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mathematics
calculus

Questions and Answers of Calculus

A population of bacteria has per capita production r < 1, and 1.0 x 106 bacteria are added each generation. What happens to the equilibrium if r = 0? What happens if r is close to 1? Do these results
There is no evaporation, and 3.0 x 106m3 of water flows out each year. Lakes receive water from streams each year and lose water to out flowing streams and evaporation. The following values are based
1.5 x 106m3 of water flows out each year, and 1.5 x 106m3 evaporates. No salt is lost through evaporation. Lakes receive water from streams each year and lose water to out flowing streams and
A total of 3.0 x 106m3 of water evaporates, and there is no outflow. Lakes receive water from streams each year and lose water to out flowing streams and evaporation. The following values are based
Assume instead that 2.0 x 106m3 of water evaporates and there is no outflow. The volume of this lake is increasing. Lakes receive water from streams each year and lose water to out flowing streams
The situation described in Exercise 41.Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt level
The situation described in Exercise 42.Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt level
The situation described in Exercise 43. Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt
The situation described in Exercise 44.Find the equilibrium concentration of salt in a lake in the above case. Describe the result in words by comparing the equilibrium salt level with the salt level
Suppose that r = 1.5 and h = l.0 x 106 bacteria. Sketch the updating function, and find the equilibrium both algebraically and graphically. A lab is growing and harvesting a culture of valuable
1.0 L of water at temperature T1 is mixed with 2.0 L of water at temperature T2. What is the temperature of the resulting mixture? Set T1 = 30 and T2 = 100 and compare with the result of Exercise
Without setting r and h to particular values, find the equilibrium algebraically. Does the equilibrium get larger when h gets larger? Does it get larger when r gets larger? If the answers seem odd
V1 liters of water at 30°C is mixed with V2 liters of water at 100°C. What is the temperature of the resulting mixture? Set V1 = 1.0 and V2 = 2.0 and compare with the result of Exercise 1. Express
V1 liters of water at temperature T1 is mixed with V2 liters of water at temperature T2. What is the temperature of the resulting mixture? Express the above weighted averages in terms of the given
V1 liters of water at temperature T1 is mixed with V2 liters of water at temperature T2 and V3 liters of water at temperature T3. What is the temperature of the resulting mixture? Express the above
1.0 L of water at 30°C is to be mixed with 2.0 L of water at 100°C, as in Exercise 1. Before mixing, however, the temperature of each moves half-way to 0 °C (so the 30°C water cools to 15°C).
A culture of bacteria has mass 3.0 × 10−3 grams and consists of spherical cells of mass 2.0 × 10−10 grams and density 1.5 grams/cm3. a. How many bacteria are in the culture? b. What is the
A person develops a small liver tumor. It grows according toS(t) = S(0)eαtwhere S(0) = 1.0 gram and α = 0.1/day. At time t = 30 days, the tumor is detected and treatment begins. The size of the
Two similar objects are left to cool for one hour. One starts at 80oC and cools to 70oC and the other starts at 60oC and cools to 55oC. Suppose the discrete-time dynamical system for cooling objects
A culture of bacteria increases in area by 10% each hour. Suppose the area is 2.0 cm2 at 2:00 P.M. a. What will the area be at 5:00 P.M.? b. Write the relevant discrete-time dynamical system and
Candidates Dewey and Howe are competing for fickle voters. A total of 100,000 people are registered to vote in the election, and each will vote for one of these two candidates. Voters often switch
A certain bacterial population has the following odd behavior. If the population is less than 1.5 × 108 in a given generation, each bacterium produces two offspring. If the population is greater
An organism is breathing a chemical that modifies the depth of its breaths. In particular, suppose that the fraction q of air exchanged is given bywhere Î³ is the ambient concentration and ct is
Lint is building up in a dryer. With each use, the old amount of lint xt is divided by 1 + xt and 0.5 lintons (the units of lint) are added.a. Find the discrete-time dynamical system and graph the
Suppose people in a bank are waiting in two separate lines. Each minute several things happen: some people are served, some people join the lines, and some people switch lines. In particular, suppose
A gambler faces off against a small casino. She begins with $1000, and the casino with $11,000. In each round, the gambler loses 10% of her current funds to the casino, and the casino loses 2% of its
Suppose the number of bacteria in a culture is a linear function of time.a. If there are 2.0 × 108 bacteria in your lab at 5 P.M. on Tuesday, and 5.0 × 108 bacteria the next morning at 9 A.M., find
Let V represent the volume of a lung and c the concentration of some chemical inside. Suppose the internal surface area is proportional to volume and that a lung with volume 400 cm3 has a surface
Suppose a person's head diameter D and height H grow according to D(t) = 10.0e0.03t H(t) = 50.0e0.09t during the first 15 years of life. a. Find D and H at t =0, t =7.5, and t =15. b. Sketch graphs
On another planet, people have three hands and like to compute tripling times instead of doubling times.a. Suppose a population follows the equation b(t) = 3.0 × 103e0.333t where t is measured in
A Texas millionaire (with $1,000,001 in assets in 2010) got rich by clever investments. She managed to earn 10% interest per year for the last 20 years and plans to do the same in the future.a. How
A major university hires a famous Texas millionaire to manage its endowment. The millionaire decides to follow this plan each year: • Spend 25% of all funds above $100 million on university
Another major university hires a different famous Texas millionaire to manage its endowment. This millionaire starts with $340 million, brings back $355 million the next year, and claims to be able
A heart receives a signal to beat every second. If the voltage when the signal arrives is below 50 micro volts, the heart beats and increases its voltage by 30 micro volts. If the voltage when the
Suppose vehicles are moving at 72 kph (kilometers per hour). Each car carries an average of 1.5 people, and all are carefully keeping a 2- second following distance (getting no closer than the
Suppose traffic volume on a particular road has been as follows:Year Vehicles1970 ...... 40,0001980 ...... 60,0001990 ...... 90,0002000 ......135,000a. Sketch a graph of traffic over time.b. Find
In order to improve both the economy and quality of life, policies are designed to encourage growth and decrease traffic flow. In particular, the number of vehicles is encouraged to increase by a
Consider the functions f (x) = e−2x and g(x) = x3 + 1. a. Find the inverses of f and g, and use these to find when f (x) = 2 and when g(x) = 2. b. Find f ο g and g ο f and evaluate each at x =
A lab has a culture of a new kind of bacteria where each individual takes 2 hours to split into three bacteria. Suppose that these bacteria never die and that all offspring are OK. a. Write an
The number of bacteria (in millions) in a lab are as followsTime, t (h)Number, bt0.0 ........................1.51.0 ........................3.02.0 ........................4.53.0
The number of bacteria in another lab follows the discrete-time dynamical systemwhere t is measured in hours and bt in millions of bacteria.a. Graph the updating function. For what values of bt does
Convert the following angles from degrees to radians and find the sine and cosine of each. Plot the related point both on a circle and on a graph of the sine or cosine.a. θ = 60ο.b. θ = −60ο.c.
Suppose the temperature H of a bird follows the equationWhere t is measured in days and H is measured in degrees C.a. Sketch a graph of the temperature of this bird.b. Write the equation if the
The butterflies on a particular island are not doing too well. Each autumn, every butterfly produces on average 1.2 eggs and then dies. Half of these eggs survive the winter and produce new
A continuous function that is -1 for x ≤ -0.1, 1 for x ≥ 0.1, and is linear for -0.1 < x < 0.1. We can build different continuous approximations of signum (the function giving the sign of a
A continuous function that is -1 for x ≤ -0.01, 1 for x ≥ 0.01, and is linear for -0.01 < x < 0.01.We can build different continuous approximations of signum (the function giving the sign
Suppose the mass of an object as a function of volume is given by M = pV. If p = 2.0 g/cm3, how close must V be to 2.5 cm3 for M to be within 0.2 g of 5.0 g? Find the accuracy of input necessary to
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. f(x) = ex / x + 1.
The area of a disk as a function of radius is given by A = πr2. How close must r be to 2.0 cm to guarantee an area within 0.5 cm2 of 4π?Find the accuracy of input necessary to achieve the desired
The flow rate F through a vessel is proportional to the fourth power of the radius, or F(r) = ar4. Suppose a = 1.0/cm s. How close must r be to 1.0 cm to guarantee a flow within 5% of 1 mL/s? Find
Consider an organism growing according to S(t) = S(0)eαt. Suppose α = 0.001/s, and S(0) = 1.0 mm. At time 1000 s, S(t) = 2.71828 mm. How close must t be to 1000 s to guarantee a size within 0.1 mm
What values of b9 produce a result within the desired tolerance? What is the input tolerance?Suppose a population of bacteria follows the discrete-time dynamical systembt+1 = 2.0btand we wish to have
What values of b5 produce a result within the desired tolerance? What is the input tolerance? Why is it harder to hit the target from here?Suppose a population of bacteria follows the discrete-time
What values of b0 produce a result within the desired tolerance? What is the input tolerance?Suppose a population of bacteria follows the discrete-time dynamical systembt+1 = 2.0btand we wish to have
How would your answers differ if the discrete-time dynamical system were bt+1 = 5bt? Would the tolerances be larger or smaller? Why? Suppose a population of bacteria follows the discrete-time
What values of T9 produce a result within the desired tolerance? What is the input tolerance? Suppose the amount of toxin in a culture declines according to Tt+1 = 0.5Tt and we wish to have a
What values of T5 produce a result within the desired tolerance? What is the input tolerance? Suppose the amount of toxin in a culture declines according to Tt+1 = 0.5Tt and we wish to have a
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. h(y) = y2ln(y - l) for y > l.
How would your answers differ if the discrete-time dynamical system were Tt+1 = 0.1Tt? Would the tolerances be larger or smaller? Why? Suppose the amount of toxin in a culture declines according to
Suppose that k = 2.0, V0 = 50, and V* = 80. Write and graph the function giving output in terms of input as a function defined in pieces. Suppose a neuron has the following response to inputs. If it
Suppose that k = 1.5, V0 = 60, and V* = 100. Write and graph the function giving output in terms of input as a function defined in pieces. Suppose a neuron has the following response to inputs. If it
If k = 2.0 and V0 = 50, what would V* have to be for the function to be continuous? Graph the resulting function. Suppose a neuron has the following response to inputs. If it receives a voltage input
If V0 = 50 and V* = 80, what would k have to be to make the function continuous? Graph the resulting function. Suppose a neuron has the following response to inputs. If it receives a voltage input V
A child outside is swinging on a swing that makes a horrible screeching noise. Starting from when the swing is furthest back, the pitch of the screeching noise increases as it swings forward and then
Little Billy walks due east to school, but must cross from the south side to the north side of the street. Because he is a very careful child, he crosses quickly at the first possible opportunity. a.
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. g(z) = ln (z - 1) / z2 for z > 1.
Describe how the following functions are built out of the basic continuous functions. Identify points where they might not be continuous. a(t) = t2 if t > 0 and 0 if t = 0.
For each of the following quadratic functions, find the slope of the secant line connecting x = 1 and x = 1 + Δx, and the slope of the tangent line at x = 1 by taking the limit. f(x) = 4 - x2.
For each of the following quadratic functions, find the slope of the secant line connecting x = 1 and x = 1 + Δx, and the slope of the tangent line at x = 1 by taking the limit. g(x) = x + 2x2.
For each of the following quadratic functions, find the slope of the secant line connecting x and x + Δx, and the slope of the tangent line as a function of x. Write your result in both differential
For each of the following quadratic functions, find the slope of the secant line connecting x and x + Δx, and the slope of the tangent line as a function of x. Write your result in both differential
For each of the following quadratic functions, graph the function and the derivative. Identify critical points, points where the function is increasing, and points where the function is
For each of the following quadratic functions, graph the function and the derivative. Identify critical points, points where the function is increasing, and points where the function is
On the figures, label the following points and sketch the derivative.a. One point where the derivative is positive.b. One point where the derivative is negative.c. The point with maximum
On the figures, label the following points and sketch the derivative.a. One point where the derivative is positive.b. One point where the derivative is negative.c. The point with maximum
On the figures, identify which of the curves is a graph of the derivative of the other.
On the figures, identify which of the curves is a graph of the derivative of the other.
On the figures, identify which of the curves is a graph of the derivative of the other.
On the figures, identify which of the curves is a graph of the derivative of the other.
The absolute value function f(x) = |x|. The following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute the derivative as the
The square root function f(x) = √x. (Because this function is only denned for x ≥ 0, you can only use Δx > 0.) The following functions all fail to be differentiable at x = 0. In each case, graph
The Heaviside function (Section 2.3, Exercise 25), defined byThe following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute
The signum function defined byThe following functions all fail to be differentiable at x = 0. In each case, graph the function, see what happens if you try to compute the derivative as the limit of
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is
The following graphs show the temperature of different solutions with chemical reactions as functions of time. Graph the rate of change of temperature in each case, and indicate when the solution is
Both move at constant speed, but the bear is faster and eventually catches the hiker. A bear sets off in pursuit of a hiker. Graph the position of a bear and hiker as functions of time from the above
Both increase speed until the bear catches the hiker. A bear sets off in pursuit of a hiker. Graph the position of a bear and hiker as functions of time from the above descriptions.
The bear increases speed and the hiker steadily slows down until the bear catches the hiker. A bear sets off in pursuit of a hiker. Graph the position of a bear and hiker as functions of time from
The bear runs at constant speed, the hiker steadily runs faster until the bear gives up and stops. The hiker slows down and stops soon after that. A bear sets off in pursuit of a hiker. Graph the
On Earth, where a = 9.78m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of the above planet with
On the moon, where a = 1.62m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of the above planet
On Jupiter, where a = 22.88m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of the above planet
On Mars' moon Deimos, where a = 2.15 × 10-3m/sec2. An object dropped from a height of 100 m has distance above the ground of M(t) = 100 - a/2t2 where a is the acceleration of gravity. For each of
On the following graphs, identify points wherea. The function is not continuous,b. The function is not differentiable (and say why),c. The derivative is zero (critical points).
On the following graphs, identify points wherea. The function is not continuous,b. The function is not differentiable (and say why),c. The derivative is zero (critical points).
Find the derivatives of the following polynomial functions. Say where you used the sum, constant product, and power rules. s(x) = 1 - x + x2 - x3 + x4.
Find the derivatives of the following polynomial functions. Say where you used the sum, constant product, and power rules. g(z) = 3z3 + 2z2.
Find the derivatives of the following polynomial functions. Say where you used the sum, constant product, and power rules. p(x) = 1 + x + x2/2 + x3/6 + x4/24.

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