dp ( ). This DEQ can be solved as follows: dp P M = kt dt P M = Ce kt P = M + Ce kt Note that in this model, when t = 0, there is a lot of room for growth so the growth is fast. Later, there is not as much room for growth, so the growth is smaller. As t approaches infinity, there is almost no room for growth so the growth is almost zero. dt = k M P A third model has the growth of a population P is proportional to both P and the carrying capacity, M P. This type of growth is called logistic growth. Let s write and solve the DEQ: Growth of Population is proportional to both population and room for growth: Separate the variables: Generate partial fractions and integrate: Perform integration: Multiply and use ln rules: Write exponentially: Solve for P: In logistic growth, the growth of the population is proportional to the population P and the carrying capacity M : dp dt = kp( M P). Its solution is P = M 1+ Ce where C and k are Mkt constants.7c An alternate DEQ describing logistic growth is dp dt = kp 1 P M M and the solution is P = 1+ Ce. kt - 7 -

2 Rumors and logistic growth: A rumor usually spreads logistically. A rumor spreads as tellers pass it on to hearers. Once told the rumor, a hearer becomes a teller. The rumor spreads slowly at first when tellers are few and the room for growth is large. As time passes, there are more tellers and the room for growth is smaller, but the product of these two numbers is larger than it was before so growth of the rumor is faster. As time increases further, the number of tellers is plentiful but the room for growth is now small so their product is small. So the growth of the rumor is now slow again. 1) A school has 1,300 students. 5 students start a rumor that the school is going to be replaced with a new one. A day later, 65 students know the rumor. Write the differential equation that describes the situation and solve it. Then determine how many students have heard the rumor after 7 days. We are usually interested when the rumor spreads the fastest. We know the spread that is dp. We wish to dt maximize it which means we need to take its derivative. In problem 1) above, find the population when the rumor is growing the fastest. In logistic growth, the population grows the fastest when the population is half the carrying capacity. However that does not tell you when the population grows the fastest. In order to find that, we need to set the logistic function equal to half the carrying capacity. For the problem above, find when the rumor spreads the fastest. Then sketch the logistic curve that describes this problem

3 ) The growth of the percentage of algae on the surface of a lake is proportional to the current percentage of algae on the lake and the percentage of the lake not yet affected. On June 30, the surface of the lake is 0% algae and on July 5, it has gone up to 5%. Determine the day when the lake will be 90% covered. 3) The population of beavers B in a park changes according to the logistic equation db dt = k ( B 80 )(40 B). The slope field is to the right. What are all possible values for lim B( t)? Find B when growth is a maximum. t 4) When a new computer operating system (OS) comes out, it is known that 30% of users will not upgrade. The percentage P of people who will upgrade spreads at the rate dp = 0.P( 70 P) where t is measured in years. dt The OS was pre-released and 5% of users are already using the OS at the time of actual release. At what month after the OS comes out will the spread of the OS be at a maximum if Euler s method is used to approximate the solution to the DEQ?

4 48. Logistic Growth (BC) Homework 1. Because of limited food and space, a squirrel population cannot exceed 1,600. It grows at a rate proportional to both the existing population and to the attainable additional population. a. Write and find the general solution to the differential equation that describes this situation. b. If there were 100 squirrels two years ago and 400 one year ago, how many squirrels are there now? c. Determine when the squirrel population is growing the fastest and verify graphically.. Suppose a flu-like virus is spreading through a population of 50,000 at a rate proportional to both the number already sick and to the number still unaffected. a. Write and find the general solution to the differential equation that describes this situation. b. If 100 people were sick yesterday and 15 are sick today, how many will be sick a week from today? c. Determine the day when the spread is at a maximum and verify graphically

5 3. A new book is published and appears on the bestseller list. The growth of readership of the book can be described by the DEQ dr dt = 4R R 00 where R represents the number of people having read the book in a particular office building and t is measured in days. a. Find lim R( t) b. Find the value of R for which the readership is growing the fastest t 4. When information is spread, it happens in one of two ways: Using social media (Facebook, Twitter, Instagram), information is spread proportionally to the number of people not having the information at that time. Using word of mouth, information is spread proportionally to the number of people who know the information and those who don t. In two football stadiums, each with 60,000 fans, a popular player is injured and will not play anymore. In one stadium, news of his unavailability spreads by word of mouth. In the second stadium, that information is put out on social media. In both stadiums, 1,000 fans know the news initially and 5 minutes later, 10,000 fans know the news. How fast is the information spreading when half of the fans in each of the stadiums know the news? Use your calculator to graph both growth curves

6 49. Curves Defined by Parametric Equations (BC) Classwork Until now, we have been representing graphs by single equations involving variable x and y. We have also examined straight-line motion of a particle moving along a horizontal line whose distance traveled is a function of time. However, we now wish to examine problems that uses a third variable, usually time, to represent curves. These equations are called Parametric Equations. Suppose a golfer strikes a golf ball that is propelled into the air at an angle of 45. If the initial velocity of the ball is 64 feet per second, it can be shown that the ball follows the parabolic path given by y = x x 18. This tells us the height of the ball y when it is a given distance x away from where it was struck. However, although we have the path of the ball, we are much more interested in knowing the height of the ball at a specific time t from when it was struck. We introduce a third variable t, called a parameter. We write both variables x and y as functions of t and we obtain the parametric equations: x = 3t and y = 3t 16t. From this set of equations, we can determine where the ball is at any value of t. When t = 0, the ball is at ( ). the point (0, 0) and at t = 1, the ball is at the point 3,3 16 ( ), then the equations x = f ( t) and y = g( t) are called ( ) obtained as t varies over the interval ( ) is called the graph of the parametric equations. The parametric equations and their graph together is If f and g are continuous functions of t on an interval t 1,t parametric equations and t is called the parameter. The set of points x, y t 1,t called a plane curve. When sketching a curve by hand represented by parametric equations, you use increasing value of t. Thus the curve will be traced out in a specific direction. This is called the orientation of the curve. You use arrows to show the orientation. Examples: Complete the table for the following parametric equations and graph the plane curve for each. 1) x = t 1 and y = 3t, t 3 ) x = 4t 1 and y = 3t, 1 t 1.5 t x y t x y

7 Many times when a parametric equation is given, we wish only to sketch the general shape of the plane curve. In that case, we wish to eliminate the parameter to create a rectangular equation in the form of y = f ( x). The technique to accomplish this is to solve for the parameter in one of the parametric equations (choosing the easiest one to do so) and then replacing the result in the other equation. Generate rectangular equations for the following parametric equations. 1 3) x = t + 3, y = t 4) x = t + 4, y = t t + 4, t > 4 5) x = 5sinθ, y = 4cosθ Remember that eliminating the parameter is an aid to curve sketching. If the parametric equations represent the path of a moving object, the graph alone is not sufficient to describe the object s motion. You still need the parametric equations to tell you the position, direction and speed at a given time. We can represent a rectangular equation with parametric equations in an infinite number of ways. The easiest way is to simply let t = x and then representing y by a function of t. 6) Represent the equation y = x x by using the parameters: a) t = x, b) m = dy dx You can sketch parametric equations on your calculator by changing the mode to parametric. Use the X,T,θ,n button for the variable t. You can specify Tmin, Tmax and TStep to control the resolution of the graph. The graph to the right is x = t sint, y = cost, t = [ 0,10]. It should be obvious that this curve is not a function as some x-values have several y-values. The points shown on the graph are at equal time intervals but not at equal distances from each other. This would indicate that the speed of an object traversing this curve is not constant. Parametric curves can have loops, cusps, vertical tangents and other peculiar features that could never be graphed in rectangular form. Hypocycloid: a point on a circle of radius 3 which is rolling around the inside a circle of radius 5. x = cost + 3cos t 3 y = sint 3sin t 3, 0 t 0 Lissajou Curve: used in the study of harmonics. x = sin5t y = sin6t 0 t π

8 49. Curves Defined by Parametric Equations (BC) Homework 1. Consider the parametric equations x = 3 t, y = t 1 a. Complete the table b. Plot the plane curve e. Eliminate the parameter t x y. Consider the parametric equations x = 6cos θ, y = 4sinθ a. Complete the tables b. Plot the plane curve e. Eliminate the parameter t x y π π 3 π 4 π 6 t 0 x y π 6 π 4 π 3 π 3. In the following problems, eliminate the parameter and confirm graphically that the rectangular equation gives the same graph as the parametric equations. Be sure to take the domain of the parametric equation into account in your rectangular equation. a. x = 4t 1, y = t + 3 b. x = t, y = t c. x = t, y = 3 t t d. x = t t, y = t + t e. x = t, y = t t f. x = t 3, y = t

9 g. x = sec θ, y = tan θ h. x = secθ, y = sinθ 4. In the following functions, convert to parametric equations by first letting t = x and second, letting m = dy dx. a. y = x b. y = e x 5. Use your calculators to graph the plane curve represented by the parametric equations. Indicate the orientation of the curve. Since these equations involve trig functions, it makes sense to let t run from 0 to π or 4π or 6π (to give a sense of the curve). a. Cycloid b. Prolate cycloid x = ( θ sinθ ), y = ( 1 cosθ ) x = ( θ sinθ ), y = 1 cosθ ( ) c. Hypocycloid d. Witch of Agnesi x = 3cos 3 θ, y = 3sin 3 θ x = cotθ, y = sin θ 6. The curve intersects itself at the times a and b when x( a) = x( b) and y( a) = y( b). Find these times for a. x = t 1, y = t 3 5t +1 b. x = t t +1, y = t 3 t

10 50. Calculus and Parametric Equations (BC) Classwork When we studied functions, we were able to determine the slope of the tangent to a curve at a point by taking the derivative. We need to be able to do the same when functions are defined parametrically. If a smooth curve C is given by the equations x = f t ( ) and y = g( t), then the slope of C is given by dy dy dt = dx dx dt, dx 0. Note that this formula is in terms of t, not x, so to find the slope of the tangent dt line to a parametric equation at a point x, y ( ), you need to find the corresponding value of t when the curve goes through the point ( x, y). The curve has a horizontal slope if dy dx = 0 and dt dt 0. The curve has a vertical slope if dx dt = 0 and dy dt 0. 1) Find dy dx for the curve given by x = cost, y = sint at a) t = π, b) the point (1, 0), c) the point (0, 1) 3 Because dy is a function of t, you can use the rule above repeatedly to find higher order derivatives: dx d d y dx = d dx [ ] dy dy dx dx = dt dx dt d 3 y dx 3 = d dx d y dx = d dt d y dx dx dt ) Find d y for the curve given by x = cost, y = sint at a) t = π, b) the point (0, 1) dx 3 3) For the curve given by x = t, y = 1 ( t t), find the slope and concavity at (, 4)

11 4) The cycloid based on the parametric equations x = t sin t and y = cost, t 0 is to the right. Find a) dy dx dy dx = sint 1 cost b) the equation of the tangent line c) values of t where d) values of t where at t = π 4 the curve is horizontal the curve is vertical The arc length S of a continuous curve C given by y = h( x) over the interval [ x 1, x ] is given by x S = 1+ h ( x ) dx = 1+ dy dx x 1 x x 1 dx. Representing h( x) as the parametric equations x = f ( t), y = g( t) on [ t 1,t ], the arc length of the continuous and differentiable plane curve that does not intersect itself is: t dy dt S = 1+ dx dt dx = t 1 t t 1 ( dx dt) + dy dt ( ) ( dx dt) dx dt dt = t t 1 ( dx dt) + ( dy dt) dx dt dx dt dt = t t 1 dx dt + dy dt dt 5) Find the arc length of the parametric equations on the given t-interval: a) x = t t, y = t 3 [ 0, ] b) x = lnt, y = e t [ 1, e] 6) Show that the circumference of a circle of radius r is πr representing the equation as a) rectangular x + y = r ( ) b) parametric x = r cosθ, y = rsinθ ( ) - 8 -

12 7) A circle of radius 1 rolls around the circumference of a larger circle of radius 5. The epicycloid traced by a point on the circumference of the smaller circle is given by x = 5sint sin5t, y = 5cost cos5t as shown by the figure below. Find the arc length of the epicycloid. 8) A bicycle race-course is in the shape of a spiral whose parametric equations are given by x = t π cost, y = t sint, where x and y are measured in miles, as shown below. The race starts at the origin, π does 3 spiral revolutions, and then goes back to the start. What is the distance the bikers ride? If a smooth curve C given by x = f ( t) and y = g( t), a t b and C doesn t intersect itself, then the area of the lateral surface of revolution about the coordinate axes is given by: About x-axis with g t ( ) 0 : S = π g( t) b dx dt + dy dt dt About y-axis with f t a ( ) 0 : S = π f ( t ) b a dx dt + dy dt dt 9) A wedge of an orange is formed by rotating the parametric curve x = 4cost, y = 4sint, 0 t π 3 about the x-axis. Find the total surface area of the orange

13 50. Calculus and Parametric Equations (BC) Homework In the following problems, find dy dx and d y and evaluate each at the indicated value of t. dx 1. x = 3t, y = 4t +1 t =. x = t, y = 8 3 t3 1 t = 4 3. x = t, y = t 4t t = 1 4. x = sint, y = cost t = 5π 4 5. x = t, y = t t = 1 6. x = sin 3 t, y = cos 3 t t = 3π 4 In the following two exercises, find the equation of the tangent line at the indicated point on the curve

14 In the following problems, find all points of horizontal and vertical tangency to the curve. 9. x = t, y = t 10. x = t + 4, y = t + 6t x = 8cos θ, y = 4sinθ 1. x = θ, y = ( 1 cosθ ) 13. x = t + t 4, y = t 3 6t 14. x = cost, y = sint In the following exercises, determine the t intervals in which the curve is concave up or concave down. 15. x = t lnt, y = t + lnt 16. x = sint, y = cost 0 t π

15 Find the arc length of the given curve on the indicated t interval. Calculators permitted on 19 and x = sin 3 t, y = cos 3 t 0, π 18. x = t 3, y = 1+ ( 8 t) 3 [ 0, 4] 19. x = t t, y = 4t 3 + [ 1,1 ] 0. x = t sint, y = 3cost [ 0, π ] 1. x = e t sint, y = e t cost [ 0,π ]. x = t 5 [ ] t 3, y = t 1,3-86 -

16 3. The path of a soccer ball is modeled by the parametric equations x = 80t cos30 and y = 80t sin30 16t where x and y are measured in feet. Graph the path of the ball and find the difference between the horizontal distance the ball travels and the arc length of the path of the ball. 4. Given the parametric equations x = 3t 1+ t, y = 3t, t 0, sketch it on your t calculator. This graph is called a folium. It looks like a leaf (foliage). a. Other than t = 0, the graph has one horizontal b. Approximate the arc length of the closed loop. tangent and a vertical tangent. Find the difference in t-values between these two points. 5. Find the area of the surface generated by rotating the curve about the x-axis and y-axis. a. x = t, y = 8 4t, 0 t b. x = 6cosθ, y = 6sinθ, 0 θ π

17 51. Polar Coordinates and Graphs (BC) Classwork The rectangular coordinate system we use is sometimes called the Cartesian system (names after French mathematician René Descartes). It is based on a grid of perpendicular lines. The polar coordinate system is based on a circular model. The city of Paris is modeled on the polar coordinate system with the Arc de Triomphe as the center. To form the polar coordinate system, we start with a point O called the pole or the origin. Each point P in the plane is assigned polar coordinates r,θ ( ) as follows. r is the directed distance from O to P and θ is the directed angle, measured counterclockwise from the polar axis (the line θ = 0 ) to segment OP. θ can be measured in degrees or radians although for calculus purposes, we usually use radians. In the rectangular system, every point has one and only one representation. In the polar system, every point has an infinite number of ways to name it. We see four points on a polar graph on the figure to the right. Name each in a variety of ways: 1) Pt A: Degrees: ( 3,30 ), ( 3, 330 ), ( 3, 10 ), ( 3, 150 ) Radians: 3, π 6, 3, 11π 6, 3, 7π 6, 3, 5π 6 ) Pt B: Degrees: Radians: 3) Pt C: Degrees: Radians: 4) Pt D: Degrees: Radians: To convert rectangular equations from polar coordinates and vice versa, learn the following relationships: Polar Rectangular ( ) Given r,θ x = r cos θ, y = rsin θ Rectangular Polar ( ) Given x, y y r = x + y, θ = tan 1 x 5) Convert the following polar points to rectangular points: ( ) b) 3, π 6 a) 5,π c) 8, 5π 4 6) Convert the following rectangular points to polar points: a) ( 3, 3) b) ( 0, ) c) ( 4, 4 3)

18 Since points can be written in both rectangular and polar format, we can also write equations with both formats. Graphs that are linear are best written in rectangular format while graphs that have curves are best written in polar form. Non-functions with loops are very difficult to write in rectangular form and usually need to be written as a piecewise function while in many cases, they are very easily written in polar form. To convert from one form to another, we use the same relationships above. For example, to convert the rectangular equation of a circle x + y = 4 to polar form, we realize that since r = x + y, we get r =. We can verify this graphically. Using function mode, we graph y = ± 4 x. We see below that the calculator has problems when the graph becomes vertical because of the resolution of the screen. But when we graph this in polar mode, we get a perfect circle. Note also that we can graph this polar equation using either radian mode or degree mode. In addition to changing θmin, θmax and θstep, you must also change the Mode from radians to degrees or vice versa. If your r equation involves operations on θ that are not trigonometric, you must be in radian format and when we use the calculator for calculus purposes, we must be in radian format. 7) Convert the following rectangular equations to polar equations. Verify with the calculator. a) y = 6 b) 4x y 1 = 0 c) y = 4x 8) Convert the following polar equations to rectangular equations. Verify with the calculator. 3 a) r = cscθ b) r = c) r = sinθ + 3cosθ 1+ cosθ To convert a graph in polar form to parametric form, we use the fact that x = r cosθ and y = r cosθ. 9) Convert the following polar equations to parametric equations: a) r = 4 b) r = 4sinθ c) r = 4 cosθ

19 To find the slope of a tangent line to a polar graph, suppose we have a differential polar function r = f θ Changing the polar equation to a parametric equation: x = r cosθ = f ( θ )cosθ and y = rsinθ = f ( θ )sinθ. Using the parametric form of dy in the previous chapter, the slope of a differentiable function of θ is: dx dy dy dθ = dx dx dθ = f ( θ )cosθ + f ( θ )sinθ dx provided that 0 at f ( θ )sinθ + f ( ( r,θ ) θ )cosθ dθ Using this formula, we make these observations: to find horizontal tangents to polar graphs, set dy dx = 0 provided that dθ dθ 0. to find vertical tangents to polar graphs, set dx dy = 0 provided that dθ dθ 0. if dx dy and are simultaneously zero, we can make no conclusions about tangent lines. dθ dθ 10) Find values of θ for which the polar graph has horizontal and vertical tangent lines: Verify graphically. a) r = sinθ, 0 θ π b) r = ( 1 sinθ ), 0 θ π ( ). If r = f θ ( ) = 0 at θ = α and f ( α ) 0, then we say the line θ = α is tangent to the pole. 11) Find the tangent lines to the pole at the following polar curves. Confirm graphically. a) r = 4cos3θ b) r = 3 1 sinθ ( )

20 51. Polar Coordinates and Graphs (BC) Homework 1. For each of the following polar coordinates, find the corresponding rectangular coordinates. a. 6, π b. 1, 7π 4 c. 1, 7π 4 d. 4, π 3. For each of the following rectangular coordinates, find two corresponding polar coordinates, one with a positive radius and the other with a negative radius. 1 a. ( 3, 3) b., 3 c. ( 0, 4) d. ( 3 3, 3) 3. For each of the following rectangular equations, change it to polar form and confirm using your calculator. a. 5x y = 7 b. xy = 16 c. ( x 1) + y = 1 d. ( x + y ) = ( x y ) For each of the following polar equations, change it to rectangular form and confirm using your calculator. a. r = 4 b. tan θ = 9 c. r = 4sinθ d. r = 1 1 cosθ

21 5. For r = + 3sinθ, find dy dx and the slope of the tangent line to the curve at the following polar points. a. 5, π b.,π ( ) c. 1, 3π d. 7, π 6 6. For each of the following, find the polar points of horizontal and vertical tangency. a. r = 4 1+ cosθ ( ) b. r = sinθ cos θ, 0 θ π 7. Find the lines tangent to the pole for the following polar curves defined on 0 θ π. a. r = 4 1 cosθ ( ) b. r = 3sin( θ ) - 9 -

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