MTH 217 CMA01 Answer Key
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.
Excellent partial differentiation. Keep it up!
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.
Excellent!
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.
Very Good!
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.
Bravo! Your optimization concept is strong,
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.
You are brilliant!
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.
Excellent!
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.
Fantastic! You got it.
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.
Incorrect! Exercise more care and patience in numerical computation.
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.
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.
Brilliant! Your understanding of global error is commendable.
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.
Fantastic! You can write the even extension. I'm proud of you.
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.
Excellent! You knew the meaning of odd extension.
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.
Excellent! You fully understood the summation of infinite series.
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.
Brilliant! You are a Fourier expert.
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.
Excellient!
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.
Excellent! Your understanding of PDE is sound. Keep it up!
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.
Brilliant! Your PDE concept is strong.
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.
Very good, you did it correctly.
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.
Excellent!
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.
Fantastic! You made it in Fourier Series.
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