Advance Calculus

Numerial methods and advance calculus

Q1

A function of two variables is given by,
f (x,y) = 4×3 + 7xy4 – 5y2 + 8.

Determine,  fxx + fyx at x = 6.62 and y =

1.

21,
giving your answer to 3 decimal places.

Answer:

Q2

Consider the initial value problem,

 f(x,y) = y(18.17 – y),           y(0) = 1

2.

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.

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

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 Valu

e.

Answer:

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.

Answer:

Q8

A function is given by,
f(x,y) = x4 – y2 – 2×2 + 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.

1.	(0,1) inconclusive, (-1,1) local minimum.
2.	(0,1) local maximum, (-1,1) local minimum.
3.	(0,1) inconclusive, (-1,1) local maximum.
4.	(0,1) local minimum, (-1,1) local maximum.
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 minimize

d.

Find the dimensions of the box such that the amount of material is minimized.

a.	x = 4 cm, y = 2 cm and z = 4 cm
b.	x = 2 cm, y = 8 cm and z = 2 cm
c.	x = 4 cm, y = 4 cm and z = 2 cm
d.	x = 2 cm, y = 4 cm and z = 4 cm
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.

Answer: