EBK BUSINESS MATH
11th Edition
ISBN: 8220103632072
Author: NOBLE
Publisher: Pearson Education (US)
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Chapter 3.3, Problem 15SE
To determine
To calculate: The value of decimal for the fraction
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Consider the initial value problem
mx" + cx' + kx = F(t),
x(0) = 0, x'(0) = 0
modeling the motion of a damped mass-spring system initially at rest and subjected to an
applied force F(t), where the unit of force is the Newton (N). Assume that m = = 2
kilograms, c = 8 kilograms per second, k 80 Newtons per meter, and F(t) = 20e¯*
=
Newtons.
Solve the initial value problem.
x(t) =
=
help (formulas)
Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0
t→∞
? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive
values of t.
For very large positive values of t,
x(t) ≈ x sp(t)
=
help (formulas)
Book: Section 2.6 of Notes on Diffy Qs
Chapter 3 Solutions
EBK BUSINESS MATH
Ch. 3.1 - Prob. 1-1SCCh. 3.1 - Prob. 1-2SCCh. 3.1 - Prob. 1-3SCCh. 3.1 - Prob. 1-4SCCh. 3.1 - Prob. 1-5SCCh. 3.1 - Prob. 1-6SCCh. 3.1 - Prob. 2-1SCCh. 3.1 - Prob. 2-2SCCh. 3.1 - Prob. 2-3SCCh. 3.1 - Prob. 2-4SC
Ch. 3.1 - Prob. 1SECh. 3.1 - Prob. 2SECh. 3.1 - Prob. 3SECh. 3.1 - Prob. 4SECh. 3.1 - Prob. 5SECh. 3.1 - Prob. 6SECh. 3.1 - Prob. 7SECh. 3.1 - Prob. 8SECh. 3.1 - Prob. 9SECh. 3.1 - Prob. 10SECh. 3.1 - Prob. 11SECh. 3.1 - Prob. 12SECh. 3.1 - Prob. 13SECh. 3.1 - Prob. 14SECh. 3.1 - Prob. 15SECh. 3.1 - Prob. 16SECh. 3.1 - Prob. 17SECh. 3.1 - Prob. 18SECh. 3.1 - Prob. 19SECh. 3.1 - Prob. 20SECh. 3.1 - Prob. 21SECh. 3.1 - Prob. 22SECh. 3.1 - Prob. 23SECh. 3.1 - Prob. 24SECh. 3.1 - Prob. 25SECh. 3.1 - Prob. 26SECh. 3.2 - Prob. 1-1SCCh. 3.2 - Prob. 1-2SCCh. 3.2 - Prob. 1-3SCCh. 3.2 - Prob. 1-4SCCh. 3.2 - Prob. 1-5SCCh. 3.2 - Prob. 1-6SCCh. 3.2 - Prob. 2-1SCCh. 3.2 - Prob. 2-2SCCh. 3.2 - Prob. 2-3SCCh. 3.2 - Prob. 2-4SCCh. 3.2 - Prob. 2-5SCCh. 3.2 - Prob. 2-6SCCh. 3.2 - Prob. 2-7SCCh. 3.2 - Prob. 2-8SCCh. 3.2 - Prob. 2-9SCCh. 3.2 - Prob. 2-10SCCh. 3.2 - Prob. 2-11SCCh. 3.2 - Prob. 2-12SCCh. 3.2 - Prob. 3-1SCCh. 3.2 - Prob. 3-2SCCh. 3.2 - Prob. 3-3SCCh. 3.2 - Prob. 3-4SCCh. 3.2 - Prob. 3-5SCCh. 3.2 - Prob. 3-6SCCh. 3.2 - Prob. 3-7SCCh. 3.2 - Prob. 1SECh. 3.2 - Prob. 2SECh. 3.2 - Prob. 3SECh. 3.2 - Prob. 4SECh. 3.2 - Prob. 5SECh. 3.2 - Prob. 6SECh. 3.2 - Prob. 7SECh. 3.2 - Prob. 8SECh. 3.2 - Prob. 9SECh. 3.2 - Prob. 10SECh. 3.2 - Prob. 11SECh. 3.2 - Prob. 12SECh. 3.2 - Prob. 13SECh. 3.2 - Prob. 14SECh. 3.2 - Prob. 15SECh. 3.2 - Prob. 16SECh. 3.2 - Prob. 17SECh. 3.2 - Prob. 18SECh. 3.2 - Prob. 19SECh. 3.2 - Prob. 20SECh. 3.2 - Prob. 21SECh. 3.2 - Prob. 22SECh. 3.2 - Prob. 23SECh. 3.2 - Prob. 24SECh. 3.2 - Prob. 25SECh. 3.2 - Prob. 26SECh. 3.2 - Prob. 27SECh. 3.2 - Prob. 28SECh. 3.2 - Prob. 29SECh. 3.2 - Prob. 30SECh. 3.2 - Prob. 31SECh. 3.2 - Prob. 32SECh. 3.2 - Prob. 33SECh. 3.2 - Prob. 34SECh. 3.2 - Prob. 35SECh. 3.2 - Prob. 36SECh. 3.2 - Prob. 37SECh. 3.2 - Prob. 38SECh. 3.3 - Prob. 1-1SCCh. 3.3 - Prob. 1-2SCCh. 3.3 - Prob. 1-3SCCh. 3.3 - Prob. 1-4SCCh. 3.3 - Prob. 1-5SCCh. 3.3 - Prob. 2-1SCCh. 3.3 - Prob. 2-2SCCh. 3.3 - Prob. 2-3SCCh. 3.3 - Prob. 2-4SCCh. 3.3 - Prob. 2-5SCCh. 3.3 - Prob. 2-6SCCh. 3.3 - Prob. 1SECh. 3.3 - Prob. 2SECh. 3.3 - Prob. 3SECh. 3.3 - Prob. 4SECh. 3.3 - Prob. 5SECh. 3.3 - Prob. 6SECh. 3.3 - Prob. 7SECh. 3.3 - Prob. 8SECh. 3.3 - Prob. 9SECh. 3.3 - Prob. 10SECh. 3.3 - Prob. 11SECh. 3.3 - Prob. 12SECh. 3.3 - Prob. 13SECh. 3.3 - Prob. 14SECh. 3.3 - Prob. 15SECh. 3.3 - Prob. 16SECh. 3.3 - Prob. 17SECh. 3.3 - Prob. 18SECh. 3.3 - Prob. 19SECh. 3.3 - Prob. 20SECh. 3 - Prob. 1ESCh. 3 - Prob. 2ESCh. 3 - Prob. 3ESCh. 3 - Prob. 4ESCh. 3 - Prob. 5ESCh. 3 - Prob. 6ESCh. 3 - Prob. 7ESCh. 3 - Prob. 8ESCh. 3 - Prob. 9ESCh. 3 - Prob. 10ESCh. 3 - Prob. 11ESCh. 3 - Prob. 12ESCh. 3 - Prob. 13ESCh. 3 - Prob. 14ESCh. 3 - Prob. 15ESCh. 3 - Prob. 16ESCh. 3 - Prob. 17ESCh. 3 - Prob. 18ESCh. 3 - Prob. 19ESCh. 3 - Prob. 20ESCh. 3 - Prob. 21ESCh. 3 - Prob. 22ESCh. 3 - Prob. 23ESCh. 3 - Prob. 24ESCh. 3 - Prob. 25ESCh. 3 - Prob. 26ESCh. 3 - Prob. 27ESCh. 3 - Prob. 28ESCh. 3 - Prob. 29ESCh. 3 - Prob. 30ESCh. 3 - Prob. 31ESCh. 3 - Prob. 32ESCh. 3 - Prob. 33ESCh. 3 - Prob. 34ESCh. 3 - Prob. 35ESCh. 3 - Prob. 36ESCh. 3 - Prob. 37ESCh. 3 - Prob. 38ESCh. 3 - Prob. 39ESCh. 3 - Prob. 40ESCh. 3 - Prob. 41ESCh. 3 - Prob. 42ESCh. 3 - Prob. 43ESCh. 3 - Prob. 44ESCh. 3 - Prob. 45ESCh. 3 - Prob. 46ESCh. 3 - Prob. 47ESCh. 3 - Prob. 48ESCh. 3 - Prob. 49ESCh. 3 - Prob. 50ESCh. 3 - Prob. 51ESCh. 3 - Prob. 52ESCh. 3 - Prob. 53ESCh. 3 - Prob. 54ESCh. 3 - Prob. 55ESCh. 3 - Prob. 56ESCh. 3 - Prob. 57ESCh. 3 - Prob. 58ESCh. 3 - Prob. 59ESCh. 3 - Prob. 60ESCh. 3 - Prob. 61ESCh. 3 - Prob. 62ESCh. 3 - Prob. 63ESCh. 3 - Prob. 64ESCh. 3 - Prob. 65ESCh. 3 - Prob. 66ESCh. 3 - Prob. 67ESCh. 3 - Prob. 68ESCh. 3 - Prob. 69ESCh. 3 - Prob. 70ESCh. 3 - Prob. 71ESCh. 3 - Prob. 72ESCh. 3 - Prob. 73ESCh. 3 - Prob. 74ESCh. 3 - Prob. 75ESCh. 3 - Prob. 76ESCh. 3 - Prob. 77ESCh. 3 - Prob. 78ESCh. 3 - Prob. 79ESCh. 3 - Prob. 80ESCh. 3 - Prob. 1PTCh. 3 - Prob. 2PTCh. 3 - Prob. 3PTCh. 3 - Prob. 4PTCh. 3 - Prob. 5PTCh. 3 - Prob. 6PTCh. 3 - Prob. 7PTCh. 3 - Prob. 8PTCh. 3 - Prob. 9PTCh. 3 - Prob. 10PTCh. 3 - Prob. 11PTCh. 3 - Prob. 12PTCh. 3 - Prob. 13PTCh. 3 - Prob. 14PTCh. 3 - Prob. 15PTCh. 3 - Prob. 16PTCh. 3 - Prob. 17PTCh. 3 - Prob. 18PTCh. 3 - Prob. 19PTCh. 3 - Prob. 20PTCh. 3 - Prob. 21PTCh. 3 - Prob. 22PTCh. 3 - Prob. 23PTCh. 3 - Prob. 24PTCh. 3 - Prob. 25PTCh. 3 - Prob. 26PTCh. 3 - Prob. 27PTCh. 3 - Prob. 28PTCh. 3 - Prob. 29PTCh. 3 - Prob. 1CTCh. 3 - Prob. 2CTCh. 3 - Prob. 3CTCh. 3 - Prob. 4CTCh. 3 - Prob. 5CTCh. 3 - Prob. 6CTCh. 3 - Prob. 7CTCh. 3 - Prob. 8CTCh. 3 - Prob. 9CTCh. 3 - Prob. 10CTCh. 3 - Prob. 1CPCh. 3 - Prob. 2CPCh. 3 - Prob. 1CS1Ch. 3 - Prob. 2CS1Ch. 3 - Prob. 3CS1Ch. 3 - Prob. 4CS1Ch. 3 - Prob. 1CS2Ch. 3 - Prob. 2CS2
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- Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F(t) = 100 cos(8t) Newtons. Solve the initial value problem. x(t) = help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0 t→∞ ? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t)≈ x sp(t) = help (formulas) Book: Section 2.6 of Notes on Diffy Qsarrow_forwardConsider the initial value problem mx" cx' + kx F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 80 Newtons per meter, and F(t) = 20 sin(6t) kilograms, c = 8 kilograms per second, k = Newtons. Solve the initial value problem. x(t) = help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0 0047 ? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) ≈ x sp(t) = ☐ help (formulas) Book: Section 2.6 of Notes on Diffy Qsarrow_forwardConsider the differential equation y' = - 4xy with initial condition y(0) = 1.9. Recall that Runge-Kutta method has the following formula for computing the next step, where h is the step size: f(xi, Yi) = fx i + (++) k1 = h k2 2 ¯‚ Yi + k₁ h h k3 = fxi 2 `, Yi + k₂· 2 k4 = f(xi+h, yikзh) i+1=i+h k12k22k3 + k4 Yi+1 Yi + h 6 Using Runge-Kutta step size h = 0.4: Estimate y(0.4) ≈ help (numbers) Estimate y(0.8) ≈ help (numbers) Book: Section 1.7 of Notes on Diffy Qsarrow_forward
- Determine which differential equation corresponds to each phase diagram. You should be able to state briefly how you know your choices are correct. х x 4 4 4 4 3 3 3 3 2 2 2 2 dx ? ✰ dt = 1. = x² - 3x 1 1 1 1 ? ◇ 2. dx dt = x(x − 2) - 0 0 0 0 ? ◇ 3. dx dt = x(2 − x)² -1 -1 -1 -1 Q -2 -2 -2 dx ? ◇ 4. ༤་ dt = = 3x - x² -3 -3 -3 -3 x³- 4x = x²|x − 2| ? ◇ 5. ம் dx dt བི་ dx ? ◇ 6. dt ཝེ་ dx ? 7. dt ཝེ་ dx ? ◇ 8. ཝེ་ dt -4 -4 -4 -4 A B 0 D = = 2x = x² * x * * x * K 4 4 4 4 = 4x - x³ 3 3 3 • 3 Book: Section 1.6 of Notes on Diffy Qs dit for this problem 2 2 2 2 1 1 1 1 0 0 0 8 -1 -1 -1 -1 N 心 -2 -2 -3 -3 -3 -4 -4 -4 -4 E FL G Harrow_forwardDear expert Chatgpt gives wrong answer Plz don't use chat gptarrow_forwardAn improved method that is similar to Euler's method is what is usually called the Improved Euler's method. It works like this: Consider an equation y' = f(x, y). From (xn, Yn), our approximation to the solution of the differential equation at the n-th stage, we find the next stage by computing the x-step Xn+1 = xn +h, and then k1, the slope at (xn, Yn). The predicted new value of the solution . İs Zn+1 = Yn + h · k₁. Then we find the slope at the predicted new point k₁ = f(xn+1, Zn+1) and get the corrected point by averaging slopes h Yn+1 = = Yn + 1½ ½ (k1 + k₂). Suppose that we use the Improved Euler's method to approximate the solution to the differential equation dy dx = x - 0.5y, y(0.5) = 9. We let xo = 0.5 and yo 9 and pick a step size h = 0.25. Complete the following table: n xn Yn k1 Zn+1 k₂ 0 0.59-48 -3.25 ♡ <+ help (numbers) The exact solution can also be found for the linear equation. Write the answer as a function of x. y(x) = = help (formulas) Thus the actual value of the…arrow_forward
- Already got wrong Chatgpt answer If ur also Chatgpt user leave itarrow_forwardThe graph of the function f(x) is 1,0 (the horizontal axis is x.) Consider the differential equation x' = f(x). List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equalibria are stable, semi-stable (stable from one side, unstable from the other), or unstable. x = help (numbers) x = help (numbers) x = help (numbers) x = help (numbers) Book: Section 1.6 of Notes on Diffy Qsarrow_forward= A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 60 cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. Determine the spring constant k. k = Newtons/meter help (numbers) Hint is to use earth gravity of 9.8 meters per second squared, and note that Newton is kg meter per second squared. Formulate the initial value problem for x(t), where x(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. Give your answer in terms of x, x',x",t. Differential equation: | help (equations) Initial conditions: x (0) = and '(0) = help (numbers) Solve the initial value problem for x(t). x(t) = ☐ help (formulas) Plot the solution and determine the maximum displacement from equilibrium made by the object on the…arrow_forward
- Suppose f(x) is a continuous function that is zero when x is −1, 3, or 6 and nowhere else. Suppose we tested the function at a few points and found that ƒ(−2) 0, and f(7) < 0. Let x(t) be the solution to x' f(x) and x(0) = 1. Compute: lim x(t) help (numbers) t→∞ Book: Section 1.6 of Notes on Diffy Qsarrow_forwardConsider the initial value problem У y' = sin(x) + y(-4) = 5 4 Use Euler's Method with five steps to approximate y(-2) to at least two decimal places (but do not round intermediate results). y(-2) ≈ help (numbers) Book: Section 1.7 of Notes on Diffy Qsarrow_forwardConsider the differential equation y' = 5y with initial condition y(0) : The actual solution is y(1) = 207.78 help (numbers) = 1.4. We wish to analyze what happens to the error when estimating y(1) via Euler's method. Start with step size h = 1 (1 step). Compute y(1) Error 8.4 help (numbers) 199.38 help (numbers) Note: Remember that the error is the absolute value! Let us half the step size to h = 0.5 (2 steps). Compute y(1) ≈ 17.15 help (numbers) Error = 190.63 help (numbers) The error went down by the factor: Error Previous error Let us half the step size to h = 0.25 (4 steps). Compute y(1) 35.88046875 help (numbers) Error = 171.90 help (numbers) help (numbers) The error went down by the factor: Error Previous error help (numbers) Euler's method is a first order method so we expect the error to go down by a factor of 0.5 each halving. Of course, that's only very approximate, so the numbers you get above are not exactly 0.5. Book: Section 1.7 of Notes on Diffy Qsarrow_forward
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