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modern engineering mathematics

Questions and Answers of Modern Engineering Mathematics

1. Calculate the mean deviation of the data set representing the lengths of names for a class of pupils.2. In Practice Question 9.2 of Refresher Unit 9, data was presented on the percentage marks
1. Word lengths in newspaper crosswords are summarised in Table 10.16Calculate the mean, mode, median, mean deviation and standard deviation for this data.2. Lead levels in blood samples from 50
Determine the Laplace transform of the function f(t) = c where c is a constant.
Determine the Laplace transform of the ramp function f(t) = t.
Determine the Laplace transform of the one-sided exponential function f(t) = ekt
Determine the Laplace transforms of the sine and cosine functions f(t) = sin at, g(t) = cos at where a is a real constant.
Show that the function f(t) = t3 (t ≥ 0) is of exponential order.
Determine L{3t + 2e}.
Determine L{5 - 3t + 4 sin 2t - 6e*}.
Determine L{te-}.
Determine L {e- sin 2t}.
Determine L {t sin 3t}.
Determine L{te}.
Determine where n is a positive integer. '{ws}
Find L-1 1 [(s + 3)(s-2)]
Find p-1. +1 I + S s (5+9)
Find L-1 1 (s + 2)
Find L-1. 2 s + 6s + 13
Find L-1. S+7] s+2s +5
Find L-1 1 (s + 1) (s + 4)
Obtain off L (T3 + sin 2T)dr 0
Solve the differential equationsubject to the initial conditions x = 1 and dx/dt = 0 at t = 0. dx dt dx + 5 + 6x = 2e (t > 0) dt
Solve the differential equationsubject to the initial conditions x = 0 and dx/dt = 0 at t = 0. dx dt dx + 6 + 9x = sint (t > 0) dt
Solve the differential equationsubject to the initial conditions x = dx/dt = 1 and d2x/dt2 = 0 at t = 0. dx dt3 dx +5- dt dx + 17 +13x = 1 (t > 0) dt
Solve for t > 0 the simultaneous first-order differential equationssubject to the initial conditions x = 2 and y = 1 at t = 0. dx dy dt dt dx 2- dt + + 5x + 3y = e- dy dt + x +y = 3 (11.20) (11.21)
The LCR circuit of Figure 11.8 consists of a resistor R, a capacitor C and an inductor L connected in series together with a voltage source e(t). Prior to closing the switch at time t = 0, both the
In the parallel network of Figure 11.9 there is no current flowing in either loop prior to closing the switch at time t = 0. Deduce the currents i1(t) and i2(t) flowing in the loops at time t. Figure
A voltage e(t) is applied to the primary circuit at time t = 0, and mutual induction M drives the current i2(t) in the secondary circuit of Figure 11.10. If, prior to closing the switch, the currents
The mass of the mass–spring–damper system of Figure 11.12(a) is subjected to an externally applied periodic force F(t) = 4 sinωt at time t = 0. Determine the resulting displacement x(t) of the
Consider the mechanical system of Figure 11.13(a), which consists of two masses M1 =1 and M2 = 2, each attached to a fixed base by a spring, having constants K1 = 1 and K3 = 2 respectively, and
Find the derivatives of the following functions:(a) x5 (b) 1/x2(c) √x (d) 1/√x(e) x5/2
Find the derivatives of the following functions:(a) 6x7 (b) 4/x4(c) 6√x5
Find the derivative of the functiony = 3x3 − 2x2 + 5x + 3
Find the derivatives of the following functions:(a) y = (3x2 + 8)3(b) y = 4x - 2
Using the product rule, find the derivatives of the following functions:(a) y =(x2 + 3)(x3 − 2x2 +1)(b) y = x2 (2x2 +1)8
Using the quotient rule, find the derivatives of the following functions: (a) y = x +1 x-3 x + 2x-1 2x+5 (b) y=-
Find the second derivative d2y/dx2 for the following y: (a) 4x +2x+6 (b) X x-2
Find the slope of the tangent at the point (1, 3) on the graph of the cubic y = 2x3 − 3x2 + x + 3 What is the equation of the tangent at this point?
Find the derivatives of the following functions:(a) e2x (b) exp(x2 + 1)
Determine the derivatives of the following functions:(a) exp(6x) (b) exp(−3x) (c) exp(3x2 − 4)
Find the derivatives of the following questions:(a) x2 exp(4x)(b) exp(-2x) x2 +1 X
Find the derivatives of the following functions:(a) ln (6x) (b) ln(x2 + 2) (c) x2 ln x(d) In x
Find the derivatives of the following functions:(a) sin(5x) (b) cos(2x + 3) (c) tan(8x)(d) sin(x3 +1) (e) x2 tan x (f) exp(−2x) cos3x
Find the derivatives of the following functions:(a) sin−1 (3x)(b)(c) x2 cos−1 x tan X
Find the derivatives of the following functions: (a) x3 (e) X (b) x10 (f) x (c) 1 " (d) 6 (h) i/
Find the derivatives of the following functions: (a) 7x8 (b) 4 (c) 3x (d)
Find the derivatives of the following functions: (a) 2x - +4 X (c) x - 7/7 + 3 VX () 6x + 17 - 23/- + x - 2 - X X (b) 7x-3x+2x-24 (d) 5x 3 X +7x-8 (f) x-2x+- x"
By introducing an appropriate intermediate variable z use the chain rule to find the derivatives of the following functions: (a) (x - 5x+4) (b) 3x - 2x (c) 1 (x + 4x-5)
Using the product rule, find the derivatives of the following functions: (a) y = (x+6)(4x + 5) (b) y = (x-1)x (c) y = x(x - 2) (d) y = x(x - 2x+1)+ (e) y = x(x + 1)
Using the quotient rule, find the derivatives of the following functions: (a) y = (b) y = (d) y = x+6 x-3 (e) y = 2x+3 3x + 2 (c) y=- x - 2x+3 2x-1 2x x +2 3x-4 2x+x+5
1. Find the second derivative d2y/dx2 for the following y:2. IfHence show that: (a) x + 2x - 6x+8 (b) x + - +|+ X
1. Find the equation of the tangent to the curve y = x2 + 4 at the point where x = 2.2. Find the equation of the tangent to the graph of the cubic y = x3 − 2x2 + 2x + 4 at the point (1, 5). What is
1. A particle is thrown vertically upwards into the air. The height s metres above the ground after t seconds is given by s = 25t − 5t2(a) Find an expression for the velocity v of the particle at
Find the derivatives of the following functions:(a) exp(5x) (b) exp(−3x +1) (c) exp(x2 + 2x −1)(d) exp(√x) (e) exp(2x +1) (f) exp(−x2 + 4)
Find the derivatives of the following functions: (a) x exp(-x) (c) exp(-4x) x+1 1 (e) x+=-xexp(2x) x (g) exp(2x) - exp(-2x) (b) x + x exp(3x-2) exp(x/2) X (d) (f) x exp(3x + 2x-1) (h) exp(-x) 1+
1. Find the derivatives of the following functions:2. IfHence show that (a) In(8x) (b) In(3x-2) (c) x In(2x)
Find the derivatives of the following functions: (a) tan(3x) (d) cos 1-x (b) sin (2x) (e) x tan x (c) cos (6x) (f) 1-x sin: in x
The distance s metres travelled, from a given starting point, by a moving object at time t seconds after starting is given by s = 6t + 8t2 Determine the velocity and acceleration of the object when t
State which of the following problems are under-determined (that is, have insufficient boundary conditions to determine all the arbitrary constants in the general solution) and which are fully
What conditions on the functions g(t) and h(t) must be satisfied for the differential equationto be exact, and what is the solution of the equation when they are satisfied? d.x g(t) + h(t)x = 0 dt
To illustrate the method determine the median of the grouped golf scores given in Table 10.2. class intervals (scores) 65-69 70-74 75-79 80-84 85-89 90-94 95-99 mid-point Xi 67 72 77 82 87 92 97
Give the general solution of the following differential equations. In each case state how many arbitrary constants you expect to find in the general solution. Are your expectations confirmed in
For each of the following differential equation problems, state how many arbitrary constants you would expect to find in the most general solution satisfying the problem. Find the solution and check
A uniform horizontal beam OA, of length a and weight w per unit length, is clamped horizontally at O and freely supported at A. The transverse displacement y of the beam is governed by the
Sketch the direction field of the differential equationFind the solution of the equation. Sketch the particular solutions for which x(0) = 2, and for which x(2) = –3, and check that these are
Sketch the direction field of the differential equationVerify that x = t – 1 + Ce–t is the solution of the equation. Sketch the solution curve for which x(0) = 2, and that for which x(4) = 0, and
Draw the direction field of the equation Sketch some of the solution curves suggested by the direction field. Verify that the general solution of the equation is x = C/(t – 3)2 and check that the
Draw the direction field of the equationSketch some of the solution curves suggested by the direction field. Verify that the general solution of the equation is x = Cte–t and check that the members
Find the general solutions of the following differential equations: (a) (c) d.x dt = kx dx dt t bx d.x (b) -= 6xt dt d.x (d) - dt = a xt
Find the solutions of the following initial-value problems: (a) d.x dt (b) 72 dx dt sin t X || 1 x(0) = 4 x(4) = 9
Find the general solutions of the following differential equations: (a) r dx dt = x (b) dx dt = (1 + sin t) cotx
Find the solutions of the following initial-value problems: dx + 1 (a) = dt (b) t(t-1) = x(x+1), x(2) = 2 dx dt (c) (d) (e) +++ 2 x(0) = -2 dx d = (x dt d.x dt dx dt (x - 1) cost, x(0) = 2 e**, x(0)
A chemical reaction is governed by the differential equationwhere x(t) is the concentration of the chemical at time t. The initial concentration is zero and the concentration at time 5 s is found to
A skydiver’s vertical velocity is governed by the differential equationwhere K is the skydiver’s coefficient of drag. If the skydiver leaves her aeroplane at time t = 0 with zero vertical
A chemical A is formed by an irreversible reaction from chemicals B and C. Assuming that the amounts of B and C are adequate to sustain the reaction, the amount of A formed at time t is governed by
Find the general solutions of the following differential equations: dx (a) xt = x + t dt d.x (c) t- dt x + xt t r dx dt (b) x P + x to
Find the solution of the following initial-value problem: d.x x1 = 1 + x, x(1) = 4 dt
Find the general solutions of the following differential equations: d.x (a) 2x1 = -x - 1 dt dx (c) t- (e) dt d.x dt 3t - x t - 2x x + 1 x-t dx dt (b) t- (d) dx dt = x + tsin = x + t tan d.x (f) 1= x
Find the solutions of the following initial-value problems: x(1) = 2 zAX - EX IP xp (a)
Show that, by making the substitution y = at + bx + c, equations of the formcan be reduced to separable form. Hence find the general solutions of the following differential equations: dx dt = f(at +
For each of the following differential equations determine whether they are exact equations and, if so, find the general solutions: dx (a) x + t=0 = dt dx (b) x-t=0 dt dx (c) (x + 1)+x-t=0 dt d.x (d)
For each of the following differential equations determine whether they are exact, and, if so, find the general solution: 0 = 1 + X - dt (a) (x + t) - xp
11. Find the solutions of the following initial-value problems: (a) cos(x + t) d.x dt +1=0, x(0) = d.x (b) 3(x + 2)/2 + 6(x + 2)/2 + 1 = 0, dt x(-1) = 6
What conditions on the constants a, b, e and f must be satisfied for the differential equation to be exact, and what is the solution of the equation when they are satisfied? (ax + bt)- + ex + ft =
For what value of k is the function (x + t)k an integrating factor for the differential equation d.x [(x + 1)In(x + t) + x] + x = 0? dt
For what value of k is the function tk an integrating factor for the differential equation (tcos.xt)- + 3 sinxt + xt cos xt = 0? d.x dt
Find the solution of the following differential equations: d.x (a) + 3x = 2 dt (c) dr dt + 2x=e=4 (b) (d) dx dt dx dt - 4x = t + tx = -2t
Find the solution of the following initial-value problems: (a) (b) d.x dt dx dt - - 2x3, x(0) = 2 + 3x = 1, x(0) = 1
Find the solutions of the following differential equations: dx (a). - x = 1 + 2t dt d.x 2x (c) + -= cost dt t (e) + d.x dt dx dt - (2 cot 2t)x = cost (b) dx dt d.x (d) t- dt dx + 6tx = 1 +21 (g). 4tx
Find the solutions of the following initial-value problems: (a) (b) (c) (e) d.x dt (f) d.x dt dx dt (d) 2x - 1 + x = 0, x(2) = 2 dt dx dt 2t (2x1) = 0, x(0) = 0 dx dt = -xlnt, x(1) = 2 + 5x t = e ,
Solve (10.6), which arose from the model of the heating of the water in a domestic hot-water storage tank developed. If the water in the tank is initially at 10°C and Tin is 80°C, what is the ratio
Solve (10.7), which arose from the model of a hydroelectric power station developed. The setting of the control valve is represented in the model by the value of the parameter γ. Derive an
Find the solutions of the following differential equations: (a) d.x 1 1 dt (c) d.x dt = x -+ dx (b) + 2x = tx dt (d) dx dt t - = x - e' X 2 += x= x t
Find the solutions of the following initial-value problems: (a) d.x dt (c) dx (b). + 3x = x, x(0) = 6 dt dx dt dx dt (d). 1 { x=fx, _x(1)=1 + x = sint x, x(0) = -1 3 t X = 1 X x(-1) = 1
Find the value of X(0.3) for the initial-value problemusing Euler’s method with steps of h = 0.1. d.x dt = x - 2t, x(0) = 1
Find the value of X(0.25) for the initial-value problemusing Euler’s method with steps of h = 0.05. d.x dt = xt, x(0) = 2
Find the value of X(1) for the initial-value problemusing Euler’s method with step size h = 0.1. dx X dt 2(t+ x) x(0.5) = 1
Find the value of X(0.5) for the initial-value problemusing Euler’s method with step size h = 0.05. d.x dt || 4-t t + x x(0) = 1
Denote Euler’s method solution of the initial-value problemusing step size h = 0.1 by Xa(t), and that using h = 0.05 by Xb(t). Find the values of Xa(2) and Xb(2). Estimate the error in the value of
Denote Euler’s method solution of the initial-value problemusing step size h = 0.1 by Xa(t), and that using h = 0.05 by Xb(t). Find the values of Xa(2) and Xb(2). Estimate the error in the value

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