October 09, 2020

CMA01

Started: 10 Oct at 10:48

Quiz instructions

This online quiz tests your competency on Units 1, 2, 3 and 4. There are 20 questions: each question carries 5 marks. Answer all questions. The cut-off-date is on Friday 16 October 2020 at 2355.

Unless otherwise stated, the magnitude of the acceleration due to gravity g, is taken to be 9.81 ms-2, and angles are in radians.

The definition of n-th harmonics is that it involves cos(nt) or sin(nt) or both. Hence 5th harmonics means cos(5t) or sin(5t) or both.

Flag this question

A function of three variables is given by

f (x,y,t) = x3y2sin t + 4x2t + 5yt2 + 4xycos t

Find ft (-0.384,2.780,1.198) giving your answer correct to 3 decimal places.

Flag this question

A function of two variables is given by

f (x,y) = 4x3 + 7xy4 - 5y2 + 8

Determine fxx + fyx at x = 6.478 and y = 1.505.
Give your answer correct to 3 decimal places.

Flag this question

A function is given by,

f(x,y) = x4 - y2 - 2x2 + 2y - 7

Using the AC - B2 criterion for functions of two variables, classify the points (0,1) and (-1,1) as local maximum, local minimum or saddle-point.

Group of answer choices

(0,1) local maximum, (-1,1) saddle-point.

(0,1) saddle-point, (-1,1) local maximum.

(0,1) local maximum, (-1,1) local minimum.

(0,1) local minimum, (-1,1) local maximum.

(0,1) saddle-point, (-1,1) local minimum.

Flag this question

This is an optimization problem.

A rectangular box with no top is to be constructed with a volume of 5.09 cm3. Let x be the width, y be the length, and z be the height.

Find the minimized amount of material in cm2 , correct to 3 decimal places.

Flag this question

A function is given by,

f(x) = e-3x

Write down the third-order Taylor approximation for f(x) about x = 0.

Hence, evaluate f(0.512) giving your answer to 3 decimal places.

Flag this question

A function of two variables is given by,

f(x,y) = e2x-3y

Find the first-order Taylor approximation to f(0.763,1.279) near (0,0), giving your answer to 3 decimal places.

Flag this question

Consider the initial value problem,

f(x,y) = y(20.86 - y), y(0) = 12.

The exact solution of the problem increases from y(0) = 12 to y = 20.86 as x increases without limit.

Determine the minimum upper bound of h for the classical 4th-order Runge-Kutta method to be absolutely stable for this problem. Give your answer to 3 decimal places.

Note : You can make use of table (3.1) of Study Unit 2 Numerical Methods for Differential Equations.

Flag this question

An initial-value problem is given by the differential equation,

f(x,y) = x(1 - y2), y(1) = 0.109.

Use the Euler-trapezoidal method with a step-size of h = 0.1 to obtain the approximate value of y(1.1). Give your answer to 4 decimal places.

Flag this question

An initial-value problem is given by the differential equation,

dy/dx = f(x,y) = x + y, y(0) = 2.802

Use the Euler-midpoint method to find an approximate

value to y(0.1) with a step size of 0.1, correct to 3 decimal places.

Flag this question

An initial-value problem is given by the differential equation,

f(x,y) = x + y, y(0) = 91.15

The Euler-midpoint method is used to find an approximate value to y(0.1) with a step size of h = 0.1.

Then use the integrating factor method, to find the exact value of y(0.1).

Hence, determine the global error, giving your answer to 4 decimal places.

Note that Global Error = Approximate Value - Exact Value.

Flag this question

A function f(t) is defined as,

f(t) = π - t for 0 < t < π

Write down the even extension of f(t) for -π < t < 0.

Determine the Fourier cosine series, and hence, calculate the Fourier series approximation for f(t) up to the 5th harmonics when t = 1.73. Use π = 3.142. Give your answer to 3 decimal places.

Flag this question

A function f(t) is defined as,

f(t) = π - t for 0 < t < π

Write down the odd extension of f(t) for -π < t < 0.

Determine the Fourier sine series, and hence, calculate the Fourier series approximation for f(t) up to the 3rd harmonics when t = 2.97. Use π = 3.142. Give your answer to 3 decimal places.

Flag this question

An infinite cosine series is given by,

Compute the sum to r = 3 and t = 1.47. Use π = 3.142. Give your answer to 3 decimal places.

Flag this question

A periodic function f(t), with period 2π is defined as,

f(t) = c for 0 < t < π
f(t) = -c for -π < t < 0

where c = 1.9, Taking π = 3.142, calculate the Fourier sine series approximation up to the 5th harmonics when t = 2.23. Give your answer to 3 decimal places.

Flag this question

A periodic function f(t), with period 2π is defined as,

f(t) = 0 for -π < t < 0
f(t) = π for 0 < t < π

Taking π = 3.142, calculate the Fourier series approximation up to the 5th harmonics when t = 0.85. Give your answer to 3 decimal places.

Flag this question

Consider the temperature distribution in a perfectly insulated rod of length L, where one end, at x = 0, is maintained at a temperature of 00C and the other end, at x = L, is insulated. This is well modeled by the diffusion equation,

where a is a constant. It is subjected to boundary conditions,

The method of separation of variables, with θ(x, t) = X(x)T(t), is used to determine the boundary-value problem satisfied by X(x) which is given by.

X(x) = sin(βx) where β is in terms of L and n = 1,2,3,......

Calculate the value of β when n = 2 and L = 0.99, giving your answer to 3 decimal places. Use π = 3.142.

Flag this question

Consider the temperature distribution in a perfectly insulated rod of length L, where one end, at x = 0, is maintained at a temperature of 00C and the other end, at x = L, is insulated. This is well modeled by the diffusion equation,

where a is a constant. It is subjected to boundary conditions,

The method of separation of variables, with θ(x, t) = X(x)T(t), is used to determine the boundary-value problem satisfied by T(t) which is given by,

T(t) = exp(-kt) where k is in terms of a, L and n = 1,2,3,......

Calculate the value of T at t = 0.45, when a = 0.5, n = 1 and L = 1.85, giving your answer to 3 decimal places. Use π = 3.142.

Flag this question

Find the Fourier series expansion for the periodic function,

f(t) = t

in the interval -π < t < π. Taking π = 3.142, calculate the Fourier sine series approximation of f(t), up to the 3rd harmonics when t = 1.53. Give your answer to 3 decimal places.

Flag this question

Find the Fourier series expansion for the periodic function,

f(t) = t2

in the interval -π < t < π. Taking π = 3.142, calculate the Fourier cosine series approximation of f(t), up to the 3rd harmonics when t = 2.89. Give your answer to 3 decimal places.

Flag this question

The Fourier series expansion for the periodic function,

f(t) = |sin t|

is defined in its fundamental interval. Taking π = 3.142, calculate the Fourier cosine series approximation of f(t), up to the 6th harmonics when t = 0.84. Give your answer to 3 decimal places.

Not saved

Comments