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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

Evaluate the following: (a) sin (d) sin - 1 2 3 2 (b) cos (-1) (e) cos 3 2 (c) tan- (-1) (f) tan 3
1. Find the following logarithms without using a calculator:2. Express in terms of ln x and ln y:3. Express as a single logarithm:4. Use the change of base formula to simplify:5. Simplify the
Write down x1, x2 and x3 for the sequences defined by n n+ 2 (b) Xn+1=X+4, Xo=2 (a) x = Xn (c) Xn+1 = -Xn 4 , Xo = 256
On the basis of the evidence of the first four terms give a recurrence relation for the sequence
A sequence is defined by xn = pn + q where p and q are constants. If x2 = 7 and x8 = –11, find p and q and write down(a) The first four terms of the sequence;(b) The defining recurrence relation
Triangular numbers (Tn) are defined by the number of dots that occur when arranged in equilateral triangles, as shown in Figure 7.5. Show that Tn = 1/2n(n + 1) for every positive integer n. Figure
A detergent manufacturer wishes to forecast its future sales. The market research department assess that their ‘Number One’ brand has 20% of the potential market at present. They also estimate
(a) If xr = r(r – 1) (2r – 5), calculate(b) If xr = rr + 1 + 3(–1)r, calculate(c) If xr = r2 – 3r + 1, calculate 4 r=0 X
A precipitate at the bottom of a beaker of capacity V always retains about it a volume v of liquid.What percentage of the original solution remains about it after it has been washed n times by
A certain process in statistics involves the following steps Si (i = 1, 2, . . . . , 6):Express the final outcome algebraically using ∑ notation. S: Selecting a number from the set T = {X, X,...,
Newton’s recurrence formula for determining the root of a certain equation isTaking x0 = 3 as your initial approximation, obtain the root correct to 4sf. By setting xn + 1 = xn = α show that the
Calculate the terms of the sequenceand show them on graphs similar to Figures 7.2 and 7.3. n n+n +1 in n=0
Calculate the sequenceShow the sequence using a cobweb diagram similar to Figure 7.4.Figure 7.4 {x}=0 where
A steel ball-bearing drops onto a smooth hard surface from a height h. The time to the first impact is T = √(2h/g) where g is the acceleration due to gravity. The times between successive bounces
Consider the following puzzle: how many single, loose, smooth 30 cm bricks are necessary to form a single leaning pile with no part of the bottom brick under the top brick? Begin by considering a
(a) Find the fifth and tenth terms of the arithmetical sequence whose first and second terms are 4 and 7.(b) The first and sixth terms of a geometric sequence are 5 and 160 respectively. Find the
An individual starts a business and loses £150k in the first year, £120k in the second year and £90k in the third year. If the improvement continues at the same rate, find the individual’s total
Show thatare in arithmetical progression and find the nth term of the sequence of which these are the first three terms. 1 1+ x 1 1. 1- x 1 1- x
The area of a circle of radius 1 is a transcendental number (that is, a number that cannot be obtained by the process of solving algebraic equations) denoted by the Greek letter π. To calculate its
A harmonic sequence is a sequence with the property that every three consecutive terms (a, b and c, say) of the sequence satisfyProve that the reciprocals of the terms of a harmonic sequence form an
The price of houses increases at 10% per year. Show that the price Pn in the nth year satisfies the recurrence relationA house is currently priced at £80 000. What was its price two years ago? What
Evaluate each of the following sums: (a) 1+2+3+...+152 + 153 (b) 1 +2+3+...+ 152 + 153 +++...+(52 + (153 +(-/-)152 (c) (d) 2 +6+ 18+...+2(3) 52 +2(3)153 (e) 1.2+2.3+3.4+...+152 153 + 153-154 1 1 1 1
A certain bacterium propagates itself by subdividing, creating four additional bacteria, each identical to the parent bacterium. If the bacteria subdivide in this manner n times, then, assuming that
By considering the sumshow that n + 1)4 k41 (k k=]
The repayment instalment of a fixed rate, fixed period loan may be calculated by summing the present values of each instalment. This sum must equal the amount borrowed. The present value of an
Consider the seriesShow thatand hence thatHence sum the series. || = - + + + n 2"
Consider the general arithmetico-geometric seriesShow thatand find a simple expression for Sn. Sn = a + (a + d)r + (a + 2d)r + ... + [a + (n 1)d]r"-1
Find the general solutions of the recurrence relations (a) Xn+1=2x-3 (C) Xn+1 = -Xn+()" (b) Xn+1=3x+10n (d) x+1=2x + 3 x 2" 'n
If a debt is amortized by equal annual payments of amount B, and if interest is charged at rate i per annum, then the debt after n years, dn, satisfies dn +1 = (1 + i )dn – B, where d0 = D, the
Show, by substituting them into the recurrence relation, that xn = 2n and xn = (–1)n are two solutions of xn + 2 – xn + 1 – 2xn = 0. Verify similarly that xn = A(2n) + B(–1)n is also a
Obtain the general solutions of (a) Yn+27Y+1 +10Y = 0 (b) Un+2 Un+1 6u, =0 (c) 25T+2-T (d) Pn+25Pn+1= 5(Pn+1- 5Pn) (e) 2En+2=En+1 + En - -
Solve the nonhomogeneous problems (use parts of Question 31)Data from Question 31Obtain the general solutions of (a) Yn+27Yn+1+10Y = 1, Y = 5/4, Y = 2 (b) 2En+2 - En+1 En=1, Eo=2, E=0 - (c) Un+2
Show that the characteristic equation for the recurrence relationwhere a is a non-zero constant, has two equal roots λ = a.(a) Verify (by substituting into the relation) that xn = (A + Bn)an is a
Let x be a constant such that |x| Find T2, T3 and T4 also directly by recursion and deduce that cos(2 cos–1x) = 2x2 – 1 and express cos(3 cos–1x) and cos(4 cos–1x) as polynomials in x.
A topic from information theory: imagine an information transmission system that uses an alphabet consisting of just two symbols ‘dot’ and ‘dash’, say. Messages are transmitted by first
Calculate the first six terms of each of the following sequences {an} and draw a graph of an against 1.(Some care is needed in choosing the scale of the y axis.) What is the behaviour of an as n →
Calculate the first six terms of each of the following sequences {an} and draw a graph of an against n. What is the behaviour of an as n → ∞? n +1 n+1 (n = 0) (a) an (b) a = (sinn)" (n 1) (c) a
Find the least value of N such that when n ≥ N, (a) n + 2n > 100 (b) 2" 1 1000 v
What is the long-term share of the detergent market achieved by the brand ‘Number One’, described in Question 5.Data from Question 5A detergent manufacturer wishes to forecast its future sales.
A linearly convergent sequence has the property thatThis is known as Aitken’s estimate for the limit of a sequence. Compute the first four terms of the sequenceand estimate the limit of the
Decide which of the following geometric series are convergent. (a) 2 + 2 + 2 3 (b) 4-2+1 9 + 2 27 1 2 (c) 10 + 11 + 21 10 2 + + + 3k (-1)* 4 2k +...+10()*+.. ... +...+ 1331 100 +... 25 125 (d) 1 - 1
Show that ifDeduce thatShow that if |r| n → a/(1 – r)2 as n → ∞. Hence sum the infinite series T= a + 2ar + 3ar + 4ar+...+ nar-1 then (1r)T, = a +ar+ar+...+ar"- - nar
For each of the following series find the sum of the first N terms, and, by letting N → ∞, show that the infinite series converges and state its sum. 2 2 2 5-7 (a) 13 + + + ... (b) + + + +
Which of the following series are convergent? (2) (1) k=1 ( k=0 1 3k +1 (b)-+-+...
By comparison with the seriesandis convergent and 1/2 100 =2[1/k(k 1)]
Show thatmay be expressed asand soas a rational number. Use a similar method to express as rational numbers 0.57 (that is, 0.575 757 ...)
Consider the seriesBy means of the inequalities (p > 0)and so on, deduce that the series is convergent for p > 1. Show that it is divergent for p ≤ 1. r=
Two attempts to evaluate the sumare made on a computer working to eight digits. The first evaluates the sumfrom the left; the second evaluates it from the right. The first method yields the result
Show thatand deduce thatDeduce that the modulus of error in the estimate for the sumobtained by computing 20 k=1 (-1)+1 k4 20 - k4 k=1 k=1 -100 -
For what values of x are the following series convergent? (a) (2n 1)x" n=l 2 (b) (-1)" - (2n + 1)! n=0 n=j n(n + 1) n? {1+n 2 (d) h=|
From known series deduce the following:In each case give the general term and the radius of convergence. (a) = 1 x2 +x4 x + ... 1 1 + x (b) In (c) 1 + x 1- x 1 (1 + x) = x + x + }{x + { x + .. =
Calculate the binomial coefficients (a) (c) 5 2 1/2 3 (b) (d) -2 3 (-1/2) 4
From known series deduce the following (the general term is not required): (a) tanx=x+jx+ +... 2x4 4! 2x 3! 2x (b) cosx = 1 - + 2! (c) et cos x = 1 + x - 25x6 6! 2xx42xx5 + 4! 5! 12 4 (d) In(1 + sin
Show thatHence derive a polynomial approximation to (1 – x)–1 with an error that, in modulus, is less than 0.5 ×10–4 for 0 ≤ x ≤ 0.25. Using nested multiplication, calculate from your
Find the sums of the following power series: (a) (-1) 2k k=0 (b) 1 + {x + 1 x2 + 1.3 2-4 xk ( k=i k(k + 1) 1.3.53 2-4-6" + 1-3-5-7 2-4-6-8x+...
A regular polygon of n sides is inscribed in a circle of unit diameter. Show that its perimeter pn is given byUsing the series expansion for sine, prove thatGiven p12 = 3.1058 and p24 = 3.1326, use
Show thatHence find lim f(x) = lim f x y0+
Draw (carefully) graphs offor 0 ≤ x ≤ 5. Use the series expansion of ex to prove that (a) xe * (b) xe (c) xe-x
Use a calculator to evaluate the function f(x) = x2 for x = 1, 0.1, 0.01, . . . . , 0.000 000 001. What do these calculations suggest about lim f(x)? x0+
Draw sketches and discuss the continuity of |x| (a) X (c) tanh 1 X (b) x-1 2 - x (d) [1 - x]
Find upper and lower bounds obtained byin the appropriate domains. Draw sketches to illustrate your answers. (a) 2x - 4x +7 (0 x2) (b) x + 4x-1 (0x3)
Use the intermediate value theorem to show that the equationhas roots between 1 and 2, between –4 and –3 and between –9 and –8. Find the root between 1 and 2 to 2dp using the bisection
Show that the equation 3x = 3x has a root in the interval (0.7, 0.9). Use the intermediate value theorem and the method of regula falsa to find this root to 3dp.
Show that the equationhas three roots α, β and γ, where α 1. For which of these is the iterative schemeconvergent? Calculate the roots to 3dp. x 3x+1=0
The cubic equation x3 + 2x – 2 = 0 can be written asDetermine which of the corresponding iteration processes converges most rapidly to find the real root of the equation. Hence calculate the root
Show that the iterationconverges to the limit a1/3. Use the formula with a = 157 and x0 = 5 to compare x1 and x2.Show that the error εn in the nth iterate is given byHence estimate the error in x1
The periods of natural vibrations of a cantilever are given bywhere l, E, I and ρ are physical constants dependent on the shape and material of the cantilever and θ is a root of the equationExamine
A bank pays interest at a fixed rate of 8.5% per year, compounded annually. A customer deposits the fixed sum of £1000 into an account at the beginning of each year. How much is in the account at
A computer simulation of the crank and connecting rod mechanism considered in Example 2.44 evaluates the position of the end Q of the connecting rod at equal intervals of the angle x°. Given that
Calculate the sequenceand illustrate the answer graphically. 1+ (-1) n n 10 n=1
Calculate the sequenceand show the pointson a graph. 1/10 {nl/n n=4
Calculate the sequencewhere x0 = 1 and {Xn}=0
How many terms of the arithmetical series 11, 15, 19, etc., will give a sum of 341?
A contractor agrees to sink a well 40 metres deep at a cost of £30 for the first metre, £35 for the second metre and increasing by £5 for each subsequent metre.(a) What is the total cost of
In its publicity material an insurance company guarantees that, for a fixed annual premium payable at the beginning of each year for a period of 25 years, the return will be at least equivalent to
Consider the sum-of-squares seriesObtain an expression for the sum of this series. Sn = 1 +2+3+...+ n = [k k=1
Obtain the sum of the series Sn = 1.2 1 2.3 + 1 3.4 ... + 1 n(n + 1) n k=1 1 k(k + 1)
Obtain the sum of the series n S = 1 + 2r + 3r + 4r + ... + nr- = krk-, r #1 k=1
Sum the series S = 1 + cos 0 + cos20 + ... + cos(n-1)0 n
Calculate the fixed annual payments £B required to amortize a debt of £D over N years, when the rate of interest is fixed at 100i%.
Find the general solutions of the recurrence relations (a) Xn+1 = 3x +4 (c) Xn+1 = ax + CB" (b) Xn+1=Xn+4 (d) Xn+1 = ax + Ca" (a, B, C given constants)
Evaluate the expression E(n) = 3xn + 2 + 5xn + 1 – 2xn where xn is defined for n ≥ 0 by (a) Xn 3" (b) xn=3" (c) Xn =3(27") (d) Xn = (-2)".
(a) Show by direct sustitution into the recurrence relationthat xn = 3n and xn = (–2)n are two solutions.(b) Further verify that xn = A(–2)n + B3n, where A and B are constants, is also a
Find the solution of the Fibonacci recurrence relation xn + 2 = xn + 1 + xn given x0 = 1, x1 = 1.
Find all the solutions of (a) x+2=Xn+1 = x + 12, where x = x=1 +3(2") (c) Xn+2=X+1/x (b) Xn+2=Xn+1 = x + 12n
Show that the general solution of the recurrence relationmay be expressed in the form Xn+2= 6xn+1 - 25xn
Find the solution of the recurrence relationwhich satisfies x0 = 1, x1 = 2. Xn+2 + 2x = 0
Find the limits of the sequencedefined by {Xn}n=0
Show that the ratio xn of successive terms of the Fibonacci sequence satisfies the recurrence relationCalculate the first few terms of this sequence and find the value of its limit. Xn+1=1+1/xn Xo =
Examine the convergence of the sequence {a}, a=(1 + 1/n)".
Examine the following series for convergence: (a) 1+3+5+7+9+...+(2k + 1) +... (b) 1 +2+ 3 +4 +5 +...+k +... 1 (c) 1 + + 1 (d) + 1.2 1 4 1 2.3 + + 1 1 + 16 8 1 3.4 + + + .. 1 4.5 + 1 2k + + 1 (k +
Examine for convergence the series (a) 1 + (b) 1+ 1! 12 + 1 + 2! 3! 4! 1 3 + 1 4 + ... + 1 - n n! (the factorial series) (the harmonic series)
Use d’Alembert’s test to determine whether the following series are convergent. (a) 8 k=0 k! (b) k=0 2k (k + 1)
Obtain the power series expansions of (a) 1 (1-x) (b) 1 (1-x)(1 + 3x) (c) In(1 + x) 1 + x
Sum the series(a) 12 + 22x + 32x2 + 42x3 + 52x4 +. . . .(b) 1 + 2! x4 + 4! 6! 8! +
Sum the seriesand show that e- (a) S() = r! r=0 Ne-a (b) T() = de-1 r! 0
Using a calculator, examine the values of f(x) near x = 0 whereWhat is the value of f(x)= X 1- (1 + x)' x = 0
The volume of a sphere of radius a is 4πa3/3. Show that the volume of material used in constructing a hollow sphere of interior radius a and exterior radius a + t isDeduce that the surface area of
Sketch the graph of the function f(x) where and show thatdoes not exist. f(x) = (x - x) X x = 0 and x < 1
Show that f(x) = 2x/(1 + x2) for x ∈ R is continuous on its whole domain. Find its maximum and minimum values and show that it attains every value between these extrema.

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