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mathematics
applied calculus

Questions and Answers of Applied Calculus

The density function for heights of American men, in inches is p(x). What is the meaning of the statement p(68) = 0.2?
Calculate the value of c if p is a density function. 0.01 P(1) с
Let p(t) = 0.1e−0.1t be the density function for the waiting time at a subway stop, with t in minutes, 0 ≤ t ≤ 60.(a) Graph p(t). Use the graph to estimate visually the median and the mean.(b)
The speeds of cars on a road are approximately normally distributed with a mean μ = 58 km/hr and standard deviation σ = 4 km/hr.(a) What is the probability that a randomly selected car is going
The distribution of IQ scores can be modeled by a normal distribution with mean 100 and standard deviation 15.(a) Write the formula for the density function of IQ scores.(b) Estimate the fraction of
In an agricultural experiment, the quantity of grain from a given size field is measured. The yield can be anything from 0 kg to 50 kg. For each of the following situations, pick the graph that best
The distribution of the heights, x, in meters, of trees is represented by the density function p(x). In each case, calculate the fraction of trees which are:(a) Less than 5 meters high(b) More than 6
(a) Use the cumulative distribution function in Figure 7.27 to estimate the median.(b) Describe the density function: For what values is it positive? Increasing? Decreasing? Identify all local
The distribution of the heights, x, in meters, of trees is represented by the density function p(x). In each case, calculate the fraction of trees which are:(a) Less than 5 meters high(b) More than 6
Show that the area under the fishing density function in Figure 7.12 on page 329 is 1. Why is this to be expected? fraction of days per ton of caught fish 0.24 0.12 0.08 +-P(x) x (tons of fish) 2 5 6
Over the past fifty years the carbon dioxide level in the atmosphere has increased. Carbon dioxide is believed to drive temperature, so predictions of future carbon dioxide levels are important. If
The distribution of the heights, x, in meters, of trees is represented by the density function p(x). In each case, calculate the fraction of trees which are:(a) Less than 5 meters high(b) More than 6
Figure 7.15 shows a density function and the corresponding cumulative distribution function.(a) Which curve represents the density function and which represents the cumulative distribution function?
A quantity x has density function p(x) = 0.5(2 − x) for 0 ≤ x ≤ 2 and p(x) = 0 otherwise. Find the mean and median of x.
The distribution of the heights, x, in meters, of trees is represented by the density function p(x). In each case, calculate the fraction of trees which are:(a) Less than 5 meters high(b) More than 6
Find an antiderivative and use differentiation to check your answer. n(x) = x √x
Find ∫ 4x(x2 + 1) dx using two methods:(a) Do the multiplication first, and then anti differentiate.(b) Use the substitution w = x2 + 1.(c) Explain how the expressions from parts (a) and (b) are
(a) Find ∫ sinθ cos θ dθ.(b) You probably solved part (a) by making the substitution w = sin θ or w = cos θ. (If not, go back and do it that way.) Now find ∫ sin θ cos θ dθ by making the
At the start of 2014, the world’s known copper reserves were 690 million tons. With t in years since the start of 2014, copper has been mined at a rate given by 17.9e0.025t million tons per
Find an antiderivative and use differentiation to check your answer.f(x) = 2/x + x/2
Find an antiderivative and use differentiation to check your answer.g(x) = x√x
Find an antiderivative and use differentiation to check your answer.p(x) = e2x − e−2x
Find an antiderivative and use differentiation to check your answer.q(x) = 7 sin x − sin(7x)
The marginal revenue function of a monopolistic producer is MR = 20 − 4q.(a) Find the total revenue function.(b) Find the corresponding demand curve.
A firm’s marginal cost function is MC = 3q2 +4q +6. Find the total cost function if the fixed costs are 200.
Find an antiderivative F(x) with F'(x) = f(x) and F(0) = 0. Is there only one possible solution?f(x) = ex
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. 3 2x x² + 1 dx
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. px/2 e-cose sine de
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. x x dx
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. -3 10 Vt + 1 dt
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. [² 0 x (1 + x²)² dx
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. [' 2te-² dt
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. e-2 1 [²²= -dt -1 1+2
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. L -1 √x + 2dx
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. 3. dt (t + 7)²
Find the exact area.Under f(x) = xex2 between x = 0 and x = 2.
Find the exact area.Under f(x) = 1∕(x + 1) between x = 0 and x = 2.
Find the exact average value of f(x) = 1∕(x + 1) on the interval x = 0 to x = 2. Sketch a graph showing the function and the average value.
(a) Find ∫ (x + 5)2 dx in two ways:(i) By multiplying out(ii) By substituting w = x + 5(b) Are the results the same? Explain.
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in Problems. [²x 0 x(x² + 1)² dx
A machine lasts up to 10 years. Figure 7.7 shows the density function, p(t), for the length of time it lasts.(a) What is the value of C?(b) Is a machine more likely to break in its first year or in
Students at the University of California were surveyed and asked their grade point average. (The GPA ranges from 0 to 4, where 2 is just passing.) The distribution of GPAs is shown in Figure 7.16.(a)
Find a density function p(x) such that p(x) = 0 when x ≥ 5 and when x < 0, and is decreasing when 0 ≤ x ≤ 5.
Let P(x) be the cumulative distribution function for the household income distribution in the US in 2009. Values of P(x) are in the following table:(a) What percent of the households made between
A person who travels regularly on the 9:00 am bus from Oakland to San Francisco reports that the bus is almost always a few minutes late but rarely more than five minutes late. The bus is never more
Calculate the value of c if p is a density function. p(x) = cx 2 X
The cumulative distribution function for heights (in meters) of trees in a forest is F(x).(a) Explain in terms of trees the meaning of the statement F(7) = 0.6.(b) Which is greater, F(6) or F(7)?
Find the median of the density function given by p(t) = 0.04 − 0.0008t for 0 ≤ t ≤ 50 using the Fundamental Theorem of Calculus.
The density function and cumulative distribution function of heights of grass plants in a meadow are in Figures 7.17 and 7.18, respectively.(a) There are two species of grass in the meadow, a short
Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance.All yields from 0 to 100 kg are equally likely; the field never yields more than 100
A congressional committee is investigating a defense contractor whose projects often incur cost overruns. The data in Table 7.7 show y, the fraction of the projects with an overrun of at most C%.(a)
Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance.High yields are more likely than low. The maximum yield is 200 kg.
Let p(t) = −0.0375t2 + 0.225t be the density function for the shelf life of a brand of banana, with t in weeks and 0 ≤ t ≤ 4. See Figure 7.19.Find the probability that a banana will last(a)
Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance.A drought makes low yields most common, and there is no yield greater than 30 kg.
Let p(t) = −0.0375t2 + 0.225t be the density function for the shelf life of a brand of banana, with t in weeks and 0 ≤ t ≤ 4. See Figure 7.19.(a) Sketch the cumulative distribution function for
Which of the following functions makes the most sense as a model for the probability density representing the time (in minutes, starting from t = 0) that the next customer walks into a store?(a) p(t)
An experiment is done to determine the effect of two new fertilizers A and B on the growth of a species of peas. The cumulative distribution functions of the heights of the mature peas without
A group of people have received treatment for cancer. Let t be the survival time, the number of years a person lives after the treatment. The density function giving the distribution of t is p(t) =
The probability of a transistor failing between t = a months and t = b months is given by c ∫ba e−ct dt, for some constant c.(a) If the probability of failure within the first six months is 10%,
While taking a walk along the road where you live, you accidentally drop your glove, but you don’t know where. The probability density p(x) for having dropped the glove x kilometers from home
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
One or more initial conditions are given for each differential equation in the following exercises. Use the qualitative theory of autonomous differential equations to sketch the graphs of the
Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f
Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f
Let c be the concentration of a solute outside a cell that we assume to be constant throughout the process, that is, unaffected by the small influx of the solute across the membrane due to a
A person deposits an inheritance of $100,000 in a savings account that earns 4% interest compounded continuously. This person intends to make withdrawals that will increase gradually in size with
After depositing an initial amount of $10,000 in a savings account that earns 4% interest compounded continuously, a person continued to make deposits for a certain period of time and then started to
Solve the initial-value problem. y' + y 1 + t = 20, y(0) = 10, t≥ 0
Verify that the function f (t) = 2e-t + t - 1 is a solution of the initial-value problem y′ = t - y, y(0) = 1.
Sketch the solutions of the differential equations in Exercises. In each case, also indicate the constant solutions.y′ = y3 - 6y2 + 9y, y(0) = - 1/4, y(0) = 1/4, y(0) = 4
Solve the following differential equations with the given initial conditions.y′ = -y2 sin t, y(π/2) = 1
Figure 4 contains the solution of the initial-value problem in Exercise 23.(a) With the help of the graph, approximate how long it will take before the account is depleted.(b) Solve the initial-value
Express f (x) + g(x) as a rational function. Carry out all multiplications. f(x)= = 9 + x x - 6' g(x) = = 9-x x+6
Solve the initial-value problem.y′ = 2(10 - y), y(0) = 1
Use the integral test to determine whether the infinite series is convergent or divergent. 1 Σ k=2 k(In k)2
Determine the domains of the following functions. f(x) = 1 V3x
Use intervals to describe the real numbers satisfying the inequalities.x ≥ 12
Factor the polynomials.5x3 + 15x2 - 20x
Solve the following differential equations with the given initial conditions. dN dt = 2tN², N(0) = 5
Match the graphs of the functions in Exercises to the systems of level curves shown in Figs. 8(a)–(d).a.b.c.d. N y X
On the slope field in Fig. 5(a) or a copy of it, draw the solution of the initial-value problem y′ = .0002y (5000 - y), y(0) = 500. Y 5000 4000 3000 2000 1000 0 0 1 2 3 4 5 6 t (a) Slope field of y
Sketch the graph of y = x3 + 2x + 2, and use the Newton– Raphson algorithm (three repetitions) to approximate all x-intercepts.

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