Question: In this project we explore three of
In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002).
1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is, p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F (t) that is a continuous function of time.
(a). Show that the change in momentum over a time interval [t0, t1] is equal to the integral of F from to t0; that is, show that
![In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002).
1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is, p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F (t) that is a continuous function of time.
(a). Show that the change in momentum over a time interval [t0, t1] is equal to the integral of F from to t0; that is, show that
This integral is called the impulse of the force over the time interval.
(b). A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g where g = 32 ft/s2.
(i). Find the change in the ball’s momentum.
(ii). Find the average force on the bat.
2. In this problem we calculate the work required for a pitcher to throw a 90-mi/h fastball by first considering kinetic energy.
The kinetic energy k of an object of mass m and velocity v is given by K = 1/2mv2. Suppose an object of mass m, moving in a straight line, is acted on by a force F = F (s) that depends on its position s. According to Newton’s Second Law
where a and v denote the acceleration and velocity of the object.
(a). Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that
where v0 = v (s0) and v1 = v (s1) are the velocities of the object at the positions s0 and s1. Hint: By the Chain Rule,
(b) How many foot-pounds of work does it take to throw a baseball at a speed of 90 mi/h?
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity of the ball after seconds satisfies the differential equation because of air resistance.
How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.)
(b). The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft/s. The manager clocks the relay time of the shortstop (catching, turning, throwing) at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft/s?
(c). For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?](../images/in-this-project-we-explore-three-of-the-many-applications-of-calculus-189129-1.jpg)
This integral is called the impulse of the force over the time interval.
(b). A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g where g = 32 ft/s2.
(i). Find the change in the ball’s momentum.
(ii). Find the average force on the bat.
2. In this problem we calculate the work required for a pitcher to throw a 90-mi/h fastball by first considering kinetic energy.
The kinetic energy k of an object of mass m and velocity v is given by K = 1/2mv2. Suppose an object of mass m, moving in a straight line, is acted on by a force F = F (s) that depends on its position s. According to Newton’s Second Law
![In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002).
1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is, p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F (t) that is a continuous function of time.
(a). Show that the change in momentum over a time interval [t0, t1] is equal to the integral of F from to t0; that is, show that
This integral is called the impulse of the force over the time interval.
(b). A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g where g = 32 ft/s2.
(i). Find the change in the ball’s momentum.
(ii). Find the average force on the bat.
2. In this problem we calculate the work required for a pitcher to throw a 90-mi/h fastball by first considering kinetic energy.
The kinetic energy k of an object of mass m and velocity v is given by K = 1/2mv2. Suppose an object of mass m, moving in a straight line, is acted on by a force F = F (s) that depends on its position s. According to Newton’s Second Law
where a and v denote the acceleration and velocity of the object.
(a). Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that
where v0 = v (s0) and v1 = v (s1) are the velocities of the object at the positions s0 and s1. Hint: By the Chain Rule,
(b) How many foot-pounds of work does it take to throw a baseball at a speed of 90 mi/h?
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity of the ball after seconds satisfies the differential equation because of air resistance.
How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.)
(b). The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft/s. The manager clocks the relay time of the shortstop (catching, turning, throwing) at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft/s?
(c). For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?](../images/in-this-project-we-explore-three-of-the-many-applications-of-calculus-189129-2.jpg)
where a and v denote the acceleration and velocity of the object.
![In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002).
1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is, p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F (t) that is a continuous function of time.
(a). Show that the change in momentum over a time interval [t0, t1] is equal to the integral of F from to t0; that is, show that
This integral is called the impulse of the force over the time interval.
(b). A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g where g = 32 ft/s2.
(i). Find the change in the ball’s momentum.
(ii). Find the average force on the bat.
2. In this problem we calculate the work required for a pitcher to throw a 90-mi/h fastball by first considering kinetic energy.
The kinetic energy k of an object of mass m and velocity v is given by K = 1/2mv2. Suppose an object of mass m, moving in a straight line, is acted on by a force F = F (s) that depends on its position s. According to Newton’s Second Law
where a and v denote the acceleration and velocity of the object.
(a). Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that
where v0 = v (s0) and v1 = v (s1) are the velocities of the object at the positions s0 and s1. Hint: By the Chain Rule,
(b) How many foot-pounds of work does it take to throw a baseball at a speed of 90 mi/h?
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity of the ball after seconds satisfies the differential equation because of air resistance.
How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.)
(b). The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft/s. The manager clocks the relay time of the shortstop (catching, turning, throwing) at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft/s?
(c). For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?](../images/in-this-project-we-explore-three-of-the-many-applications-of-calculus-189129-3.jpg)
(a). Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that
![In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002).
1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is, p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F (t) that is a continuous function of time.
(a). Show that the change in momentum over a time interval [t0, t1] is equal to the integral of F from to t0; that is, show that
This integral is called the impulse of the force over the time interval.
(b). A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g where g = 32 ft/s2.
(i). Find the change in the ball’s momentum.
(ii). Find the average force on the bat.
2. In this problem we calculate the work required for a pitcher to throw a 90-mi/h fastball by first considering kinetic energy.
The kinetic energy k of an object of mass m and velocity v is given by K = 1/2mv2. Suppose an object of mass m, moving in a straight line, is acted on by a force F = F (s) that depends on its position s. According to Newton’s Second Law
where a and v denote the acceleration and velocity of the object.
(a). Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that
where v0 = v (s0) and v1 = v (s1) are the velocities of the object at the positions s0 and s1. Hint: By the Chain Rule,
(b) How many foot-pounds of work does it take to throw a baseball at a speed of 90 mi/h?
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity of the ball after seconds satisfies the differential equation because of air resistance.
How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.)
(b). The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft/s. The manager clocks the relay time of the shortstop (catching, turning, throwing) at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft/s?
(c). For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?](../images/in-this-project-we-explore-three-of-the-many-applications-of-calculus-189129-4.jpg)
where v0 = v (s0) and v1 = v (s1) are the velocities of the object at the positions s0 and s1. Hint: By the Chain Rule,
![In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002).
1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum.
The momentum p of an object is the product of its mass m and its velocity v, that is, p = mv. Suppose an object, moving along a straight line, is acted on by a force F = F (t) that is a continuous function of time.
(a). Show that the change in momentum over a time interval [t0, t1] is equal to the integral of F from to t0; that is, show that
This integral is called the impulse of the force over the time interval.
(b). A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m = w/g where g = 32 ft/s2.
(i). Find the change in the ball’s momentum.
(ii). Find the average force on the bat.
2. In this problem we calculate the work required for a pitcher to throw a 90-mi/h fastball by first considering kinetic energy.
The kinetic energy k of an object of mass m and velocity v is given by K = 1/2mv2. Suppose an object of mass m, moving in a straight line, is acted on by a force F = F (s) that depends on its position s. According to Newton’s Second Law
where a and v denote the acceleration and velocity of the object.
(a). Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that
where v0 = v (s0) and v1 = v (s1) are the velocities of the object at the positions s0 and s1. Hint: By the Chain Rule,
(b) How many foot-pounds of work does it take to throw a baseball at a speed of 90 mi/h?
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity of the ball after seconds satisfies the differential equation because of air resistance.
How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.)
(b). The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft/s. The manager clocks the relay time of the shortstop (catching, turning, throwing) at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft/s?
(c). For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?](../images/in-this-project-we-explore-three-of-the-many-applications-of-calculus-189129-5.jpg)
(b) How many foot-pounds of work does it take to throw a baseball at a speed of 90 mi/h?
3. (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity of the ball after seconds satisfies the differential equation because of air resistance.
How long does it take for the ball to reach home plate? (Ignore any vertical motion of the ball.)
(b). The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft/s. The manager clocks the relay time of the shortstop (catching, turning, throwing) at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft/s?
(c). For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?
Transcribed Image Text:
p(t1) – p(to) = [" F(t) dt dv F(s) = ma = m- dt Batter's box W = [" F(s) ds = }mvỉ – Įmv3 dv m dt dv ds dv m ds dt mv ds
> What can you say about the solutions of the equation y' = x2 + y2 just by looking at the differential equation?
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> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. у — ху — х', (О, 1) y' = xy
> Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. у — у — 2х, (1, 0)
> A function satisfies the differential equation dy/dt = y4 – 6y3 + 5y2 (a). What are the constant solutions of the equation? (b). For what values of y is increasing? (c). For what values of y is decreasing?
> A direction field for the differential equation y' = x cos Ï€y is shown. (a). Sketch the graphs of the solutions that satisfy the given initial conditions. (b). Find all the equilibrium solutions. ~~ーー|/ン //|\~~~+N ~ \|/ンノン ///|\\\ ////
> A sphere with radius 1 m has temperature 150C. It lies inside a concentric sphere with radius 2 m and temperature 250C. The temperature T (r) at a distance r from the common center of the spheres satisfies the differential equation If we let S = dT/dr,
> In contrast to the situation of Exercise 40, experiments show that the reaction H2 + Br2 → 2HBr satisfies the rate law and so, for this reaction the differential equation becomes where x = [HBr] and a and b are the initial concentr
> In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B → C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A
> Find the first 40 terms of the sequence defined by and a1 = 11. Do the same if a1 = 22. Make a conjecture about this type of sequence. Sta. if a, is an even number as+1 3a, + 1 if a, is an odd number
> In Exercise 15 in Section 7.1 we formulated a model for learning in the form of the differential equation dP/dt = k (M – P) where P (t) measures the performance of someone learning a skill after a training time t, M is the maximum level of performance, a
> In Exercise 28 in Section 7.2 we discussed a differential equation that models the temperature of a cup of coffee in a room. Solve the differential equation to find an expression for the temperature of the coffee at time t.
> Solve the initial-value problem in Exercise 27 in Section 7.2 to find an expression for the charge at time t. Find the limiting value of the charge.
> Find a function f such that f (3) = 2 and (t2 + 1) f'(t) + [f (t)]2 + 1 = 0, t ≠ 1 [Hint: Use the addition formula for on Reference Page 2.]
> An integral equation is an equation that contains an unknown function f (x) and an integral that involves y (x). Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] y(x) = 4 + 2tvy(1) dt
> An integral equation is an equation that contains an unknown function f (x) and an integral that involves y (x). Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] dt y(x) = 2 + o x>0 ty(t)'
> An integral equation is an equation that contains an unknown function f (x) and an integral that involves y (x). Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] y(x) = 2 + [t – ty(1)] dt
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y = x/1 + kx
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y = k/x
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. .3 y? = kx %3D
> A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month and the farmer harvests 300 catfish per month. (a). Show that the catfish population Pn after n months is given recursively by (b). How many catfish are in the
> Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. x² + 2y? = k?
> In Exercise 14 in Section 7.1 we considered a 950C cup of coffee in a 200C room. Suppose it is known that the coffee cools at a rate of 10C per minute when its temperature is 700C. (a). What does the differential equation become in this case? (b). Sketch
> (a). Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b). Solve the differential equation. (c). Use the CAS to draw
> (a). Program your computer algebra system, using Euler’s method with step size 0.01, to calculate y (2), where y is the solution of the initial-value problem (b). Check your work by using the CAS to draw the solution curve. y' = x
> (a). Program a calculator or computer to use Euler’s method to compute y (1), where y (x) is the solution of the initial value problem (b). Verify that y = 2 + e-x3 is the exact solution of the differential equation. (c). Find the er
> (a). Use Euler’s method with step size 0.2 to estimate y (0.4), where y (x) is the solution of the initial-value problem y' = x + y2, y (0) = 0. (b). Repeat part (a) with step size 0.1.
> (a). Solve the differential equation y' = 2x√1 – y2. (b). Solve the initial-value problem y' = 2x√1 – y2, y (0) = 0, and graph the solution. (c). Does the initial-value problem y' = 2x√1 – y2, y (0) = 2, have a solution? Explain.
> Let c be a positive number. A differential equation of the form dy/dt = ky1+c where is a positive constant, is called a doomsday equation because the exponent in the expression is larger than the exponent 1 for natural growth. (a). Determine the solution
> Consider a population P = P (t) with constant relative birth and death rates α and β, respectively, and a constant emigration rate m, where α, β, and m are positive constants. Assume that α
> (a). How long will it take an investment to double in value if the interest rate is 6% compounded continuously? (b). What is the equivalent annual interest rate?
> Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula William Gosper used this series in 1985 to compute the first 17 million digits of π. (a). Verify that the series is convergent. (b). How many correct decimal pl
> (a). If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b). If A (T) is the amount of t
> (a). If $1000 is borrowed at 8% interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b). Suppose $1000 is borrowed an
> The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At 150C the pressure is 101.3kPa at sea level and 87.14 kPa at h = 1000m. (a). What is the pressure at an altitude of
> A freshly brewed cup of coffee has temperature 950C in a 200C room. When its temperature is 700C, it is cooling at a rate of 10C per minute. When does this occur?
> When a cold drink is taken from a refrigerator, its temperature is 50C. After 25 minutes in a 200C room its temperature has increased to 100C. (a). What is the temperature of the drink after 50 minutes? (b). When will its temperature be 150C?
> Find the solution of the differential equation that satisfies the given initial condition. xy sin x y' y +1 y(0) = 1 '
> A roast turkey is taken from an oven when its temperature has reached 1850F and is placed on a table in a room where the temperature is 750F. (a). If the temperature of the turkey is 1500F after half an hour, what is the temperature after 45 minutes? (b
> A curve passes through the point (0, 5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
> (a). Show that if P satisfies the logistic equation (1), then d2P/dt2 = k2P (1 – P/M) (1 – 2P/M). (b). Deduce that a population grows fastest when it reaches half its carrying capacity.
> Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,000. The number of fish tripled in the first year. (a). Assuming that the size of the fish population
> (a). Show that ∑∞n=0 xn/n! converges for all x. (b). Deduce that limn→∞ xn/n! = 0 for all x.
> In Example 1(b) we showed that the rabbit and wolf populations satisfy the differential equation By solving this separable differential equation, show that where C is a constant. It is impossible to solve this equation for W, as an explicit function
> Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory. y species 1 1200- 1000 800 600 400 species 2 200 5 10 15
> Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory. y. species 1 200+ species 2 150+ 100+ 50 1
> Solve the differential equation. du/dr = 1 + √r/1 + √u
