Advanced Engineering Mathematics Problems Set
Published:
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' with two rows. The first row 'X' contains integer values from 1 to 8. The second row 'f(X)' contains the following values corresponding to X: 2, 3, 18, 31, 54, 83, 118, 159. The image is a composite of several text-based math problems on a dark background.
Animated Video Solution
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Step by Step Written Solution
Let's prove the first trigonometric identity from the list step by step. We want to show that sine theta over one plus cosine theta, plus one plus cosine theta over sine theta, equals two cosecant theta.
Proving a Trigonometric Identity
A standard algebraic strategy for verifying identities is to start with the more complicated side and simplify it. Let's define our starting point as the left-hand side.
We have a sum of two rational expressions. To combine them, we need a common denominator, which will be the product of one plus cosine theta, and sine theta.
By multiplying the numerators together across both terms, we can now write this entire expression as a single large fraction over our new common denominator.
Let's carry this combined expression over to a clean board and focus our attention entirely on expanding the terms in the numerator.
Expanding the Numerator
For the first term in the numerator, sine theta multiplied by sine theta simplifies directly to sine squared theta.
Next, we need to carefully distribute the binomials multiplied by each other. Expanding one plus cosine theta squared gives four terms in the numerator.
Inside those grouping parentheses, we have two isolated cosine theta terms. We can add them together right away to get a middle term of two cosine theta.
Now, let's purposely rearrange the terms in the numerator so that sine squared theta and cosine squared theta sit right next to each other.
Do you notice this familiar pair of squared trigonometric terms? Let's quickly pause and recall the most fundamental Pythagorean identity.
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