Advance Calculus
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tutorial_5.docxQ 1
A function of two variables is given by, f (x,y) = 4x3 + 7xy4 - 5y2 + 8.
Determine, fxx + fyx at x = 6.62 and y = 1.21, giving your answer to 3 decimal places.
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Q2
Consider the initial value problem,
f(x,y) = y(18.17 - y), y(0) = 12.
The exact solution of the problem increases from y(0) =12 to y = 18.17 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.
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Q 3
An initial-value problem is given by the differential equation,
f(x,y) = –20xy2, y(1) = 1.
Use the classical fourth-order Runge-Kutta method with a step-size of h = 0.02, to obtain the approximate value of y(1.02). Give your answer to 6 decimal places.
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Q4
A function of three variables is given by, f (x,y,t) = x3y2sin t + 4x2t + 5yt2 + 4xycos t. Find ft (8.18,0.58,3.16) giving your answer to 3 decimal places.
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Q5
An initial-value problem is given by the differential equation,
f(x,y) = x(1 - y2), y(1) = 0.48.
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.
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Q6
An initial-value problem is given by the differential equation, f(x,y) = x + y,
y(0) = 0.45
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 5 decimal places.
Note that Global Error = Approximate Value - Exact Value.
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Q7
A function of two variables is given by, f(x,y) = e2x-3y
Find the tangent approximation to f(0.778,0.647) near (0,0), giving your answer to 4 decimal places.
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Q8
A function is given by, f(x,y) = x4 - y2 - 2x2 + 2y - 7 Using the second derivative test for functions of two variables, classify the points (0,1) and (-1,1) as local maximum, local minimum or inconclusive.
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1. |
(0,1) inconclusive, (-1,1) local minimum. |
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2. |
(0,1) local maximum, (-1,1) local minimum. |
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3. |
(0,1) inconclusive, (-1,1) local maximum. |
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4. |
(0,1) local minimum, (-1,1) local maximum. |
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5. |
(0,1) local maximum, (-1,1) inconclusive. |
Q9
This is an optimization problem. A rectangular box with no top is to be constructed to have a volume of 32 cm3. Let x be the width, y be the length and z be the height. The amount of material used to construct the box is to be minimized. Find the dimensions of the box such that the amount of material is minimized.
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a. |
x = 4 cm, y = 2 cm and z = 4 cm |
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b. |
x = 2 cm, y = 8 cm and z = 2 cm |
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c. |
x = 4 cm, y = 4 cm and z = 2 cm |
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d. |
x = 2 cm, y = 4 cm and z = 4 cm |
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e. |
x = 2 cm, y = 16 cm and z = 1 cm |
Q10
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(0.235) giving your answer to 4 decimal places.
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