Engineering Mathematics Problem Set
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Given vector $\mathbf{a} = 4\mathbf{i} - \mathbf{j} + 3\mathbf{k}$, find the magnitude of vector $\mathbf{a}$.
Given that $(a + b) + j(a - b) = 9 + j10$, find the values of $a$ and $b$.
Given that $\sin A = 12/13$ and $\cos B = 7/25$ and $A$ and $B$ are acute angles, determine the values of:
* $\sin(A + B)$
* $\cos(A - B)$
Prove the identity: $(\sin \theta)/(1 + \cos \theta) + (1 + \cos \theta)/(\sin \theta) = 2 \csc \theta$
Use implicit differentiation to determine the equation of normal to the curve $x^2 + y^2 - 4xy + 6x + 4y = 8$ at point $(1, 1)$.
Given the matrices $A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$, find $AB$.
Prove that $\cosh^2 x - \sinh^2 x = 1$
Expand $(3 - 2x)^5$ in ascending powers of $x$, up to and including the term in $x^3$.
Show that the equation has a better approximation to the root $2x^3 - 7x^2 - x + 12 = 0$
Resolve into partial fractions: $(11 - 3x)/(x^2 + 2x - 3)$
Given that $x_n$ is an approximation to the root of the equation $2x^3 + x^2 + x + 3 = 0$, use the Newton Raphson method to show that a better approximation is given by $x_{n+1} = (4x_n^3 + x_n^2 - 3)/(6x_n^2 + 2x_n + 1)$
The table in figure 1 represents a polynomial function and an error is suspected in one of the effects:
Table
X: 1, 2, 3, 4, 5, 6, 7, 8
f(X): 2, 3, 18, 31, 54, 83, 118, 159
* Use the finite table to locate and correct the errors
* Use the Newton Gregory interpolation formula to determine $f(3.7)$
Find the inverse Laplace transform of: $(s + 3)/(s^2 + 2s + 5)$
Determine the half-range Fourier Cosine Series for $f(x) = x$ in the range $0 < x < \pi$
Find the stationary points of the surface $Z = x^3 - xy + y^3$ and distinguish them
The area $A$ of a triangle is given by $A = (1/2)ac \sin B$, where $B$ is the angle between sides $a$ and $c$. If $a$ is increasing at $0.4$ units/s, $c$ is decreasing at $0.8$ units/s and $B$ is increasing at $0.2$ units/s, find the rate of change of the area of the triangle correct to 3 significant figures, where $a$ is 3 units, $c$ is 4 units and $B$ is $\pi/6$ radians.
This question includes visual content: A table labeled 'Table 1' containing two rows: 'X' and 'f(X)'. X values range from 1 to 8. Corresponding f(X) values are: 2, 3, 18, 31, 54, 83, 118, 159. This table is used for numerical analysis and interpolation exercises.
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In this problem, we are asked to use the Newton-Raphson method to derive a specific iterative formula for the roots of a given cubic equation.
Newton-Raphson Derivation
The Newton-Raphson method is a powerful numerical algorithm that uses the tangent lines of a function's graph to find increasingly accurate approximations of its roots.
We begin by recalling the general iteration formula. The next approximation, x sub n plus one, is found by subtracting the ratio of the function and its derivative from the current approximation, x sub n.
The Iteration Formula
Let's first define our function, f of x, based on the cubic equation provided in the problem.
To use the method, we must compute the first derivative of this function with respect to x.
By the linearity of differentiation, we can take the derivative of each term individually.
Applying the power rule, the derivative of two x cubed is six x squared. x squared becomes two x, x becomes one, and the constant three becomes zero.
Simplifying this, we have our derivative function completely defined.
Since our iterative formula requires these functions evaluated at x sub n, we simply replace every x with x sub n.
Now we are ready to substitute these expressions back into our Newton-Raphson iteration formula.
Substituting the Expressions
To simplify this into a single fraction, we need a common denominator. We achieve this by multiplying the isolated x sub n term by our denominator divided by itself.
Since both parts now share the same denominator, we can combine their numerators into a single rational expression.
The denominator is now in its final form, so let's extract the numerator and simplify it step by step.
Simplifying the Numerator
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