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mathematics
calculus

Questions and Answers of Calculus

(a) Use the binomial series to expand 1/√1 + x2. (b) Use part (a) to find the Maclaurin series for sinh –1x.
(a) Expand 3√1 + x as a power series. (b) Use part (a) to estimate correct to four decimal places.
(a) Expand 1/4√1 + x as a power series. (b) Use part (a) to estimate 1/4√1 + x correct to three decimal places.
(a) Find the Taylor polynomials up to degree 6 for f(x) = cos x centered at a = 0. Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at x = π/4, π/2,
(a) Find the Taylor polynomials up to degree 3 for f(x) = 1/x centered at a = 1. Graph f and these polynomials on a common screen.(b) Evaluate f and these polynomials at x = 0.9 and 1.3.(c) Comment
Use a computer algebra system to find the Taylor polynomials Tn at a = 0 for the given values of n. Then graph these polynomials and f on the same screen.
Use the information from Exercise 5 to estimate sin 35o correct to five decimal places.
Use the information from Exercise 16 to estimate cos 69o correct to five decimal places.
Use Taylor’s Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.
How many terms of the Maclaurin series for In (1 + x) do you need to use to estimate In 1.4 to within 0.001?
Use the Alternating Series Estimation Theorem or Taylor€™s Inequality to estimate the range of values of for which the given approximation is accurate to within the stated error. Check your
A car is moving with speed 20 m/s and acceleration 2 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use
The resistivity p of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (Ω-m). The resistivity of a given metal depends on the temperature according
An electric dipole consists of two electric charges of equal magnitude and opposite signs. If the charges are and €“ q and are located at a distance from each other, then the electric field
(a) Derive Equation 3 for Gaussian optics from Equation 1 by approximating cos Φ in Equation 2 by its first-degree Taylor polynomial. (b) Show that if cos Φ is replaced by its third-degree
If a water wave with length L moves with velocity across a body of water with depth d, as in the figure, then(a) If the water is deep, show that v ≈ √gL / (2π).(b) If the water is
If a surveyor measures differences in elevation when making plans for a highway across a desert, corrections must be made for the curvature of the Earth.(a) If R is the radius of the Earth and L is
Show that Tn and f have the same derivatives at up to order n.
In Section 4.9 we considered Newton’s method for approximating a root r of the equation f(x) = 0, and from an initial approximation x1 we obtained successive approximations x2, x3, . . . , where
Suppose Σ an = 3 and sn is the nth partial sum of the series. What is lim n→∞ an? What is lim n→∞ sn?
(a) What is an absolutely convergent series?(b) What can you say about such a series?(c) What is a conditionally convergent series?
(a) Write the general form of a power series.(b) What is the radius of convergence of a power series?(c) What is the interval of convergence of a power series?
(a) Write an expression for the th-degree Taylor polynomial of f centered at a.(b) Write an expression for the Taylor series of f centered at a.(c) Write an expression for the Maclaurin series of f.
An ODE may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution.The ODE y’2 – xy’ + y = 0 is of the kind. Show by
If in Prob.17 the stone starts at t = 0 from initial position y0 with initial velocity v = v0, show that the solution is y = gt2/2 + v0t + y0. How long does a fall of 100 m take if the body falls
The efficiency of the engines of subsonic airplanes depends on air pressure and usually is maximum near about 36000ft. Find the air pressure y(x) at this height without calculation. Physical
Show by algebra that the investment y(t) from a deposit y0 after t years at an interest rate r is y0(t) = y0[1 + r]2 (Interest compounded annually) y0(t) = y0[1 + (r/365)]365t (Interest compounded
Velocity equal to the reciprocal of the distance, y(1) = 1
Velocity plus distance equal to the square of time, y (0) = 6.
Discuss direction fields as follows.(a) Graph a direction field for the ODE y’ = 1 – y and in it the solution satisfying y(0) = 5 showing exponential approach. Can you see the limit of any
Introduce limits of integration in (3) such that y obtained from (3) satisfies the initial condition y(x0) = y0. Try the formula out on Prob. 19.
Show that any (nonvertical) straight line through the origin of the xy-plane interects all solution curves of y' = g(y/x) at the same angle.
If in a population of bacteria the birth rate and death rate are proportional to the number of individuals present, what is the population as a function of time? Figure out the limiting situation for
The Gompertz model is y' = – Ay In y (A > 0), where y(t) is the mass of tumor cells at time t. The model agrees well with clinical observations. The declining growth rate with increasing y > 1
Jack, arrested when leaving a bar, claims that he has been inside for at least half and hour (which would provide him with an alibi) the police check the water temperature of his car (parked near
How does the answer Example 5 (the time when the tank is empty) change if the diameter of the hole is doubled? First guess.
To tie a boat in a harbor, how many times must a rope be wound around a bollard (a vertical rough cylindrical post fixed on the ground) so that a man holding one end of the rope can resist a force
Suppose that the tank is Example 5 is hemispherical, of radius R, initially full of water, and has an outlet of 5 cm2 cross-sectional area at the bottom. (Make a sketch). Set up the model for
Graph particular solutions of the following ODE, proceeding as explained. (21) y cos x dx + 1/y dy = 0.(a) Test for exactness. If necessary, find an integrating factor. Find the general solution u(x,
Show this as indicated. Compare the amount of work. (a) ey (sin h x dx + cos h x dy) = 0 as an exact ODE and by separation.(b) (1 + 2x) cos y dx + dy/cos y = 0 by Theorem 2 and by separation.(c) (x2
A tank (as in Fig. 9 in Sec. 1.3) contains 1000 gal of water in which 200lb of salt is dissolved. 50 gal of brine, each gallon containing (1 + cos t) lb of dissolved salt, runs into the tank per
Heating and cooling of a building can be modeled by the ODE T' = k1(T – Ta) + k2(T – Tw) + P, where T = T (t) is the temperature in the building at time t, Ta the outside temperature, Tw the
A model for the spread of contagious diseases is obtained by assuming that the rate of spread is proportional to the number of contacts between infected any non-infected persons, who are assumed to
Suppose that the population y(t) of a certain kind of fish is given by the logistic equation (8), and fish are caught at a rate Hy proportional to y. Solve this so-called Schaefer model. Find the
In Prob. 32 assume that you fish for 3 years then fishing is banned for the next 3 years. Thereafter you start again. And so on. This is called intermittent harvesting. Describe qualitatively how the
Do you save work in Prob. 43 if you first transform the ODE to a linear ODE? Do this transformation. Solve the resulting ODE. Does the resulting y(t) agree with that in Prob. 34?
Y = 0 (that is, y(x) = 0 for all x, also written y(x) ≡ 0) is a solution of (2) [not of (1) if r(x) ≠ 0!], called the trivial solution.
The difference to two solutions of (1) is a solution of (2).
If y1 and y2 are solutions of y'1 + py1 = r1 and y'2 + py2 = r2, respectively (with the same p!), what can you say about the sum y1 + y2?
Show that if a family is given as g(x, y) = c, then the orthogonal trajectories can be obtained from the following ODE, and use the latter to solve Prob.6 written in the form g(x, y) =c.
Suppose that the streamlines of the flow (paths of the particles of the fluid) in Fig. 24 are ? (x, y) = xy = const. Find their orthogonal trajectories (called equipotential lines, for reasons given
The lines of electric force of two opposite charges of the same strength at (?? 1, 0) and (1, 0) are the circles through (?? 1, 0) and (1, 0). Show that these circles are given by x2 + (y ?? c)2 = 1
The lines of electric force of two opposite charges of the same strength at (?? 1, 0) and (1, 0) are the circles through (?? 1, 0) and (1, 0). Show that these circles are given by x2 + (y ?? c)2 = 1
Does the initial value problem (x – 1) y' = 2y, y(1) = 1 have solution? Does your result contradict our present theorems?
What happens in Prob. 2 if you replace y(1) = 1 with y(1) = k?
Find all initial conditions such that (x2 – 4x)y' = (2x – 4)y has no solution, precisely one solution, and more than one solution.
Find all initial conditions such that (x2 – 4x)y' = (2x – 4)y has no solution, precisely one solution, and more than one solution. Discuss.
A thermometer showing 10oC is brought into a room whose temperature is 25oC. After 5 minutes it shows 20oC. When will the thermometer practically reach the room temperature, say, 24.9oC?
In a room containing 20 000 ft3 of air, 600ft3 of fresh air flows in per minute, and the mixture (made practically uniform by circulating fans) is exhausted at a rate of 600 cubic feet per minute
In a bimolecular reaction A + B → M, a moles per liter of a substance A and b moles per liter of a substance B are combined. Under constant temperature the rate of reaction is y' = k(a – y)
Find all curves in the first quadrant of the xy-plane such that for every tangent, the segment between the coordinate axe s is bisected by the point of tangency. (Make a sketch).
If 10% of a radioactive substance if after 5 days, 0.020g is present and after 10 days, 0.015g
It can be shown that the curve y(x) of an inextensible flexible homogenouse cable hanging between two fixed points is obtained by solving y" = k √1 + y'2, where the constant k depends on the
(a) Coefficient formulas, show how a and b in (1) can be expressed in terms of λ1 and λ2. Explain how these formulas can be used in constructing equations for given bases. (b) Root zero,
If D2 + aD + bl has distinct roots µ and λ, show that a particular solution is y = (eµx – eλx) / (µ – λ). Obtain from this a solution xeλx by letting µ → λ and
Show that the definition of linearity in the text is equivalent to the following. If L[y] and L[w] exist, then L[y + w] exists and L[cy] and L[y + w] = L[y] + L[w] as well as L[cy] = cL[y] and L[kw]
Find the frequency of vibration of a ball of mass m = 3 kg on a spring of modulus (i) k1 = 27 nt/m, (ii) k2 = 75 nt/m, (iii) on these springs in parallel (see Fig. 40), (iv) in series, that is, the
What is the frequency of a harmonic oscillation if the static equilibrium position of the ball is 10 cm lower than the lower end of the spring before the ball is attached?
This principle states that the buoyancy force equals the weight of the water displaced by the body (partly or totally submerged). The cylindrical buoy of diameter 60cm in Fig. 42 is floating in water
Different physical or other systems may have the same or similar models, thus showing the unifying power of mathematical methods. Illustrate this for the system in Figs. 43-45(c) Water in a tube
(Beats) How does the graph of the solution in Prob. 20 change if you change (a) y (0), (b) The frequency of the driving force?
Show that the ratio of two consecutive maximum amplitudes of a damped oscillation (10) is constant, and the natural logarithm of this ratio, called the logarithmic decrement, equals ∆ =
Consider an under damped motion of a body of mass m = 2kg. If the time between two consecutive maxima is 2sec and the maximum amplitude decreases to ¼ of its initial value after 15 cycles, what is
Find the over damped motion (7) that starts from y0 with initial velocity v0.
Study this transition in terms of graphs of typical solutions (Cf. Fig. 46)(a) Avoiding unnecessary generality is part of good modeling. Decide that the initial value problems (A) and (B).(A) y" +
Study this transition in terms of graphs of typical solutions (Cf. Fig. 46)(a) Avoiding unnecessary generality is part of good modeling. Decide that the initial value problems (A) and (B).(A) y" +
This concerns some noteworthy general properties of solutions. Assume that the coefficients p and q of the ODE (1) are continuous on some open interval I, to which the subsequent statements refer.(a)
Team project extensions of the Method of Undetermined Coefficients(a) Extend the method to products of the function in Table 2.1 (b) Extend the method to Euler-Cauchy equations. Comment on the
Solve this models an un-damped system on which a force F acts during some interval of time (see Fig. 59), for instance, the force on a gun barrel when a shell is fired, the barrel being braked by
Solve Prob.1 when E = E0 sin wt and R, L, E0, w are arbitrary. Sketch a typical solution
Find the current in the RC-circuit in Fig. 66 with E = E0 sin wt and arbitrary R, C, E0, andw.
Find the current when L = 0.5H, C = 8 ∙ 10–4 F, E = t2 V and initial current and charge zero.
Transient current probe the claim in the text that if R ≠ 0 (hence R > 0), then the transient current approaches Ip as t → ∞.
Find the steady-state current in the RLC-circuit in Fig. 60 on p. 92 for the given data. (Show the details of your work. R = 8 ?, L = 0.5 H, C = 0.1 F, E = 100 sin 2tV
Find the steady-state current in the RLC-circuit in Fig. 60 on p. 92 for the given data. (Show the details of your work.R = 2 ?, L = 1 H, C = 0.05 F, E = 157/9 sin 3tV
Find the transient current (a general solution) in the RLC-circuit in Fig. 60 for the given data. (Show the details of your work.)R = 0.2 ?, L = 0.1 H, C = 2 F, E = 754 sin 0.5t V
Solve the initial value problem for the RLC-circuit in Fig.60 with the given data, assuming zero initial current and charge. Graph or sketch the solution (Show the details of your work.)R = 4 ?, L =
Solve the initial value problem for the RLC-circuit in Fig.60 with the given data, assuming zero initial current and charge. Graph or sketch the solution (Show the details of your work.) R = 3.6 ?,
Find the steady-state solution of the system in Fig. 70 when m = 4, c = 4, k = 17 and the driving force is 202 cos3t.
In Prob.26 find the solution corresponding to initial displacement 10 and initial velocity 0.
In Fig. 70 let m = 2, c = 6, k = 27, and r(t) = 10 cos wt. For what w will you obtain the steady-state vibration of maximum possible amplitude? Determine this amplitude. Then use this w and the
Find the current in the RLC-circuit in Fig. 71 when L = 0.1 H, R = 20 ?, C = 2 ? 10??4 F, and E (t) = 110 sin 415t V (66 cycles/sec.)
Find a particular solution in Prob. 33 by the complex method.
(a) Investigate the given question about a set S of functions on an interval I. Give an example. Prove your ansnwer.(1) If S contains the zero functions, can S be linearly independent?(2) If S is
(a) Investigate the given question about a set S of functions on an interval I. Give an example. Prove your ansnwer.(1) If S contains the zero functions, can S be linearly independent?(2) If S is
Cas Experiment, Undetermined Coefficients Since variation of parameters is generally complicated; it seems worthwhile to try to extend the other method. Find out experimentally for what ODEs this is
What happens in Example 1 if we replace T2 by a tank containing 500 gal of water and 150lb of fertilizer dissolved in it?
In Example 1 find a “general solution” for any ratio a = (flow rate) (tank size), tank sized being equal, Comment on the result.
Find a “general solution” of the system in Prob. 5.
Find the currents in Example 2 if the resistance of R1 and R2 is doubled (general solution only). First, guess.

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