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mathematics
calculus

Questions and Answers of Calculus

Lengths of human pregnancies are normally distributed with mean 268 days and standard deviation 15 days. What percentage of pregnancies last between 250 days and 280 days?
The length of time spent waiting in line at a certain bank is modeled by an exponential density function with mean 8 minutes.(a) What is the probability that a customer is served in the first 3
Find the area of the region S = {(x, y) x > 0, y < 1, x2 + y2 < 4y}.
Find the centroid of the region enclosed by the loop of the curve y2 = x3 – x4.
If a sphere of radius is sliced by a plane whose distance from the center of the sphere is d, then the sphere is divided into two pieces called segments of one base. The corresponding surfaces are
(a) Show that an observer at height H above the north pole of a sphere of radius can see a part of the sphere that has area(b) Two spheres with radii and R are placed so that the distance between
Suppose that the density of seawater, p = p (z), varies with the depth below the surface.(a) Show that the hydrostatic pressure is governed by the differential equationwhere is the acceleration due
The figure shows a semicircle with radius 1, horizontal diameter PQ, and tangent lines at P and Q. At what height above the diameter should the horizontal line be placed so as to minimize the shaded
Let P be a pyramid with a square base of side 2b and suppose that is a sphere with its center on the base of P and is tangent to all eight edges of P. Find the height of P. Then find the volume of
Consider a flat metal plate to be placed vertically under water with its top 2 m below the surface of the water. Determine a shape for the plate so that if the plate is divided into any number of
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 triangle with area 30cm2 is cut from a corner of a square with side 10 cm, as shown in the figure. If the centroid of the remaining region is 4 cm from the right side of the square, how far is it
This ratio is the probability that the needle intersects a line. Find the probability that the needle will intersect a line if h = L. What if h = L/2?
If the needle in Problem 11 has length h > L, it’s possible for the needle to intersect more than one line.(a) If L = 4, find the probability that a needle of length 7 will intersect at least one
Show that y = x – x–1 is a solution of the differential equation xy` + y = 2x.
Verify that y = sin x cos x – cos x is a solution of the initial value problem y` + (tan x)y = cos2x y(0) = –1 on the interval – π/2< x < π/2.
(a) For what nonzero values of does the function y = sin kt satisfy the differential equation yn + 9y = 0?(b) For those values of k, verify that every member of the family of functions y = A sin kt +
For what values of does the function y = ert satisfy the differential equation yn + y’ – 6y = 0?
Which of the following functions are solutions of the differential equation yn + 2y’ + y = 0?(a) y = et (b) y = e–t (c) y = te–t (d) y = t2 e–t
(a) Show that every member of the family of functions y = Cex2/2 is a solution of the differential equation yt = xy.(b) Illustrate part (a) by graphing several members of the family of solutions on a
(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
(a) What can you say about the graph of a solution of the equation y’ = xy3 when is close to 0? What if is large?(b) Verify that all members of the family y = (c – x2)–½ are solutions of the
A population is modeled by the differential equation(a) For what values of is the population increasing?(b) For what values of is the population decreasing?(c) What are the equilibrium solutions?
A function y(t) satisfies the differential equation(a) What are the constant solutions of the equation?(b) For what values of is increasing?(c) For what values of is decreasing?
Explain why the functions with the given graphs can€™t be solutions of the differential equation
The function with the given graph is a solution of one of the following differential equations. Decide which the correct equation and justify your answer.
Psychologists interested in learning theory study learning curves. A learning curve is the graph of a function P(t), the performance of someone learning a skill as a function of the training time t.
Suppose you have just poured a cup of freshly brewed coffee with temperature 95oC in a room where the temperature is 20oC.(a) When do you think the coffee cools most quickly? What happens to the rate
A direction field for the differential equation y€™ = y(1 €“ 1/4y2) is shown.(a) Sketch the graphs of the solutions that satisfy the given initial conditions.(i) y(0) = 1 (ii)
A direction field for the differential equation y€™ = x sin y is shown.(a) Sketch the graphs of the solutions that satisfy the given initial conditions.(i) y (0) = 1(ii) y (0) = 2(iii) y
Match the differential equation with its direction field (labeled I€“IV). Give reasons for your answer.
Use the direction field labeled I (for Exercises 3–6) to sketch the graphs of the solutions that satisfy the given initial conditions.(a) y(0) = 1 (b) y(0) = 0 (c) y(0) = –1
Repeat Exercise 7 for the direction field labeled III.
Sketch a direction field for the differential equation. Then use it to sketch three solution curves.
Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.
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 (0, 1). Then use the CAS to draw
Use a computer algebra system to draw a direction field for the differential equation yt = y3 – 4y. Get a printout and sketch on it solutions that satisfy the initial condition y (0) = c for
Make a rough sketch of a direction field for the autonomous differential equation y1 = f(y), where the graph of f is as shown. How does the limiting behavior of solutions depend on the value of y (0)?
(a) Use Euler’s method with each of the following step sizes to estimate the value of y (0, 4), where is the solution of the initial-value problem y’ = y, y(0) = 1.(i) h = 0.4 (ii) h = 0.2
A direction field for a differential equation is shown. Draw, with a ruler, the graphs of the Euler approximations to the solution curve that passes through the origin. Use step sizes h = 1 and h =
Use Euler’s method with step size to compute the approximate -values of the solution of the initial value problem yt = y - 2x, y(1) = 0.
Use Euler’s method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem y’ = 1 – xy, y(0) = 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) Use Euler’s method with step size 0.2 to estimate y(1.4) where y(x) is the solution of the initial-value problem y’ = x – xy, y(1) = 0.(b) Repeat part (a) with step size 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(i) h = 1 (ii) h = 0.1(iii) h = 0.01 (iv) h =
(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 y’ = x3 – y3 y (0) = 1(b) Check your
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
In Exercise 14 in Section 9.1 we considered a 95oC cup of coffee in a room. 20oC Suppose it is known that the coffee cools at a rate of 1oC per minute when its temperature is 70oC.(a) What does the
Find an equation of the curve that satisfies dy/dx = 4x3y and whose -intercept is 7.
Find an equation of the curve that passes through the point (1, 1) and whose slope at (x, y) is y2/x3.
(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
Solve the equation e–y y’ + cos x = 0 and graph several members of the family of solutions. How does the solution curve change as the constant C varies?
Solve the initial-value problem y’ = (sin x)/ sin y, y (0) = π/2, and graph the solution (if your CAS does implicit plots).
Solve the equation y’ = x√x2 + 1 / (ye y) and graph several members of the family of solutions (if your CAS does implicit plots). How does the solution curve change as the constant C varies?
(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
Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.
Solve the initial-value problem in Exercise 27 in Section 9.2 to find an expression for the charge at time t. Find the limiting value of the charge.
In Exercise 28 in Section 9.2 we discussed a differential equation that models the temperature of a 95oC cup of coffee in a 20oC room. Solve the differential equation to find an expression for the
In Exercise 13 in Section 9.1 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
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
A sphere with radius 1 m has temperature 15oC. It lies inside a concentric sphere with radius 2 m and temperature 25oC. The temperature T(r) at a distance from the common center of the spheres
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 certain small country has $10 billion in paper currency in circulation, and each day $50 million comes into the country’s banks. The government decides to introduce new currency by having the
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 tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the
When a raindrop falls, it increases in size and so its mass at time is a function of t, m (t). The rate of growth of the mass is km (t) for some positive constant k. When we apply Newton’s Law of
An object of mass m is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is,where v = v(t) and s = s (t) represent the velocity and
Let A (t) be the area of a tissue culture at time 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
According to Newton€™s Law of Universal Gravitation, the gravitational force on an object of mass m that has been projected vertically upward from Earth€™s surface is F = mgR2/(x
A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.
A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a
A bacteria culture starts with 500 bacteria and grows at a rate proportional to its size. After 3 hours there are 8000 bacteria.(a) Find an expression for the number of bacteria after hours.(b) Find
A bacteria culture grows with constant relative growth rate. After 2 hours there are 600 bacteria and after 8 hours the count is 75,000.(a) Find the initial population.(b) Find an expression for the
The table gives estimates of the world population, in millions, from 1750 to 2000:(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900
The table gives the population of the United States, in millions, for the years 1900€“2000.(a) Use the exponential model and the census figures for 1900 and 1910 to predict the population
Experiments show that if the chemical reaction N2O5 → 2NO2 + ½ O2 takes place at 45oC, the rate of reaction of Dinitrogen Pentoxide is proportional to its concentration as follows:
Bismuth-210 has a half-life of 5.0 days.(a) A sample originally has a mass of 800 mg; find a formula for the mass remaining after days.(b) Find the mass remaining after 30 days.(c) When is the mass
The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample.(a) Find the mass that remains after years.(b) How much of the sample remains after 100 years?(c) After how long will only 1
After 3 days a sample of radon-222 decayed to 58% of its original amount.(a) What is the half-life of radon-222?(b) How long would it take the sample to decay to 10% of its original amount?
Scientists can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon,
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 roast turkey is taken from an oven when its temperature has reached 185oF and is placed on a table in a room where the temperature is 75oF.(a) If the temperature of the turkey is 150oF after half
A thermometer is taken from a room where the temperature is 20oC to the outdoors, where the temperature is 5oC. After one minute the thermometer reads 12oC.(a) What will the reading on the
When a cold drink is taken from a refrigerator, its temperature is 5oC. After 25 minutes in a 20oC room its temperature has increased to 10oC.(a) What is the temperature of the drink after 50
A freshly brewed cup of coffee has temperature 95oC in a 20oC room. When its temperature is 70oC, it is cooling at a rate of 1oC per minute. When does this occur?
The rate of change of atmospheric pressure P with respect to altitude is proportional to P, provided that the temperature is constant. At 15oC the pressure is 101.3 kPa at sea level and 87.14 kPa at
(a) If $500 is borrowed at 14% interest, find the amounts due at the end of 2 years if the interest is compounded (i) Annually, (ii) Quarterly, (iii) Monthly, (iv) Daily, (v) Hourly and (vi)
(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,
(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?
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.
Let 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 ky1+c is larger than
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 k?(b) A
Suppose that a population grows according to a logistic model with carrying capacity 6000 and k = 0.0015 per year.(a) Write the logistic differential equation for these data.(b) Draw a direction
The Pacific halibut fishery has been modeled by the differential equation dy/dt = ky (1 – y/K) where y (t) is the biomass (the total mass of the members of the population) in kilograms at time
The table gives the number of yeast cells in a new laboratory culture.(a) Plot the data and use the plot to estimate the carrying capacity for the yeast population.(b) Use the data to estimate the
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
(a) Make a guess as to the carrying capacity for the U.S. population. Use it and the fact that the population was 250 million in 1990 to formulate a logistic model for the U.S. population.(b)
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
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
(a) Show that if P satisfies the logistic equation (1), then(b) Deduce that a population grows fastest when it reaches half its carrying capacity.

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