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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

A manufacturer produces an article in batches of N items. Each production run has a set-up cost of £100 and each item costs an additional £0.05 to produce. The weekly storage costs are a basic
The gravitational attraction at the point (x, y) in the x–y plane due to point masses in the plane isShow that G(x, y) has a stationary value of 9. 1 4 G(x, y) = = + - + x y 9 4--
Find constants a and b such thatis a minimum. 0 [sinx - (ax + bx)]-dx
A tank has the shape of a cuboid and is open at the top and has a volume of 4m3. If the base measurements (in m) are x by y, show that the surface area (in m2) is given byand find the dimensions of
A flat circular metal plate has a shape defined by the region x2 + y2 ≤ 1. The plate is heated so that the temperature T at any point (x, y) on it is given byFind the temperatures at the hottest
A metal channel is formed by turning up the sides of width x of a rectangular sheet of metal through an angle θ. If the sheet is 200mm wide, determine the values of x and θ for which the
Find the extremum of x2 – 2y2 + 2xy + 4x subject to the constraint 2x = y and verify that it is a maximum value.
Find the extremum of 3x2 + 2y2 + 6z2 subject to the constraint x + y + z = 1 and verify that it is a minimum value.
The equation 5x2 + 6xy + 5y2 – 8 = 0 represents an ellipse whose centre is at the origin. By considering the extrema of x2 + y2, obtain the lengths of the semi-axes.
Which point on the sphere x2 + y2 + z2 = 1 is at the greatest distance from the point having coordinates (1, 2, 2)?
Find the maximum and minimum values ofwhere (x, y) lies on the circle x2 + y2 + 2x + y = 1. f(x, y) = 4x + y +y
Obtain the stationary value of 2x + y + 2z + x2 – 3z2 subject to the two constraints x + y + z = 1 and 2x – y + z = 2.
Determine the position and nature of the stationary points on the surface z = e-(x+y)(3x + y)
A nonlinear spring has a restoring force which is proportional to the cube of the displacement x. The period T of oscillation from an initial displacement a is given byUse the substitution x2 = a2
1) The diagram below shows a lawn, which is rectangular with semi-circular ends, each of radius 8 m. A circular rose bed, with diameter 6 m has been planted in the middle of the lawn. Calculate(a)
Evaluate the following improper integrals: S. (a) (xlnx)dx 0 (b) (c) S. 0 x exp(-x)dx (d) xe-2 dx 0 S e exp(-e)dx (e) (f) (h) (i) S 0 x(1 - x)/dx So kla (x - 1)/x dx sin x cos x [ta cos x sin-1/3x dx
By means of sketches of the graphs y = 1/x and y = tan x, show that the equation x tan x = 1 has a root between x = 0 and x = 1/2π and an infinity of roots near x = kπ, where k = 1, 2, 3, . . . . .
If α = f(α) but the iteration xn+1 = f (xn) fails to converge to the root a, under what condition on f(x) will the iteration xn+1 = f−1(xn) converge?
Show the cubic equation x3 – 2x – 1 = 0 has a root near x = 2. Prove that the iterationfails to converge to that root. Devise a simple iteration formula for the root of the equation, and use it
The equation f(x) = 0 has a root at x = α. Show that rewriting the equation as x = x + λf(x), where λ is a constant, yields a convergent iteration for a if λ = –1/f´(x0) and x0 is sufficiently
Consider the iteration defined byShow that(a) if 0 0 (b) if x0 > 1 then the iteration is divergent. Explain this behaviour. (7 + x) = +x (z
Consider the iterationWorking to 2dp, obtain the first three iterates. Then continue to obtain the following six iterates. From the numerical evidence what do you estimate as the limit of the
Show that if f(x) = ecos x thenand find f(0) and f´(0). Differentiating the expression for f´(x), obtain f"(x) in terms of f(x) and f´(x), and find f"(0). Repeating the process, obtain f(n)(0) for
A function y = y(x) satisfies the equationwith y = 1 when x = 0. By repeated differentiation, show that y(n)(0) = 1 (n ≥ 2), and find the Maclaurin series for y. dy dx =y=x+1
An alternative approach to Question 9 uses the method of successive approximation, rewriting the equation asPutting y0(x) = 1 into the integral, show thatand find y3 and y4. Yn+1(x) = 1+ [y(t)- t +
Show that the binomial expansion of (1 + x)−1 isHence find the Maclaurin series expansion of tan−1x. (1+x)=1-x+x-x+... (-1
Use the series for sin x and cos x to obtain the Maclaurin series for tan x as far as the term in x7. Deduce the series for ln cos x.
Show that coth x = (1 + x = 1/3 x + 25x6 - ...) 945 X
The field strength H of a magnet at a point on the axis at a distance x from its centre is given bywhere 2l is the length of the magnet and M is its moment. Show that if l is very small compared with
Using the Maclaurin series expansions of ex and cos x, show that e* + e*- 2 lim xo 2 cos 2x -2 -1
Show thatif powers of x greater than x5 are neglected. In sin x X =-=-x-x 180
By expanding e−x2 as a Maclaurin series, show that 1/2 edx 0.461
Considering the problem of Example 9.13, for what values of l does the approximationhave a percentage error of less than 0.05% when R = 5 and r = 4?Data from Example 9.13The continuous belt has
Using L’Hôpital’s rule, find the following limits: (a) lim x 3x-2 x-8 sin 3x * sin 2x (c) lim- x cos x sin x x (e) lim- x-0 (b) lim 3 xx-1 (d) lim 1- (1 - x)/4 X 1 - sin x x-T/2 In sin x (f)
Consider again the design of the milk carton discussed in Example 8.35. Show that if the overlap used in its construction is x mm instead of 5 mm, the objective function that must be minimized isShow
A table for ex is required for use with linear interpolation to 6dp. It is tabulated for values of x from x = 0 to x = X at intervals of 0.001. What is the largest possible value of X?
A table for tan x is required for use with linear interpolation to 6dp. It is tabulated for values of x from x = 0 to x = 1 at intervals of h rad. What is the largest possible value of h?
We discussed the process of numerical differentiation using the approximationUsing the Taylor series for f(a + h) and f(a – h) about x = a, show thatto f´(a) with truncation error O(h6). Apply
Given below are three methods for calculating √2 by iteration. Find the order of each process and discuss their numerical properties. "1 (c) xn+1 = (3x + 12x2 - 4)/(8x) ("x/7 + "x) = +x (q) (x +
Use the Newton–Raphson iteration procedure to find the real root of x3 – 6x2 + 9x + 1 = 0 to 4dp.
Use the Newton–Raphson method to find the two positive roots of x4 – 4x3 – 12x2 + 32x + 28 = 0.
The iteration may be used to calculate the reciprocal of a; that is, to solve ax = 1. Show that this is a third-order process with Apply the iteration with a = 1.735, starting with x0 = 0.5, and
Simpson’s rule for the numerical evaluation of an integral iswhere n is an even number. The global truncation error isIf f(x) = ln cosh x and a = 0, b = 0.5, show that |f(4)(x)| 4). If f(x) is
(a) Use the trapezium rule with h = 0.25 to evaluate ∫10√x dx. Compare your answer with the exact value, 2/3 .(b) Put x = t2 in the integral and again evaluate it using the trapezium rule with
The trapezium rule estimate for ∫10ex2dx with h = 0.25 is 1.490 68 to 5dp. Estimate the size of the global truncation error in this approximation and show thatWhat value of h will give an answer
Show that the composite trapezium rule with step length h yields the approximationUsing the series expansion for coth xCompare this answer with the Euler–Maclaurin theorem. e dxh(e-1) coth 0 2
If r = (t, t2, t3), find r(t) and r(t).
Given the vectorevaluate dr/dt and write it in the formwhere T̂ is the unit tangent direction. Calculate dT̂/dt in its simplest form and show that it is perpendicular to T̂. r = (1 +t)i + tj + tk
In polar coordinates (r, θ), the unit vectors r̂ and θ̂ are defined as in Figure 9.11. Show thatHence from the definition r = rr̂ show that f = cose i sinej = sin0 i + cose j
Show that if the vector a(t) = f (t)i + g(t) j has constant magnitude, then a and da/dt are perpendicular.
A curve is given parametrically by r(t) = f(t) i + g(t) j. Show that, if s is the length of an arc measured from a fixed point P0 on the curve so that s increases as t increases, thenDeduce that
Obtain from first principles the partial derivatives ∂f/∂x and ∂f/∂y of the function f(x, y) at the point (1, 2), where f(x, y) = 2x2 – xy + y2
Obtain from first principles the partial derivatives ∂f/∂x and ∂f/∂y of the function f(x, y) at the general point (x, y) where f(x, y) = x cos y
Find ∂f/∂x and ∂f/∂y when f(x, y) is (a) xy + 2x +9y + xy + 10 (b) (x + y) (c) (3x + y + 2xy)1/2
Find ∂f/∂x and ∂f/∂y when f(x, y) is (a) e cos x (b) x x + y x + y 2 x + 2y + 6 (c)-
Find ∂z/∂x and ∂z/∂y when z(x, y) satisfies (a) x + y + 2 = 10 (b) xyz = x - y +z
Show that z = x2y2/(x2 + y2) satisfies the differential equation z - + z = 2z
Find fx, fy and fz when f(x, y, z) is(a) x2y + 3yxz – 2z3x2y(b) e2zcos xy
Show thatsatisfies f(x, y, z) = (x + y + 2)-1/2
Show thatsatisfies - f(x, y, z) = x + x y y-z
Find the gradient of f(x, y) = x2 + 2y2 – 3x + 2y at the point (x, y) in the direction making an angle α with the positive x direction. What is the value of the gradient at (2, –1) when α =
Find dA/dt where A = r tan−1(r tan θ) and r = 2t + 1, θ = πt.
Find ∂f/∂s and ∂f/∂t when f(x, y) = excos y,x = s2 – t2 and y = 2st.
Find dz/dt when (a) z = x + y x = f + 1 and y = t - 1 (b) z = xt and x + 3x + 2t = 1
Show that if u = x + y, v = xy and z = f(u, v) then z (a) x-- z = z (x-y)-
Show that if z = xnf(u), where u = y/x, thenVerify this result for z = x4 + 2y4 + 3xy3. z z x+y= + y= = nz
Show that, if f is a function of the independent variables x and y, and the latter are changed to independent variables u and v where u = ey/x and v = x2 + y2, then af af (a) x- + y (b) x3 af af
In a right-angled triangle a cm and b cm are the sides containing the right angle. a is increasing at 2cms−1 and b is increasing at 3cm s−1. Calculate the rate of change of (a) The
Show that the total surface area S of a closed cone of base radius r cm and perpendicular height h cm is given byIf r and h are each increasing at the rate of0.25cms−1, find the rate at which S is
A particle moves such that its position at time t is given by r = (t, t2, t3). Find the rate of change of the distance |r| of the particle from the origin.
Find ∂f/∂s and ∂f/∂t where f(x, y) = x + 2y and x = + e and y = e - e*.
Find all the second partial derivatives of f(x, y) = xexy.
Find all the second partial derivatives of f(x, y, z) = (x + 2y)cos3z.
Verify thatsatisfies the equation f(x, y) = X 2 x + y
Find the value of the constant a if V(x, y) = x3 + axy2 satisfies + = 0
Verify thatin the cases(a) f(x, y) = x2cos y (b) f(x, y) = sinh x cos y a2f a2f
Show thatsatisfies the differential equation V(x, y, z) = - exp Z x + y 4z
Prove that z = xf (x + y) + yF(x + y), where f and F are arbitrary functions, satisfies the equation Zxx + Zyy = 22xy
If u = ax + by and v = bx – ay, where a and b are constants, obtain ∂u/∂x and ∂v/∂y. By expressing x and y in terms of u and v, obtain ∂x/∂u and ∂y/∂v and deduce that = Show
Find the values of the constants a and b such that u = x + ay, v = x + by transforms into 2 22f v 22f 9- - + 20f = 0 = 0
Regarding u and v as functions of x and y and defined by the equations x = eucos v, y = eusin v show thatwhere z is a twice-differentiable function of u and v. (a) = cos'v= dz Jz (b) + 2 -2u
The function z is defined byFind Δz and dz when x = 4, y = 3, Δx = –0.01 and Δy = 0.02. Z(x, y) = xy - 3y
An open box has internal dimensions 2 m × 1.25 m × 0.75 m. It is made of sheet metal 4 mm thick. Find the actual volume of metal used and compare it with the approximate volume found using the
The angle of elevation of the top of a tower is found to be 30° ± 0.5° from a point 300 ± 0.1 m on a horizontal line through the base of the tower. Estimate the height of the tower.
The equationsdefine u as a function of y if x and z are eliminated. Find du/dy when x = 1, y = –1, z = 2, u = –2. x + 2y + 3z + 4u = -3 x + y+z+u =10 x + y + z + u =0
Use the Newton–Raphson method to find the root ofin the interval 0 ≤ x ≤ 1. Start with x = 0.5 and give the root correct to 4dp. ex + 3x - 2 = 0
The deflection at the midpoint of a uniform beam of length l, flexural rigidity EI and weight per unit length w, subject to an axial force P, iswhere m2 = P/EI. On making the substitution θ = 1/2ml,
Using the Maclaurin series expansion of ex, determine the Maclaurin series expansion of x/(ex – 1) as far as the term in x4, and hence obtain the approximation X e' - 1 0 -dx= 311 400
Use L’Hôpital’s rule to find In x lim- x1x-1
Determinewhere k is a constant. lim x->2 2 sin kx - xsin 2k 2(4- x)
Show that the equation x3 – 2x – 5 = 0 has a root in the neighbourhood of x = 2 and find it to three significant figures using the Newton–Raphson method.
(a) Obtain the Maclaurin series expansions of sinh x and cosh x.(b) A telegraph wire is stretched between two poles at the same height and a distance 2l apart. The sag at the midpoint is h. If the
Prove thatand deduce roo 0 sech xdx =
Evaluate (a) 5. (c) -d.x [*20 0 xe-4x dx ofe (e) e-2t cos.x dx (b) (d) (f) 0 1 x + 2x + 2 In x x3 [e -d.x e-2t cosh xdx d.x
Evaluatestating in each case the value of x for which the integrand becomes unbounded. (6 (0) [" xin de (b) (21-35dx X-1/3 -1/3 dx 0 Jo (c) 0 In x dx 3/2
Use the Taylor series to show that the principal term of the truncation error of the approximationConsider the function f(x) = xex. Estimate f´´(1) using the approximation above with h = 0.01, and
A particle moves in three-dimensional space such that its position at time t (seconds) is given by the vector (4 cos t, 4 sin t, 3) where distance is measured in metres. Find the magnitude of its
The acceleration a (m s−2) of a particle at time t (s) is given by a = (1 + t)i + t2j + 2k. At t = 0 its displacement r is zero and its velocity V (m s−1) is i – j. Find its displacement at
The temperature gradient u at a point in a solid is u(x, t) = t−1/2e−x2/4kt where k is a constant. Verify that nze xe || 1 k dt
Show that the surfaces defined by z2 = 1/2(x2 + y2) – 1 and z = 1/xy intersect, and that they do so orthogonally.

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