All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Ask a Question
AI Study Help New

Search
Sign In
Register

study help
mathematics
calculus

Questions and Answers of Calculus

A uniform disk with radius 1 m is to be cut by a line so that the center of mass of the smaller piece lies halfway along a radius. How close to the center of the disk should the cut be made? (Express
(a) Write an expression for the surface area of the surface obtained by rotating the curve y = f(x), a ≤ x ≤ b, about the -axis. (b) What if x is given as a function of y? (c) What if the curve
Describe how we can find the hydrostatic force against a vertical wall submersed in a fluid.
What does the Theorem of Pappus say?
Suppose f(x) is the probability density function for the weight of a female college student, where is measured in pounds.(a) What is the meaning of the integral(b) Write an expression for the mean of
Show that y = 2/3ex + e-2x is a solution of the differential equation y' + 2y = 2ex.
Explain why the functions with the given graphs can't be solutions of the differential equation dy/dt = et(y - 1)2 (a) (b)
Match the differential equations with the solution graphs labeled I-IV. Give reasons for your choices. (a) y' = 1 + x + y2 (b) y' = xe-x2 - y2 (c) y' = 1 + ex2 + y2 (d) y' = sin(xy) cos(xy)
Psychologists interested in learning theory study learning curves. A learning curve is the graph of a function, the performance of someone learning a skill as a function of the training time t. The
(a) For what values of does the function y = erx satisfy the differential equation y'' + y' - y = 0?
Which of the following functions are solutions of the differential equation y'' + y = sin x? (a) y = sin x (b) y = cos x (c) y = 1/2 x sin x (d) = y 1/2 x cos x
(a) What can you say about a solution of the equation y' = - y2 just by looking at the differential equation? (b) Verify that all members of the family y = 1/(x + C) are solutions of the equation in
A population is modeled by the differential equation dP/dt = 1.2P(1 - P/4200) (a) For what values of P is the population increasing? (b) For what values of P is the population decreasing?
A direction field for the differential equation is y' = x cos π y is shown.
Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. (a) y' = y - 2x, (1, 0) (b) y' = y + xy, (0, 1)
Use a computer algebra system to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through. Then use the CAS to draw the
Use a computer algebra system to draw a direction field for the differential equation y' = y3 - 4y. Get a printout and sketch on it solutions that satisfy the initial condition y(0) = c for various
(a) Use Euler's method with each of the following step sizes to estimate the value of y(0.4), where y is the solution of the initial-value problem y' = y, y(0) = 1. (i) h = 0.4 (ii) h = 0.2 (iii) h =
Use Euler's method with step size 0.5 to compute the approximate -values y1, y2, y3, of the solution of the initial-value problem y' = y - 2x,(1) = 0.
Use Euler 's method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem y' = y + xy, y(0) = 1.
(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 dy/dx + 3x2y = 6x2 y(0) = 3 (i) h = 1 (ii) h = 0.1 (iii) h =
The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (Ω). The voltage drop across the capacitor is
Use the direction field labeled II (above) to sketch the graphs of the solutions that satisfy the given initial conditions. (a) y(0) = 1 (b) y(0) = 2 (c) y(0) = - 1
Sketch a direction field for the differential equation. Then use it to sketch three solution curves. y' = 1/2y
(a) dy / dx = xy2(b) xy2 y' = x + 1(c) (y + sin y)y' = x + x3
Find the solution of the differential equation that satisfies the given initial condition.(a) dy/dx = x/y' y(0) = -3(b) du/dt = 2t + sec2t/2u, u(0) = -5(c) x 1n x = y(1 + √3 + y2)y', y(1) = -1
Find an equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is xy.
Solve the differential equation y' = x + y by making the change of variable u = x + y.
(a) Solve the differential equation y' = 2x√1 - y2.
Solve the initial=value problem y' = (sin x) / sin y, y(0) = π / 2, and graph the solution (if your CAS does implicit plots).
(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)
Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. (a) x2 + 2y2 = k2 (b) y = k/x
An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation.(a)(b)
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
In contrast to the situation of Exercise 40, experiments show that the reaction H2 + Br2 → 2HBr satisfies the rate lawD[HBr/dt] = k[H2][Br2]1/2And so for this reaction the differential equation
A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate
A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How
A vat with 500 gallons of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the vat at a rate of 5 gal / min and the mixture is pumped out at the same rate. What is the
When a raindrop falls, it increases in size and so its mass at time t is a function of t, namely m(t). The rate of growth of the mass is km(t) for some positive constant k. When we apply New ton's
Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance). If L1(t) and L2(t) are the sizes of two organs in an organism
Let be the area of a tissue culture at time t and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on
Suppose that a population develops according to the logistic equation dP/dt = 0.05P - 0.0005P2 where t is measured in weeks. (a) What is the carrying capacity? What is the value of?
(a) Show that if satisfies the logistic equation 4, then d2P / dt2 = k2P(1 - P / M) (1 - 2P / M) (b) Deduce that a population grows fastest when it reaches half its carrying capacity.
The table gives the midyear population of Japan, in thousands, from 1960 to 2005.Use a graphing calculator to fit both an exponential function and a logistic function to these data. Graph the data
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 > β.
Let's modify the logistic differential equation of Example 1 as follows: dP / dt = 0.08P(1 - P / 1000) - 15 (a) Suppose represents a fish population at time t, where is measured in weeks. Explain the
There is considerable evidence to support the theory that for some species there is a minimum m population such that the species will become extinct if the size of the population falls below m. This
In a seasonal-growth model, a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be caused by seasonal changes in
Graphs of logistic functions (Figures 2 and 3) look suspiciously similar to the graph of the hyperbolic tangent function. Explain the similarity by showing that the logistic function given by
The Pacific halibut fishery has been modeled by the differential equation dy/dt = ky(1 - y/M) Where y(t) is the biomass (the total mass of the members of the population) in kilograms at time
Suppose a population grows according to a logistic model with initial population 1000 and carrying capacity 10,000. If the population grows to 2500 after one year, what will the population be after
The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the
One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction of the population who have heard the rumor and the fraction who have not heard the
Determine whether the differential equation is linear. a. x - y' = xy b. y' = 1 / x + 1 / y
Solve the initial value problem. a. x2y' + 2xy = ln x, y(1) = 2 b. t du = t2 + 3u, t > 0, y(2) = 4 c. (x2 + 1) dy / dx + 3x(y - 1) = 0, y(0) = 2
Solve the differential equation and use a graphing calculator or computer to graph several members of the family of solutions. How does the solution curve change as C varies? xy' + 2y = ex
A Bernoulli differential equation (named after James Bernoulli) is of the form dy / dx + P(x) y = Q(x)yn Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, show
Use the method of Exercise 23 to solve the differential equation. y' + 2 / x y = y3 / x2
In the circuit shown in Figure 4, a battery supplies a constant voltage of 40 V, the inductance is 2 H, the resistance is 10 Ω, and I(0) = 0. (a) Find I(t) = 0. (b) Find the current after 0.1 s.
The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (Î©). The voltage drop across the
Let P(t) be the performance level of someone learning a skill as a function of the training time t. The graph of P is called a learning curve. In Exercise 15 in Section 9.1 we proposed the
In Section 9.3 we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable equations. (If the rates of flow into and out of the
An object with mass is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s(t)
(a) Show that the substitution z = 1 / P transforms the logistic differential equation P' = kP(1 - P / M) into the linear differential equation z' + kz = k / M (b) Solve the linear differential
Determine whether the differential equation is linear. a. y' + y = 1 b. y' = x - y c. xy' + y = √x d. sin x dy / dx (cos x) y = sin(x2) e. (1 + t) du / dt + u = 1 + t, t > 0
For each predator-prey system, determine which of the variables, x or y, represents the prey population and which represents the predator population. Is the growth of the prey restricted just by the
In Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let's modify those equations as follows: dR/dt = 0.08R(1 - 0.0002R) - 0.001RW dW/dt = -0.02W + 0.0000RW (a)
The system of differential equations dx/dt = 0.5x - 0.004x2 - 0.001xy dy/dt = 0.4y - 0.001y2 - 0.002xy is a model for the population of two species. (a) Does the model describe cooperation, or
A phase trajectory is shown for populations of rabbits (R) and foxes (F).(a) Describe how each population changes as time goes by.(b) Use your description to make a rough sketch of the graphs of R
Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory.
In Example 1(b) we showed that the rabbit and wolf populations satisfy the differential equation dW/dR = -0.02W + 0.00002RW / 0.08R - 0.001RW By solving this separable differential equation, show
(a) A direction field for the differential equation y' = y(y - 2) (y - 4) is shown. Sketch the graphs of the solutions that satisfy the given initial conditions. (i) y(0) = -0.3 (ii) y(0) = 1 (iii)
Find the orthogonal trajectories of the family of curves. y = kex
(a) Write the solution of the initial-value problem dP/dt = 0.1P(1 - P/2000) P(0) = 100 and use it to find the population when t = 20, (b) When does the population reach 1200?
The von Bertalanffy growth model is used to predict the length L(t) of a fish over a period of time. If L∞ is the largest length for a species, then the hypothesis is that the rate of growth in
One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000
The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation dh/dt = - R/V (h/k + h) where h is the hormone concentration in the bloodstream,
Suppose the model of Exercise 22 is replaced by the equations dx/dt = 0.4 x(1 - 0.000005x) - 0.002xy dy/dt = -0.2y + 0.000008xy (a) According to these equations, what happens to the insect population
When a flexible cable of uniform density is suspended between two fixed points and hangs of its own weight, the shape y = f(x) of the cable must satisfy a differential equation of the formWhere is a
(a) A direction field for the differential equation y' = x2 - y2 is shown. Sketch the solution of the initial-value problem y' = x2 - y2 y(0) = 1 Use the graph to estimate the value of y(0.3). (b)
Solve the differential equation. a. y' = xe-sin x - y cos x b. 2yey2y' = 2x + 3√x
Solve the initial-value problem. a. dr/dt + 2tr = r, r(0) = 5 b. xy' - y = x ln x, y(1) = 2
Find all functions f such that f' is continuous and
A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles
Recall that the normal line to a curve at a point P on the curve is the line that passes through P and is perpendicular to the tangent line at P. Find the curve that passes through the point (3, 2)
Find all curves with the property that if a line is drawn from the origin to any point (x, y) on the curve, and then a tangent is drawn to the curve at that point and extended to meet the -axis, the
Let f be a function with the property that f(0) = 1, f(0) = 1, and f(a + b) = f(a)f(b) for all real numbers a and b. Show that f'(x) = f(x) for all and deduce that f(x) = ex.
Find the curve y = f(x) such that f(x) > 0, f(0) = 0, f(1), and the area under the graph of f from 0 to x is proportional to the (n + 1)st power of f(x).
A peach pie is taken out of the oven at 5:00 PM. At that time it is piping hot, 1000C. At 5:10 PM its temperature is 800C; at 5:20 PM it is 650C. What is the temperature of the room?
A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system (as shown in the figure), assume:(i) The rabbit is at the origin and the dog is
(a) What is a differential equation? (b) What is the order of a differential equation? (c) What is an initial condition?
What is a separable differential equation? How do you solve it?
What is a first-order linear differential equation? How do you solve it?
(a) Write a differential equation that expresses the law of natural growth. What does it say in terms of relative growth rate? (b) Under what circumstances is this an appropriate model for population
(a) Write the logistic equation. (b) Under what circumstances is this an appropriate model for population growth?
(a) Write Lotka-Volterra equations to model populations of food fish (F) and sharks (S). (b) What do these equations say about each population in the absence of the other?
Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as increases. a. x = t2 +t, y= t 2 - t, -2 ≤ t ≤ 2 b. x =
(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x = sin
Describe the motion of a particle with position (x, y) as t varies in the given interval. a. x = 3 + 2 cost, y = 1 + 2 sin t, π /2 ≤ t ≤ 3π /2. b. x = 5 sin t, y= 2 cos t, -π ≤ t ≤ 5π
Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 5sint, y = 2cos t, -π ≤ t ≤ 5π

Showing 3300 - 3400 of 14235

First
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
Last

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Secure Payment

Download Our App

© 2024 SolutionInn. All Rights Reserved