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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

Consider the chemical reactionLet x be the amount of product X, and a and b the initial amounts of A and B (with x, a and b in mol). The rate of reaction is proportional to the product of the uncombined amounts of A and B remaining. Express this relationship in terms of dx/dt, x, a and b. A + B X
A wire of length l metres is bent so as to form the boundary of a sector of a circle of radius r metres and angle θ radians. Show thatand prove that the area of the sector is greatest when the radius is l /4. = 1 - 2r r
A manufacturer found that the sales figure for a certain item depended on the selling price. The market research department found that the maximum number of items that could be sold was 20 000 and that the number actually sold decreased by 100 for every 1p increase in price. The total cost of
Differentiate the function f where f(x) is (a) x (b) (x) (d) 4x + 2x5 (e) 4x + x - 8 (f) 1/(2x) (g) x + x (h) 2x7/2 (i) 1/(3x) (c) 4x -
Using the product rule, differentiate the function f where f(x) is (a) (3x - 1)(x + 5x) (b) (5x + 1)(x + 3x - 6) - (c) (7x + 3)(x + 1/x) (d) (3 2x)(2x 9/x) (e) (x - 1/x)(x - 1/x) (f) (x + x + 1) (2x + x - 1)
Using the quotient rule, differentiate the function f where f(x) is (a) (3x + x + 1)/(x + 1) (b) (2x)/(x + 4) (c) (x + 1)/(x + 1) (d) x2/3/(x/3 + 1) (e) (x + 1)/(x + 1) (f) (2x-x + 1)/(x - 2x + 2)
Differentiate the function f where f(x) is(a) (ax + b)(cx + d) (b) (ax + b)/(cx + d)(c) 3ax2 + 5bx + c (d) ax2/(bx + c)
A fruit juice manufacturer wishes to design a carton that has a square face, as shown in Figure 8.17(a). The carton is to contain 1 litre of juice and is made from a rectangular sheet of waxed cardboard by folding it into a rectangular tube and sealing down the edge and then folding and sealing the
Using the method of Example 8.13, evaluate f(3), f'(3), f(–1) and f'(–1) for the polynomal function f(x) = 5x4 – 3x3 + x2 – 2Data from Example 8.13Evaluate f(2) and f'(2) for the polynomial function f(x) = 2x42xx + 3x - 2
Differentiate the function f where f(x) is (a) 5x - 2x + 1 (b) 4x + x - 8 (c) x + 3 (d) (e) (x - 3x + 1)(6x + 5) (x+x-2)(3x - 5x + 1) (f) (x 3)/(x - 2) (g) x/(x + 1) (h) 1/(x - 4x + 1) (i) x/(x + 5x + 6)
Differentiate the function f where f(x) is (a) (5x + 3) (b) (4x - 2) (c) (1 - 3x) (d) (3x - x + 1) (e) (4x - 2x + 1) (f) (1 + x-x)
Differentiate the function f where f(x) is (a) (2x + 4) (3x - 2) (c) (x + 2)(x + 3) (d) (x + x + 1)(x + 2x + 1) (b) (5x + 1)(32x)* (e) (x + 2x + 1)(2x + 3x - 1)* (f) (2x + 1)(7- x) (g) (x + 4x + 1)(3x + 1)
The algebraic functionis a root of the equationShow that x = 4y/(y – 1)2 and hence that y = (1+x)-1, x>-1 (1 + x) + 1
An open water conduit is to be cut in the shape of an isosceles trapezium and lined with material which is available in a standard width of 1 metre, as shown in Figure 8.21.To achieve maximum potential capacity, the designer has to maximize the area of cross-section A(b). Show thatand that this is
A carton is made from a sheet of A4 card (210mm × 297 mm) using the net shown in Figure 8.22. Find the dimensions that yield the largest capacity. 5 mm overlap for seal Figure 8.22 Net used
Differentiate(a) √(1 + 2x) (b) x√(x + 2) (c) √(x2 + 2x)
Differentiate(a) x√(4 + x2) (b) x√(9 – x2)(c) (x + 1)√(x2 + 2x + 3) (d) x2/3 – x1/4(e) 3√(x2 + 1) (f) x(2x – 1)1/3
Differentiate(a) 1/(x + 3)2(b)(c) x/√(x2 – 1) (d) (2x + 1)2/(3x2 + 1)3 x +. x
Differentiate with respect to x: (a) sin(3x - 2) (c) cos3x (e) x sin x (g) a cos(x + 0) (b) cosx (d) sin 2x cos 3x (f) (2 + cos 2x) (h) tan 4x
Differentiate with respect to x: (a) sin (x/2) (b) cos (5x) (c) (1 + x)tan x (d) sin ((x - 1)/2) (e) tan 3x (f) (1-x) sin x
A cone of semi-vertical angle θ is inscribed in a sphere of radius a. Show that the volume of the cone isHence prove that the cone of maximum volume that can be inscribed in a sphere of given radius is 8/27 th of the volume of the sphere. V = na sin0 cos0
Differentiate with respect to x: (a) cos(x) (c) (1 + sinx) (b) tan(tan x) (d) cos(x)
Differentiate with respect to x: (a) e (c) exp(x + x) (e) (3x + 2)e* (g) (1 + e) (b) e 2 (d) xex (f) e%(1 + e) (h) ex+b
Differentiate with respect to x: (b) In(x + 2x + 3) (d) - In x X (a) In(2x + 3) (c) In[(x - 2)/(x 3)] (e) In[(2x + 1)/(1-3x)] (f) In[(x+1)x]
Differentiate with respect to x:(a) sinh 3x (b) tanh 4x (c) x3cosh 2x(d) ln(cosh 1/2x) (e) cos x cosh x (f) 1/cosh x
Differentiate with respect to x: (a) sinh-'2x (c) tanh '(1/x) (e) (4x) - 2 cosh-(2/x) (f) tanhx/(1+x) (b) cosh (2x - 1) (d) V(1 + x)sinh 'x
Draw a careful sketch of y = e–ax sin ωx where a and ω are positive constants. What is the ratio of the heights of successive maxima of the function?
The line AB joins the points A(a, 0), B(0, b) on the x and y axes respectively and passes through the point (8, 27). Find the positions of A and B which minimize the length of AB.
Sketch the curve y = e−x2. Find the rectangle inscribed under the curve having one edge on the x axis, which has maximum area.
Show that y = 9e–9t/(10 – e–9t) satisfies the differential equation dy dt : -y(9 + y) =
A sky diver’s downward velocity v(t) is given byWhere u and α are constants. What is the terminal velocity achieved? When does the sky diver achieve half that velocity and what is the acceleration then? v(t) = u(1 e)/(1+e")
The equations x = t sin t, y = t cos t are the parametric equations for a spiral. Find dy/dx in terms of t.
A curve is defined parametrically by the equationsDraw a sketch of the curve for 0 ≤ θ ≤ 2π. Find the equation of the tangent to the curve at the point where θ = π/4. x = 2 cos 0 + cos 20 y = 2 sin sin 20
Find dy/dx when(a) x2 + y2 + 4x – 2y = 20(b) xy = 2ex+y–3
Find the equations of the tangent and normal to the curve having equationat the point (1, 3). y2y - 4x + 1 = 0
Find the equation of the tangent, at the point (0, 4), to the curve defined by yx + y + 7x = 4
Find the value of dy/dx at the point (1, –1) on the curve given by the equation 0 = x - x - - EX
Differentiate with respect to x: (a) 10 (b) 2x (c) (x - 1)7/2(x + 1) 1/ x + 2
The equation of a curve isShow that the tangent to the curve at the point (1, 2) has a slope of unity. Hence write down the equation of the tangent to the curve at this point. What are the coordinates of the points at which this tangent crosses the coordinate axes? xy 2xy + x - 1 = 0
A cycloid is a curve traced out by a point p on the rim of a wheel as it rolls along the ground. Using the coordinate system shown in Figure 8.27, show that the curve has the parametric representationwhere θ is the angle through which the wheel has turned. Draw a sketch of the curve. Find the
Use logarithmic differentiation to differentiate (a) (In x) (b) xinx (c) (1-x)/(2x + 3)-4
Using logarithmic differentiation, find the derivatives of (a) xe2 In x (b) -esin 2x X
Find d2y/dx2 when y is given by (a) x(1 + x) (b) In(x + x + 1) (c) yx + y + 7x = 4 (d) xy - xy - x = 0
Find d2y/dx2 when x and y are given by(a) x = t sin t and y = t cos t(b) x = 2 cos t + cos 2t and y = 2 sin t – sin 2t
If y = 3e2x cos(2x – 3), verify that dy dx dy 4 +8y=0 d.x
If y = (sin–1x)2, prove thatand deduce that (1-x) dy d.x = 4y
(a) If y = x2 + 1/x2, find dy/dx and d2y/dx2. Hence show that(b) If x = tan t and y = cot t, show that dy .2 d.x dy + 4x + 2y = 12x dx
If x = a(θ – sin θ) and y = a(1 – cos θ), find dy/dx and d2y/dx2.
Find dy/dx in terms of t for the curve with parametric representation x = 1-t 1 + 2t 1 - 2t 1+t y = -
Confirm that the point (1, 1) lies on the curve with equation x3 – y2 + xy – x2 = 0 and find the values of dy/dx and d2y/dx2 at that point.
Find f(4)(x) and f(n)(x) for the following functions f(x): (a) ex (c) 1 1-x (b) In(x + 2)
Find the fourth derivative of f(x) = sin(ax + b) and verify that f(n)(x) = ansin(ax + b +1/2 nπ).
Prove thatwhere d" - (e sin bx) = (a + b)/ex sin(bx + n0) d.x"
If y = u(x)v(x), prove thatHence prove Leibniz’s theorem for the nth derivative of a product: (a) y(x) = u(x)v(x) + 2u(x)v(x) + u(x)v)(x) (b) y(x) = u(x)v(x) + 3u2(x)v(x) + 3u(x)v(x) + u(x)v(x)
Use Leibniz’s theorem (Question 71) to find the following:Data from Question 71If y = u(x)v(x), prove thatHence prove Leibniz’s theorem for the nth derivative of a product: (a) d dxs (x sin x) (put u = sin .x, v = x) d dx(3x + 1)] (b) (re ) d4 d.x4
Find the radius of curvature at the point (2, 8) on the curve y = x3.
Show that the radius of curvature at the origin to the curve x3 + y3 + 2x2 – 4y + 3x = 0 is 125/64 .
Find the radius of curvature and the coordinates of the centre of curvature of the curve y = (11 – 4x)/(3 – x) at the point (2, 3).
Find the radius of curvature at the point where θ = 1/3π on the curve defined parametrically by x = 2 cos θ, y = sin θ
Find the radius of curvature at (x, y) of the curve y=tanh`'x (|x|
Find the radius of curvature at (1, 1) of the curve defined by x = t3, y = t2 (t ∈ R)
Find the stationary values of the following functions and determine their nature. In each case also find the point of inflection and sketch a graph of the function. (a) f(x) = 2x5x + 4x1 (b) f(x) = x + 6x - 15x + 51 (c) f(x) = x - 6x + 8x + 2
Find the stationary values of the following functions, distinguishing carefully between them. In each case sketch a graph of the function. 3x (x - 1)(x - 4) (b) f(x) = 2e (x - 1) (c) f(x) = xe-* (d) f(x)= (a) f(x) = - x + 8 (1-x)
Consider the can shown in Figure 8.40, which has capacity 500 ml. The cost of manufacture is proportional to the amount of metal used, which in turn is proportional to the surface area of the can. Ignoring the overlaps necessary for the manufacture of the can, find the diameter and height of the
Consider again the can shown in Figure 8.40. Allowing for an overlap of 6mm top and bottom surfaces to give a rim of 3mm on the can, show that the area A mm2 of metal used is given bywhere d cm is the diameter of the can. Show that the value of d* which minimizes the area of the can satisfies the
In an underwater telephone cable the ratio of the radius of the core to the thickness of the protective sheath is denoted by x. The speed v at which a signal is transmitted is proportional to x2ln(1/x). Show thatwhere K is some constant, and hence deduce the stationary values of v. Distinguish
A closed hollow vessel is in the form of a rightcircular cone, together with its base, and is made of sheet metal of negligible thickness. Express the total surface area S in terms of the volume V and the semi-vertical angle θ of the cone. Show that for a given volume the total area of the surface
A numerical method which is more efficient than repeated subtabulation for obtaining the optimal solution is the following bracketing method. The initial tabulation locates an interval in which the solution occurs. The optimal solution is then estimated by optimizing a suitable quadratic
A pipeline is to be laid from a point A on one bank of a river of width 1 unit to a point B 2 units downstream on the opposite bank, as shown in Figure 8.41. Because it costs more to lay the pipe under water than on dry land, it is proposed to take it in a straight line across the river to a point
Cross-current extraction methods are used in many chemical processes. Solute is extracted from a stream of solvent by repeated washings with water. The solvent stream is passed consecutively through a sequence of extractors, in each of which a crosscurrent of wash water, flowing at a determined
The management of resources often requires a chain of decisions similar to that described in Question 87. Consider the harvesting policy for a large forest. The profit produced from the sale of felled timber is proportional to the square root of the volume sold, while the volume of standing timber
Use the chord approximation to obtain two estimates for f´(1.2) using h = 0.2 and h = 0.1 where f(x) is given in the table below.Use extrapolation to obtain an improved approximation. X f(x) 1.0 1.000 1.1 1.2 1.008 1.061 1.3 1.192 1.4 1.414
Use your calculator (in radian mode) to calculate the quotient {f(x + h) – f(x – h)}/(2h) for f(x) = sin x, where x = 0(0.1)1.0 and h = 0.001. Compare your answers with cos x.
Consider the function f(x) = x ex, tabulated below:(a) Find, exactly, f´(1) and f"(1).(b) Use the tabulated values and the formulato estimate f´(1), for various h. Compute the errors involved and comment on the results.(c) Repeat (b) for f´´(1) using X 0.96 0.97 0.98 0.99 1.00 f(x) 2.5072
Use the following table of f(x) = (ex – e–x)/2 to estimate f'(1.0) by means of an extrapolation method.Compare your answer with (e + e–1)/2 = 1.5431 correct to 4dp. 0.2 f(x) 0.2013 0.6 0.8 0.6367 0.8881 1.2 1.4 1.8 1.5095 1.9043 2.9422
Investigate the effect of using a smaller value for h in Example 8.37. Show that Φ(0.1) gives a poorer estimate for f´(0.5) and the error bound for the consequent extrapolation [4Φ(0.1) – Φ(0.2)]/3 is 7 × 10–4.Data from Example 8.37.Estimate f´(0.5), where f (x) is given by the table x
Two hot-rodders, Alan and Brian, compete in a drag race. Each accelerates at a constant rate from a standing start. Alan covers the last quarter of the course in 3 s, while Brian covers the last third in 4 s. Who wins and by what time margin?
Show that the area under the graph of the constant function f(x) = 1 between x = a and x = b (a < b) is given by b – a.
Show that the area under the graph of the linear function f(x) = x between x = a and x = b (a < b) is given by 1/2(b2 – a2).
Draw the graph of the function f(x) = 2x – 1 for –3 3(2x - 1)dx.
Using n strips of equal width, show that the area under the graph y = x2 between x = 0 and x = c satisfies the inequalityand deduce -1 h? r < area
Using the method of Question 98 and the fact thatshow thatData from Question 98Using n strips of equal width, show that the area under the graph y = x2 between x = 0 and x = c satisfies the inequalityand deduce 1=4 z (I + U) u = 7
If y = f(x) = x3 + 2 find(a) f(2) (b) f(−2)(c)  (3) 2
Sketch the graph of the function (see Example 8.2) with formula y = (10 + 2x)xData from Example 8.2An isosceles trapezium has the dimensions, expressed in cms, shown in Figure 8.1. Find the formula which relates the enclosed area A cm2 to the height x cm. Evaluate A for x = 1, 2, 3, 4 and 5. 2x X
Find the equation of the linear function which takes the value 1260 when x = 1250, and the value 1345 when x = 1500.
Determine the nature of the quadratic function with formula(a) y = 15x2 − 3x + 25(b) y = 12 + 7x − 2x2
Given the quadratic function y = f(x) = 12x −11− 4x2 Decide whether the graph of f(x) has a minimum or a maximum value, and evaluate it.
Sketch the graph of the rational function 9-x-zx y = f(x) = - x - 1
Express in its partial fraction 2x (x+1)(x+2)
Obtain the inverse function of y = f(x) = 2x + 5
(a) Find the logarithm log2 32(b) Expressin terms of log10 x and log10 y(c) Express as a single logarithm ln 20 – ln 18 + 2ln 6 logio 110x
Use the substitution u = ex to reduce the equation e2x − 8ex +12 = 0 to a quadratic equation in u. Factorise this quadratic and hence find the value(s) of u which satisfy it. What are the value(s) of x, correct to two decimal places, which satisfy the original equation?
Iffind:(a) f(3)(b) f(−1) (c) f(0)2. A car costs £1200 to run in its first year and subsequently the annual cost increases by £100 each year. How much does it cost to run the second year, third year and fourth year? Write down a formula for the cost £C in the x-th year.3. A cake manufacturer
1. Draw up a table of values for the function with formula y = 1100 +100x for x values 0 to 10 in steps of 1. Hence, sketch the graph of the function 2. Draw up a table of values for the function given by y = 25000x −100x2 −1020000 for x values 25 to 200 in steps of 25. In the context of
1. Find the formulae of the linear functions such that:2. The total cost, £C, of manufacturing a batch of articles is suspected to be related to the number of articles in the batch, x, by a linear model of the form C = ax + b where a and b are constants. The following returns were noted:(a) By
1. Determine the nature of the quadratic functions with formula:2. Given the quadratic function y = f (x) = 2x2 −11x +12 Decide whether the graph of f(x) has a minimum or a maximum value, and evaluate it.3. A gun is situated at a point O on horizontal ground at sea level. It fires a shell which
1. By drawing up a short table of values, sketch the graphs of the following cubic functions:2. The cubic function f(x) is defined by(a) Sketch the graph of f (x) for x = 0, 1, 2 and 3 only, hence estimate the value of a zero of f (x) .(b) Observing that f(x) = 0 implies that (3x − 5)3 = 8 find
1. Express in partial fractions:2. (a) Expressin the formand then in the form(b) Use the result obtained in (a) to sketch the graph of the rational function3. (a) Sketch the graph of the rational function f(x) defined by(b) Using (a) as a guide, sketch the graph ofwhere p and q are positive
1. Find the inverse function of:2. Obtain the inverse function of the functionPlot graphs of both the function and its inverse on the same set of axes.3. Obtain the inverse function of the functionPlot graphs of both the function and its inverse. (a) f(x)=2x-3 (b) f(x)=- -X-4 Notice f(f(x)) = x .
1. Sketch using the same set of axes:for −2π ≤ x ≤ 2π . What can you observe?2. Sketch the graph of the functionfor −2π ≤ x ≤ 2π .3. Sketch the graph of the functionfor −2π ≤ x ≤ 2π . What is the period of the function?4. Sketch the graph of the functionfor 0 ≤ x ≤ 2π .

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