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Questions and Answers of Calculus

Find the current in Example 2 if the capacitance is changed to C = 1/5.4 F (Farad).
(a) Set up the model for the (un-damped) system in Fig. 80(b) Solve the system of ODEs obtained.(c) Describe the influence of initial conditions on the possible kind ofmotions.
Each of the two tanks contains 400 gal of water, in which initially 100 lb (Tank T1) and 40 lb (Tank T2) of fertilizer are dissolved. The inflow, circulation, and outflow are shown in Fig. 87. The
Transformation of variable what happens to the system (1) and its critical point if you introduce τ = – t as a new independent variable?
Perturbation of center if a system has a center as its critical point, what happens if you replace the matric A by Ã = A + kI with any real number k ≠ 0 (representing measurement erros in the
Write the ODE y'' – 4y + y3 = 0 as a system, solve it for y2 as a function of y1, and sketch or graph some of the trajectories in the phase plane.
In Prob 14 add a linear damping term to get y'' + 2y' – 4y + y3 = 0, Using arguments from mechanics and comparison with Prob. 14, as well as with Examples 1 and 2, guess the type of each critical
What is the essential difference between a limit cycle and a closed trajectory surrounding a center?
(a) Van der Pol Equation. Determine the type of the critical point at (0, 0) when µ > 0, µ = 0, µ > 0. Show that if µ → 0, the isoclines approach straight lines through the origin. Why is
Collect Maclaurin series of the function known from calculus and arrange them systematically in a list that you can use for your work.
Show that by Kirchhoff??s voltage law (Sec. 2.9), the currents in the network in Fig. 150 are obtained from the system Li'1 + T(i1 ?? i2) = v(t) R(i'2 ?? i'1) + 1/C i2 = 0 Solve this system, where R
Find and graph the charge q(t) and the current i(t) in the LC-circuit in Fig. 148, where L = 0.5H, C = 0.02 F, v(t) = 1425 sin 5t V if 0 ?, and current and charge at t = 0 are 0.
Solve the model in Example 3 with k = 4 and initial conditions y1 (0) = 1, y'1 (0) = 1, y2 (0)1, y'2 (0) = – 1 under the assumption that the force 11 sin t is acting on the first body and the force
What will happen in Example 1 if you double all flows (in particular, an increase to 12 gal/min containing 12lb of salt from the outside), leaving the size of the tanks and the initial conditions as
Solve Prob. 25 when the EMF (electromotive force) is acting from 0 to 2π only. Can you do this just by looking at Prob. 25, practically without calculation?
Team Project Useful Formulas for Two and more Vectors, prove (12) ?? (16), which are often useful in practical work, and illustrate each formula with tow examples. Show that each side of (13) then
Find the plane through (2, 1, 3), (4, 4, 5), (1, 6, 0).
Find the volume of the tetrahedron with vertices (0, 2, 1), (4, 3, 0) (6, 6, 5), (4, 7, 8).
Find the first and second derivatives of [4 cos t, 4 sin t, 2t].
Find the first partial derivatives of [sin x soch y, cos x sinh] and [ex cos y, ex sin y].
Hypocycloid r(t) = [a cos3 t, a sin3 t], total length
Velocity and Acceleration, Forces on moving objects (cars, airplanes, etc) require that the engineer knows corresponding tangential and normal acceleration. Find them, along with the velocity and
Velocity and Acceleration, Forces on moving objects (cars, airplanes, etc) require that the engineer knows corresponding tangential and normal acceleration. Find them, along with the velocity and
Find the centripetal acceleration of the moon toward the earth, assuming that the orbit of the moon is a circle of radius 239,000 miles = 3.85 ∙ 108 m, and the time for one complete revolution
A satellite moves in a circular orbit 450 miles above the earth’s surface and completes 1 revolution in 100 min. Find the acceleration of gravity at the orbit from these data and from the radius of
Using (22), show that if C is represented by r(t) with arbitrary t,then
Using b = y x p and (23), show that (23**) τ(s) = (u p p') = r' r'' r''')/k2 (k > 0).
Show that if C is represented by r(t) with arbitrary parameter t, then assuming k > 0 asbefore.
Show that the helix [a cos t, a sin t, ct] can be represented by [a cos (s/K), a sin (s/K), cs/K], where K = √a2 + c2 and s is the arc length. Show that it has constant curvature k = a/K2 and
Show that u' = κp, p' = – κu + τb, b' = – τp.
Project, Useful Formulas for Gradients and Laplacians, prove the following formulas and give for each of them two examples showing when they areadvantageous.
Let v = [x, y, v3]. Find a v3 such that (a) div v > 0 everywhere, (b) div v > 0 if |z| < 1 and div v < 0 if |z| > 1.
Consider the flow with velocity vector v = xi. Show that the individual particles have the position vectors r(t) = c1eti + c2j + c3k with constant c1, c2, c3. Show that the particles that at t = 0
Find an equation of the plane through (1, 0, 2), (2, 3, 5), (3, 5, 7).
Find the velocity, speed, and acceleration of the motion given by r(t) = [5 cos t, sin t, 2t) at the point P; [5/√2, 1/√2, π/2]. What kind of curve is the path?
(Harmonic functions) Verify Theorem 1 for f = y2 – x2 and the surface of the cylinder x2 + y2 < z < 5.
(Green’s first formula) Verify (8) for f = x, g = y2 + z2, S the surface of the box 0 < x < 1, 0 < y < 2, 0 < z < 3.
(Green’s second formula) Verify (9) for f = x4, g = y2 and the cube in Prob. 1.
The tetrahedron cut form the first octant by the plane 3/2x + 2y + z = 6. Check by vector methods.
Calculate this line integral by Stokes’s theorem, clockwise as seen by a person standing at the origin, for the following F and C. Assume the Cartesian coordinates to be right-handed. (Show the
Using (9), find a bound for the absolute value of the work W done by the force F = [x2, y] in the displacemenet along the segment from (0, 0) to (3, 4).
Condition (4) finds the points in Probs. 2 – 7 at which (4) N ≠ 0 does not hold and state whether this is owing to the shape of the surface or to the choice of the representation.
If IA is the moment of inertia of a mass distribution of total mass M with respect to an axis A through the center of gravity, show that its moment of inertia IB with respect to an axis B, which is
Let R and C be as in Green??s theorem, r' a unit tngent vector, and n the outer unit normal vector of C (Fig. 238 in Example 4) show that (1) may be written where k is a unit vector perpendicular to
Cas Experiment Graphing Write a program for graphing partial sums of the following series. Guess from the graph what f(x) the series may represent. Confirm or disprove your guess by using the
Project Euler formulas in Terms of Jumps withut Integration; show that for a function whose third derivative is identically zero, where n = 1, 2, ??? and we sum over all the jumps js, j's, j''s of f,
Integrate an d graph the integral of the product cos mx cos nx (with various integer m and n of your choice) from – a to a as a function of a and conclude orthogonality of cos mx and cos nx (m
Find the Fourier series of the function obtained by passing the voltage v(t) = V0 cos 100πt through a half-wave rectifier.
Obtain the series in Prob. 3 from that in Prob. 21 of Problem Set 11.1
The partial sums sn(x) of a Fourier series show oscillations near a discontinuity point. These oscillations do not disappear as n increases but instead become sharp "spikes". They were explained
Illustrate the formulas in the proof of Theorem 1 with examples. Prove the formulas.
(a) Are the following expressions even or odd? Sums and products of even functions and of odd functions, products of even times odd functions. Absolute values of odd functions f(x) + f(– x) and
Find the complex Fourier series of the following functions. (Show the details of your work). Convert the series in Prob. 8 to real form.
It is very interesting that the cn in (6) can be derived directly by a method similar to that for an and bn in Sec. 11.1. For this, multiply the series in (6) by e-imx with fixed integer m, and
What would happen in Example 1 if we replaced r(t) with its derivative (the rectangular wave)? What is the ratio of the new Cn to the old ones?
Find a general solution of the ODE y'' + w2y = r(t) with r(t) as given. (Show the details of yourwork.)
Find the steady-state oscillation of y'' + cy' + y = r(t) with c > 0 and r(t) as given. (Show the details of yourwork.)
Find the steady-state current I(t) in the RLC-circuit in Fig. 272, where R = 100?, L = 10H, C = 10??2 F and E(t) V as follows and periodic with period 2?. Sketch or graph the first four partial sums.
Compare the size of the minimum square error E* for functions of your choice. Find experimentally the factors on which the decrease of E* with N depends. For each function considered find the
Let f(x) = x if 0 < x < k, f(x) = 0 if x > k. Find fc(w).
Obtain formula 7 in Table III from formula 8.
Find the inverse Fourier cosine transform f(x) from the answer to Prob. 1.
Find the Fourier transform of f(x) (without using Table III in Sec. 11.10). Show thedetails.
Find the Fourier transform of f(x) = xe – x if x > 0 and 0 if x < 0 from formula 5 in Table III and (9) in the text.
Find the answer to Prob. 11 from (9b).
Let f(x) = sin x of 0 < x < π and 0 if x > π. Find ₣S (f). Compare with Prob. 6 in Sec. 11.7. Comment
Show that ₣S (x– ½) = w –½ by setting wx = t2 and using S(∞) = √π/8 in (38) of App. 3.1.
Show that ₣S (x – 3/2) = 2w½. Set wx = t2, integrate by parts, and use C(∞) = √π/8 in (38) of App. 3.1.
Obtain the result of Prob. 23 by applying Parseval's identity to Prob. 12.
Find the half-range sine series of f(x) = 0 if 0 < x < π/2, f(x) = 1 if π/2 < x < π. Compare with Prob. 12.
Show that among all rectangular membrances of the same area A = ab and the same c the square membrance is that for which u11 [see (10)] has the lowest frequency.
Changing end temperatures assume that the ends of the bar in Probs. 5 – 9 have been kept at 100oC for a long time. Then at some instant, call it t = 0, the temperature at x = L is suddenly changed
Find the temperature of the bar in Prob. 13 if the left end is kept at 0oC, the right end is insulated, and the initial temperature is U0 = const.
Bar with heat generation if heat is generated at a constant rate throughout a bar of length L = π with initial temperature f(x) and the ends at x = 0 and π are kept at temperature 0, the
Consider vt = c2uxx – v (0 < x < L, t > 0), v(0, t) = 0, v(L, t) = 0, v(x, 0) = f(x), where the term – v models heat transfer to the surrounding medium kept at temperature 0. Reduce this PDE by
Find the steady-state temperature in the plate in Prob. 31 if the lower side is kept at U0oC, the upper side at U1oC, and the other sides are kept at 0oC.
Find steady-state temperatures in the rectangle in Fig. 293 with the upper and left sides perfectly insulated and the right side radiating into a medium at 0oC according to ux(a, y) + hu(a, y) = 0 h
Find the potential exterior to the sphere in Example 2 of the text and in Prob. 15.
Show that because of the boundary conditions (2). Sec. 12.3, the function f in (13) of this section must be odd and of period 2L.
Tricomi and Airy equations2 show that the Tricomi equation yuxx + uvv = 0 is of mixed type. Obtain the Airy equation G'' – yG = 0 from the Tricomi equation by separation.
Find the type, transform to normal form, and solve, (Show the details of your work.) Uxx – 2uxy + uyy = 0
Find the type, transform to normal form, and solve, (Show the details of your work.) xuxy – yuyy = 0
Find the type, transform to normal form, and solve, (Show the details of your work.) uxx + 2uxy + 5uyy = 0
Find the type, transform to normal form, and solve, (Show the details of your work.) uxx + 4uxy + 3uyy = 0
Between two concentric spheres of radii r0 and r1 kept at the potentials u0 and u1, respectively.
In the interior of a sphere of radius 1 kept at the potential f(Ф) = cos 2 Ф + 3 cos Ф (referred to our usual spherical coordinates).
Find the surfaces on which the functions u1, u2, u3, are zero.
Let r, θ, Ф be spherical coordinates. If u(r, θ, Ф) satisfies ∆2u = 0, show that v(r, θ, Ф) = u(1/r, θ, Ф/r satisfies ∆2u = 0. What does this
Find the electronstatic potential in the semidisk r < 1, 0 < θ < π which equals 110 θ (π – θ) on the semicircle r = 1 and 0 on the segment – 1 < x < 1.
Why is A1 = A2 + ∙∙∙ = 1 in Example 1? Compute the first few partial sums untial you get 3-digit accuracy. What does this problem mean in the field of music?
Repeat the task in Prob. 23 with c2 = 1, for f(x, y) as in Prob. 22 and initial velocity 0.
Find the area of the surface generated by revolving the curve x = (ey + e(-y))/2 in the interval 0 ≤ y ≤ In 7 about the y-axis.
Find the area of the surface generated by revolving the curve y = √(2x - x2), 0.5 ≤ x ≤ 2, about the x-axis.
If f (x) = x + √2 - x and g(u) = u + √2 - u, is it true that f = g?
The graph shown gives the weight of a certain person as a function of age. Describe in words how this person's weight varies over time. What do you think happened when this person was 30 years old?
You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph
The graph shows the power consumption for a day in September in San Francisco. (is measured in megawatts; is measured in hours starting at midnight.)(a) What was the power consumption at 6 AM? At 6
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.
A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.

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