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mathematics
applied calculus
Applied Calculus 6th Edition Deborah Hughes Hallett, Patti Frazer Lock, Andrew M. Gleason, Daniel E. Flath, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale - Solutions
Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. 4 S et 1 x + 1 dx; n = 5
Determine the following integrals by making an appropriate substitution. [2x 2x cos x² dx
Determine the following integrals by making an appropriate substitution. cos x (2 + sin x)³ - dx
If k > 0, show that 8 [ k x(ln x)k+1 · dx = 1.
Determine the following integrals by making an appropriate substitution. cos Vx Vx dx
If k > 0, show that 00 k xk+1 S dx = 1.
Determine the following integrals by making an appropriate substitution. cos os³ x sin x dx
Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. -11+x² dx; n = 5
The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of the asset may be written in the formwhere r is the annual rate
Evaluate the following improper integrals whenever they are convergent. ,00 fe6-3x dx.
Determine the following integrals by making an appropriate substitution. (sin 2x)ecos 2x dx
The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of the asset may be written in the formwhere r is the annual rate
Evaluate the following improper integrals whenever they are convergent. 1 00 x + 2 2 x² + 4x - 2 dx
Determine the following integrals by making an appropriate substitution. cos 3x V2 - sin 3x | - dx
Evaluate the following improper integrals whenever they are convergent. XP ε/7-x 1 J
The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of the asset may be written in the formwhere r is the annual rate
Determine the following integrals by making an appropriate substitution. Scot cot x dx
Evaluate the following improper integrals whenever they are convergent. J fe 13 dx
Determine the following integrals by making an appropriate substitution. [₁ tan x sec² x dx
Evaluate the following improper integrals whenever they are convergent. 00 (x + 3)-5/4 dx
Determine the following integrals by making an appropriate substitution. sin x + cos x sin x - cos x dx
Evaluate the following improper integrals whenever they are convergent. L -00 8 (5 - 2x)³ dx
Determine ∫2x(x2 + 5)dx by making a substitution. Then, determine the integral by multiplying out the integrand and antidifferentiating. Account for the difference in the two results.
Find the present value of a continuous stream of income over the next 4 years, where the rate of income is 50e-0.08t thousand dollars per year at time t, and the interest rate is 12%.
Suppose that t miles from the center of a certain city the property tax revenue is approximately R(t) thousand dollars per square mile, where R(t) = 50e-t/20. Use this model to predict the total property tax revenue that will be generated by property within 10 miles of the city center.
Suppose that a machine requires daily maintenance, and let M(t) be the annual rate of maintenance expense at time t. Suppose that the interval 0 ≤ t ≤ 2 is divided into n subintervals, with endpoints t0 = 0, t1,. . . , tn = 2.(a) Give a Riemann sum that approximates the total maintenance
The capitalized cost of an asset is the total of the original cost and the present value of all future “renewals” or replacements. This concept is useful, for example, when you are selecting equipment that is manufactured by several different companies. Suppose that a corporation computes the
You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed line where the concavity of
Refer to Example 1.(a) How fast was the amount in the account growing when it reached $30,000?(b) How much money was in the account when it was growing at twice the rate of your annual contribution?(c) How long do you have to wait for the money in the account to reach $40,000?Example 1.You invest
Suppose that f (t) is a solution of the differential equation y′ = ty - 5 and the graph of f (t) passes through the point (2, 4). What is the slope of the graph at this point?
Find an integrating factor for each equation. Take t > 0.y′ - 2y = t
What is a differential equation?
Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 3; (0, 1) is on the graph; the slope is always positive, and the slope becomes less positive (as t increases).
Solve the differential equations in Exercises.y2y′ = 4t3 - 3t2 + 2
You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed line where the concavity of
Solve the differential equations in Exercises. 1+1 = y + 1
Refer to Example 2. Answer questions (a) and (b) in Exercise 1 if the interest rate is 7%. How long will it take to pay off the $25,000 loan in this case?Example 2You took a loan of $25,000 to pay for a new car. The interest rate on the loan is 5%. You arranged through your online banking to make
Suppose that f (t) is a solution of y′ = t2 - y2 and the graph of f (t) passes through the point (2, 3). Find the slope of the graph when t = 2.
Find an integrating factor for each equation. Take t > 0.y′ + ty = 6t
Solve the differential equations in Exercises. y"' = -²- 3y, - 3y, 1>0 t
What does it mean for a function to be a solution to a differential equation?
Sketch the graph of a function with the stated properties.Domain: 0 ≤ t ≤ 4; (0, 2) is on the graph; the slope is always positive, and the slope becomes more positive (as t increases).
You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed line where the concavity of
A person planning for her retirement arranges to make continuous deposits into a savings account at the rate of $3600 per year. The savings account earns 5% interest compounded continuously.(a) Set up a differential equation that is satisfied by f (t), the amount of money in the account at time
Suppose that f (t) satisfies the initial-value problem y′ = y2 + ty - 7, y(0) = 3. Is f (t) increasing or decreasing at t = 0?
Find an integrating factor for each equation. Take t > 0.t3y′ + y = 0
What is a solution curve?
Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties.Domain: 0 ≤ t ≤ 5; (0, 3) is on the graph; the slope is always negative, and the slope becomes less negative.
State the order of the differential equation and verify that the given function is a solution. (11²)y" - 2ty' + 2y = 0, y(t) = t
You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed line where the concavity of
Find an integrating factor for each equation. Take t > 0. y' y 10+ t = 2
A person deposits $10,000 in a bank account and decides to make additional deposits at the rate of A dollars per year. The bank compounds interest continuously at the annual rate of 6%, and the deposits are made continuously into the account.(a) Set up a differential equation that is satisfied by
Suppose that f (t) satisfies the initial-value problem y′ = y2 + ty - 7, y(0) = 2. Is the graph of f (t) increasing or decreasing at t = 0?
Find an integrating factor for each equation. Take t > 0.y′ + √t y = 2(t + 1)
What is a constant solution to a differential equation?
Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties.Domain: 0 ≤ t ≤ 6; (0, 4) is on the graph; the slope is always negative, and the slope becomes more negative.
Solve the differential equations in Exercises.(y′)2 = t
Twenty years ahead of her retirement, Kelly opened a savings account that earns 5% interest rate compounded continuously, and she contributed to this account at the annual rate of $1200 per year for 20 years. Ten years ahead of his retirement, John opened a similar savings account that earns 5%
Use Euler’s method with n = 2 on the interval 0 ≤ t ≤ 1 to approximate the solution f (t) to y′ = t2y, y(0) = -2. In particular, estimate f (1).
State the order of the differential equation and verify that the given function is a solution. (1 − t²)y" — 2ty' + 6y = 0, y(t) = ½ (3t² − 1) - -
What is the slope field?
Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties.Domain: 0 ≤ t ≤ 7; (0, 2) is on the graph; the slope is always positive, the slope becomes more positive as t increases from 0 to 3, and the slope becomes less
Solve the differential equations in Exercises.y = 7y′ + ty′, y(0) = 3
Answer the question in Exercise 5 if John contributed to his savings account at the annual rate of $3000 per year for 10 years.Exercise 5Twenty years ahead of her retirement, Kelly opened a savings account that earns 5% interest rate compounded continuously, and she contributed to this account at
Use Euler’s method with n = 2 on the interval 2 ≤ t ≤ 3 to approximate the solution f (t) to y′ = t - 2y, y(2) = 3. Estimate f (3).
Find an integrating factor for each equation. Take t > 0.y = t2(y + 1)
Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 8; (0, 6) is on the graph; the slope is always negative, the slope becomes more negative as t increases from 0 to 3, and the slope becomes less
Describe the separation-of-variables technique for obtaining the solution to a differential equation.
Solve the differential equations in Exercises.y′ = tet+y, y(0) = 0
For information being spread by mass media, rather than through individual contact, the rate of spread of the information at any time is proportional to the percentage of the population not having the information at that time. Give the differential equation that is satisfied by y = f (t), the
A person took out a loan of $100,000 from a bank that charges 7.5% interest compounded continuously. What should be the annual rate of payments if the loan is to be paid in full in exactly 10 years? (Assume that the payments are made continuously throughout the year.)
Use Euler’s method with n = 4 to approximate the solution f (t) to y′ = 2t - y + 1, y(0) = 5 for 0 ≤ t ≤ 2. Estimate f (2).
Is the constant function f (t) = 3 a solution of the differential equation y′ = 6 - 2y?
What is a first-order linear differential equation?
Solve the given equation using an integrating factor. Take t > 0.y′ + y = 1
World annual natural gas production, N, in billion cubic meters, is approximated by N = 2711 + 77t, where t is in years since 2004.(a) How much natural gas was produced in 2004? In 2015?(b) Estimate the total amount of natural gas produced during the 10-year period from 2004 to 2014.
A 2015 Porsche 918 Spyder accelerates from 0 to 88 ft/sec (60 mph) in 2.2 seconds, the fastest acceleration of any car available for retail sale in 2015.(a) Assuming that the acceleration is constant, graph the velocity from t = 0 to t = 2.2 seconds.(b) How far does the car travel during this time?
Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. 3 [²²/dx 1 X - dx, n = 4
If t is measured in days since June 1, the inventory I(t) for an item in a warehouse is given by I(t) = 5000(0.9)t.(a) Find the average inventory in the warehouse during the 90 days after June 1.(b) Graph I(t) and illustrate the average graphically.
The marginal cost function of producing q mountain bikes is(a) If the fixed cost in producing the bicycles is $2000, find the total cost to produce 30 bicycles.(b) If the bikes are sold for $200 each, what is the profit (or loss) on the first 30 bicycles?(c) Find the marginal profit on the 31st
Given ∫0−1 f(x) dx = 0.25 and Figure 5.46, estimate:(a) ∫10 f(x) dx (b) ∫1−1 f(x) dx(c) The total shaded area. -2 -1 -1 1 Figure 5.46 f(x) 2
Solar photovoltaic (PV) cells are the world’s fastest growing energy source. In year t since 2007, PV cells were manufactured worldwide at a rate of S = 3.7e0.61t gigawatts per year. Estimate the total solar energy generating capacity of the PV cells manufactured between 2007 and 2011.
Figure 5.15 shows the velocity, v, of an object (in meters/ sec). Estimate the total distance the object traveled between t = 0 and t = 6. v (m/sec) 40 30 20 10 1 2 3 4 5 Figure 5.15 6 1 (sec)
Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. -7 (x² + 3x) dx, n 5.5 3 =
The population of the world t years after 2010 is predicted to be P = 6.9e0.012t billion.(a) What population is predicted in 2020?(b) What is the predicted average population between 2010 and 2020?
The marginal revenue function on sales of q units of a product is R'(q) = 200 − 12√q dollars per unit.(a) Graph R¨(q).(b) Estimate the total revenue if sales are 100 units.(c) What is the marginal revenue at 100 units? Use this value and your answer to part (b) to estimate the total revenue if
Using Figure 5.47, list the following integrals in increasing order (from smallest to largest). Which integrals are negative, which are positive? Give reasons.I. ∫ba f(x) dx II. ∫ca f(x) dx III. ∫ea f(x) dxIV. ∫eb f(x) dxV. ∫cb f(x) dx a f(x) ho Figure 5.47 e X
Show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times t = 0 and t = 5 seconds. -3 v(1) 5 1 (sec)
Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff’s data follow:(a) Assuming that Roger’s speed is never increasing, give upper and lower
Match the graph with one of the following possible values for the integral ∫50 f(x) dx:I. − 10.4 II. − 2.1 III. 5.2 IV. 10.4 10 5 -5 f(x) 3 5 X
Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. 5.² sin x dx, n = 4
A car accelerates smoothly from 0 to 60 mph in 10 seconds with the velocity given in Figure 5.16. Estimate how far the car travels during the 10-second period. v (mph) 60 40 20 5 Figure 5.16 10 t (sec)
Annual income for ages 25 to 85 is given graphically. People sometimes spend less than their income (to save for retirement) or more than their income (taking out a loan). The process of spreading out spending over a lifetime is called consumption smoothing.(a) Find the average annual income for
The net worth, f(t), of a company is growing at a rate of f'(t) = 2000 − 12t2 dollars per year, where t is in years since 2005. How is the net worth of the company expected to change between 2005 and 2015? If the company is worth $40,000 in 2005, what is it worth in 2015?
Using Figure 5.45, estimate ∫5−3 f(x)dx. -3. -10- -10- -20- Figure 5.45 4. X f(x) #
Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. [²2/1 + J-4 -4 2x + 1dx, n = 4
The marginal cost function for a company is given byC'(q) = q2 − 16q + 70 dollars/unit,where q is the quantity produced. If C(0) = 500, find the total cost of producing 20 units. What is the fixed cost and what is the total variable cost for this quantity?
The value, V , of a Tiffany lamp, worth $225 in 1975, increases at 15% per year. Its value in dollars t years after 1975 is given by V = 225(1.15)t.Find the average value of the lamp over the period 1975–2010.
A bicyclist traveling at 20 ft/sec puts on the brakes to slow down at a constant rate, coming to a stop in 3 seconds.(a) Figure 5.14 shows the velocity of the bike during braking. What are the values of a and b in the figure?(b) How far does the bike travel while braking? velocity (ft/sec) b Figure
Estimate the integral using a left-hand sum and a right-hand sum with the given value of n. fiv √xdx, n=3
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