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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

The height h of the top of a pylon is calculated by measuring its angle of elevation α at a point a distance s horizontally from the base of the pylon. Find the error in h due to small errors in s
The resistance of a length of wire is given bywhere k is a constant. L is increasing at a rate of 0.4% min−1, ρ is increasing at a rate of 0.01% min−1 and D is decreasing at a rate of 0.1%
The deflection H of a metal structure can be calculated using the formulawhere I, ρ, D and L are the moment of inertia, density, diameter and length respectively, and g is the acceleration due to
In the calculation of the power in an a.c. circuit using the formula W = EI cos Φ, errors of +1% in I, –0.7% in E and + 2% in Φ occur. Find the percentage error in the calculated value of W when
(a) Prove that u = x3 – 3xy2 satisfies(b) Givenevaluatein terms of u. + = 0
Verify that z = ln √(x2 – y2) satisfies the equation ze xe SI + Zze + ze HI 1 (x - y)
(a) Find the value of the positive constant c for which the functionsatisfies the equation(b) V is a function of the independent variables x and y. Given that x = r cos θ and y = r sin θ, find
A curve C in three dimensions is given parametrically by (x(t), y(t), z(t)), where t is a real parameter, with a ≤ t ≤ b. Show that the equation of the tangent line at a point P on this curve
Show that u = f(x + y) + g(x – y) satisfies the differential equation n-e n-e 2 xe = 0
Show that ifand find a similar expression for ∂2Φ/∂x2. Deduce that if then o(x, t) = ap t f(z) t and z zf'(z) + f(z) 2tt X 21
Water waves move in the direction of the x axis with speed c. Their height h at time t is given bywhere a is a constant. A small cork floats on the water and is blown by the wind in the direction of
The components of velocity of an inviscid incompressible fluid in the x and y directions are u and v respectively, whereFind the stream function Ψ(x, y) such that U = x - y (x + y) and V = 2xy (x +
Show that the function has one maximum and four saddle points. Sketch the part of the surface z = f(x, y) that lies in the first quadrant. f(x, y) = xy? 5r 8ry 5y
A trough of capacity 1m3 is to be made from sheet metal in the shape shown in Figure 9.28. Calculate the dimensions that use the least amount of metal.Figure 9.28 2(1+Y cos 0)Y sin 0 + (2Y+1) Z [(1 +
Find the critical points of the functionand identify the character of each point. z=12xy - 3 - x3
Find the local maxima and minima of the functionsubject to the constraint f(x, y) = y - 8x + 17
The period T of oscillation of a simple pendulum of length l is given byby expanding the integrand as a power series in sin2 1/2α show that T = 4OS Jo 1 (1-sina sin0) -do
(a) An oil tanker runs aground on a reef and its tanks rupture. Assuming that the oil forms a layer of uniform thickness on the sea and that the rate of spill is constant, show that the rate at which
Working to 5dp, evaluateusing the trapezium rule with five ordinates. Evaluate the integral by direct integration and comment on the accuracy of the numerical method. xP-(x + 1) S
Find dy/dx when y is given by(a) x2ex (b) 3e–2x (c) ln x/x2(d) ln(x2 + 1) (e) e–x(sin x + cos x)
Find the slope of the tangents to the circleat the points A(1, 3), B(4, 2) and C(–2, –6). x + y - 2x + 4y - 20 = 0
Estimate f´(0.5), where f(x) is given by the table X 0.1 0.2 0.3 0.4 f(x) 0.0998 0.1987 0.2955 0.3894 0.6 0.7 0.8 0.5 0.4794 0.5646 0.6442 0.7174 0.9 0.7833
An Education Officer decides to collect data on the number of school-age children per family in a small town. Thirty families are selected at random and the number of school-age children recorded as
The data given in Table 9.8 represents the percentage marks scored by 66 candidates in an examination.(a) What is the lowest mark?(b) What is the highest mark?(c) What is the range of the data
Construct a ranked stem-and-leaf plot for the grouped data, established in Practice Question 9.2, representing the percentage marks data given in Table 9.8.Data from Practice Question 9.2The data
Construct a cumulative frequency table and a cumulative frequency curve for the grouped data, established in Practice Question 9.2, representing the percentage marks data given in Table 9.8.Data from
The duration, in seconds, of telephone calls from an office over a two-week period are recorded. The recorded data is given in Table 9.12.(a) What is the duration of the shortest call?(b) What is the
Use the given substitutions to integrate the following functions: (a) x(1 + x), with t = (1 + x) 3 1 x(x + 9) (b) (c) 1 3+ x with t = X with t = x
Differentiate the following expressions, giving your answers as simply as possible: (a) e+x (c) sin(5x - 1) (e) cos (1-x) (g) sin- (i) sin(3x + 1) 1 (1 + x) (k) (3-x) (m) (1 + cosh x) (0) x-1 (x + 2)
1. Evaluate (a) (c) (e) (i) [xmxdx 1/2 In.x dx -[x]dx xdx (x - 1)(x - 2) (4 - 3x)dx 0 J x dx (x - 1) (b) (d) (h) S (j) (2x + 3) dx x + 2x + 2 S 10/ xsin x dx (1-x) 1/2 (set x = sin t)
Find the equation of the tangent and normal at the point (1, 4) to the curve whose equation is y = 2x3x + 5x + 3x3
Find the equation of the tangent to the curve x2 – 3xy + 2y2 = 3 at the point (1, 2) and the equation of the normal to the curve y = x3 – x2 at the point (1, 0). Find the distance of the point of
With reference to Example 2.10, confirm that the functionhas maximum value when x = 2/3.Data from Example 2.10An open conical container is made from a sector of a circle of radius 10 cm as
Find the turning points on the curve y = 2x3 – 5x2 + 4x – 1 and determine their nature. Find the point of inflection and sketch the graph of the curve.
The turning moment T on the crankshaft of an engine is given byFind the maximum and minimum values of T for 0 ≤ θ ≤ 2π. T = 6 + 2.5 sin 20 - 3.8 cos 20
The deflection of a beam of length L is given bywhere w, E and I are constants. Determine(a) The maximum deflection;(b) The points along the beam at which points of inflection lie. y =wx (L x) (0xL)
A running track is set out in the form of a rectangle, of length L and width W, with two semicircular areas, of radius 1/2 W, adjoined at each end of the rectangle. If the perimeter of the whole
Find the maximum and minimum values of y wherejustifying your answers. Sketch the curve, indicating the stationary points and any asymptotes. y = x (x-2)(x-6)
Light sources are placed at two fixed points Q and R which are 1 metre apart. The source at R is twice as intense as that at Q. The total illumination at a point P on the line QR x metres distant
Using partial fractions, show that (a) (b) S. 2 S 2x + 3 -dx-In 3 - In 2 x(x - 1)(x + 2) 6x dx (x + 1)(2x - 1) - 3 In 3 - In 2 -
The parametric equations of a curve areIf ρ is the radius of curvature and (h, k) is its centre of curvature, prove that x = at, y = 2at
(a) Using the substitution u = x + 1, evaluate(b) Using the substitution u = √x + 6, evaluate(c) The region R is bounded by the x axis, the line x = 9/2 and the curve with parametric equationswhere
A curve has parametric equationsShow thatFind d2y/dx2 and d2x/dy2 in terms of t, and demonstrate that x = 2t+ sin 2t, y = cos 2t
Verify that the point (–1, 1) lies on the curveand find the values of dy/dx and d2y/dx2 there. What is the radius of curvature at that point? y(y - 3x) = y - 3x3 - =
Sketch the curve whose equation is y2 = x(x – 1)2 and find the area enclosed by the loop.
Sketch the curve whose parametric representation is x = a sin3t, y = b cos3t (0 ≤ t ≤ 2π) Find the area enclosed.
Sketch the curve whose polar equation is r = 1 + cos θ Show that the tangent to the curve at the point r = 3/2 , θ = 1/3π is parallel to the line θ = 0. Find the total area enclosed by the curve.
A curve is specified in polar coordinates (r, u) in the form r = f(θ). Show that the sectorial area bounded by the line θ = α, θ = β and the curve r = f (θ) (α ≤ θ ≤ β) is given byAlso
Find the length of the arc of the parabola y = x2 that lies between (–1, 1) and (1, 1).
The parametric equations x = t2 – 1, y = t3 – t describe a closed curve as t increases from –1 to 1. Sketch the curve and find the area enclosed.
(a) Find the area of the region bounded by the x axis and one arch of the cycloidwhere a is a positive constant.(b) Show that the radius of curvature of the cycloid defined in (a) at the point O is
Consider the integralwhere n is an integer. Using the trigonometric identity 1 + tan2x = sec2x, show thatand hence obtain the recurrence relationUse this to find(Recurrence relations of this type are
Use integration by parts (writing the integrand as sin θ sinn–1θ) to show thatsatisfies the reduction formulaHence prove thatThese results are known as Wallis’s formulae. Use them to show that
Consider the integralShow that Im,n satisfies the reduction formula m.n = 70/2 cos"x sin"x dx
Reduction formulae of the type are iteration formulae – and, like other iteration formulae, when they are used, attention must be paid to their numerical properties. This is illustrated by
The function F(r) is defined byBy considering d(cos x sinr – 1x)/dx, or otherwise, show that F(r) = 7/2 Jo sin'x dx r>-1
A solid of revolution is generated by rotating the area between the y axis, the line y = 1 and the parabola y = x2 about the y axis. Find its volume and its surface area.
The numerical procedures developed in this chapter for evaluating integrals have all used strips of equal width. An alternative procedure is to specify the number of tabular points to be used but not
The symbols Tn and Sn are defined as the estimates of the integralusing n intervals with the trapezium and Simpson’s rules respectively. Calculate T1, T2, T4, S2 and S4, working to 3dp only. Verify
(a) A curve is represented parametrically byFind the volume and the surface area of the solid of revolution generated when the curve is rotated about the x axis through 2π radians.(b) Find the
When a homogeneous bar of constant crosssectional area A (see Figure 8.85) is under uniformly distributed tensile stress, the elongation in the direction of the stress for a material obeying
Figure 8.87 shows an old cylindrical borehole that has been filled in part with silt and in part with water. Before the hole can be redrilled, the water has to be pumped to the surface. We wish to
Draw the graph of the function f(x) defined byfor the interval –5 ≤ x ≤ 5. xp{[ - x] - - - [x)} f(x): 0 ) = 0
An even function f(x) of period 2π is given on the interval [0, π] by the formula y = x/π(a) Using the even-ness property of the function, draw the graph of the function for –π ≤ x ≤ π.(b)
A frame tent has a square of side 2 m and two semicircular cross members, FBE and GBD, as shown in Figure 8.88.(a) Show that the cross-section ABC has equation 2x2 + z2 = 2(b) Show that the capacity
Using the definition of a derivative given in (8.1), find f´(x) when f(x) is(a) x2 (b) 1/x(c) mx + c (m, c constants) df d.x f x-0 = f(x) = lim = lim f(x + 4x) = f(x) . (8.1)
Consider the function f(x) = 25x – 5x2. Find(a) The derivative of f(x) from first principles;(b) The rate of change of f(x) at x = 1;(c) The equation of the tangent to the graph of f(x) at the
A particle is thrown vertically upwards into the air. Its height s (in m) above the ground after time t (in seconds) is given by s = 25t – 5t2(a) What height does the particle reach?(b) What is
Suppose that a tank initially contains 80 litres of pure water. At a given instant (taken to be t = 0) a salt solution containing 0.25 kg of salt per litre flows into the tank at a rate of 8 litres
In a suspension bridge a roadway, of length 2l, is suspended by vertical hangers from cables carried by towers at the ends of the span, as illustrated in Figure 8.9(a). The lowest points of the
A radio telescope has the shape of a paraboloid of revolution (see Figure 1.22(b)). Show that all the radio waves arriving in a direction parallel to its axis of symmetry are reflected to pass
Show that the shear force F acting in a beam is related to the bending moment M by F = dM/dx.
An open box, illustrated in Figure 8.13(a), is made from an A4 sheet of card using the folds shown in Figure 8.13(b). Find the dimensions of the tray which maximize its capacity. Figure 8.13 (a) The
Using result (8.9), find f´(x) when f (x) is(a) √x (b) 1/x5(c) 1 3x
Using the result (8.9) and the rules of Section 8.3.1, find f´(x) where f (x) isData from Rules (a) 8x4 - 4x (d) (x + 1)x (b) (2x + 5)(x + 3x + 1) (c) 4x7(x 3x) x x + 2x + 1 x + 1 x + 1 (e) (f)
If y = 2x4 – 2x3 – x2 + 3x – 2, find dy/dx.
The distance s metres moved by a body in t seconds is given by s = 2t3 – 1.5t2 – 6t + 12 Determine the velocity and acceleration after 2 seconds.
Evaluate f(2) and f´(2) for the polynomial function f(x) = 2x4 – 2x3 – x2 + 3x – 2.
Find the derivative of the following functions of x: (a) 3x + 2 2x + 1 (c) x + 2x - (b) 1 X + 2x + 3 x + x + 1 - + 3, x = 0
Find dy/dx when y is(a) (5x2 + 11)9 (b) √(3x2 + 1)
Find dy/dx when y is (a) (3x 2x + 1) (b) (c) (x + 1)(x - 1) (d) 1 (5x2 2)7 (2x + 1) (x + 1)
Find dy/dx when y is given by (a) sin(2x + 3) (d) sec 6x (g) xcos x (b) x cos x (e) x tan 2x (h) tan 2x 1 + x sin 2x x + 2 (f) sin 6x (c)
Find dy/dx when y is given by(a) sin2(x2 + 1) (b) cos–1√(1 – x2)
Find dy/dx when y is given by(a) tanh 2x (b) cosh2 x (c) e–3x sinh 3x (d) sinh 3x 4
The function y = f(x) is defined by x = t3, y = t2 (t ∈ R). Find dy/dx.
Find dy/dx when x2 + y2 + xy = 1.
Find the equations of the tangent and normal to the curve having equation x2 + y2 – 3xy + 4 = 0 at the point (2, 4).
Find the derivative of the function f(x) = (sin x)x (x ∈ (0, π))
Differentiate with respect to x y = (x - 2)(x + 3) (x + 1)
Find the second derivative of the functions given by (a) y = 3x2x + x - 1 (c) y esin 2x (b) y = x/(x + 1) In x (d) y = X
Show thatsatisfies the equation y = e(A cost + B sin t) + 2 sin 2t - cos 2t
Find d2y/dx2 when y is given by(a) y = t2, x = t3(b) x2 + y2 – 2x + 4y – 20 = 0
Determine the stationary points of the function f(x) = 4x3 – 21x2 + 18x + 6 and examine their nature.
Using the second derivative, confirm the nature of the stationary points of the function f(x) = 4x3 – 21x2 + 18x + 6 determined in Example 8.30.Data from Example 30Determine the stationary points
Two cones are made from a circular sheet of metal of radius 1. Find the sectional angle θ (see Figure 8.37) that maximizes the combined capacity. Figure 8.37 0 277 x (1-x) V
Determine the stationary values of the function 82 f(x)=x - 6x + X + 45 x2 x #0
A manufacturer has to supply N items per month at a uniform daily rate. Each time a production run is started it costs £c1, the ‘set-up’ cost. In addition, each item costs £c2 to manufacture.
A milk retailer wishes to design a milk carton that has a square cross-section, as illustrated in Figure 8.39(a), and is to contain two pints of milk (2 pints ≡ 1.136 litres). The carton is to be
For a particular model of car, bought for £14 750, the second-hand value after t years is given fairly accurately by the formulaThe running costs of the car increase as the car gets older, so after
By considering the area under the graph of y = x + 3, evaluate the integral x) 5-S 25(x + 3)dx.

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