Leaving Cert Mathematics Trigonometry: Let $h(x) = ext{cos}(2x)$, wher

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Question 3

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Let $h(x) = ext{cos}(2x)$, where $x ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ... show full transcript

Worked Solution & Example Answer:Let $h(x) = ext{cos}(2x)$, where $x ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } \ A tangent is drawn to the graph of $h(x)$ at the point where $x = \frac{ ext{π}}{3}$.\ Find the angle that this tangent makes with the positive sense of the $x$-axis.\ \ Find the average value of $h(x)$ over the interval $0 \leq x \leq \frac{ ext{π}}{4}$, $x \in ext{R}$.\ Give your answer in terms of π. - Leaving Cert Mathematics - Question 3 - 2018

Step 1

Find the angle that this tangent makes with the positive sense of the x-axis.

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Answer

To find the angle of the tangent line at $x = \frac{\pi}{3}$ , we first need to compute the derivative of $h(x)$ .\

Differentiate $h(x)$ :

The function is given as $h(x) = \cos(2x)$ .
Differentiating, we get:
[ h'(x) = -2 \sin(2x) ]\
Evaluate the derivative at $x = \frac{\pi}{3}$ :

Substitute $x = \frac{\pi}{3}$ :
[ h'\left(\frac{\pi}{3}\right) = -2 \sin\left(2 \cdot \frac{\pi}{3}\right) = -2 \sin\left(\frac{2\pi}{3}\right) = -2 \cdot \frac{\sqrt{3}}{2} = -\sqrt{3} ]\
Find the angle:

The angle $\theta$ that the tangent makes with the positive $x$ -axis can be found using the tangent function:
[ \tan(\theta) = \frac{-\sqrt{3}}{1} ]
Therefore,
[ \theta = 120^\circ ]
The tangent line makes an angle of $120^\circ$ with the positive sense of the $x$ -axis.

Step 2

Find the average value of h(x) over the interval 0 ≤ x ≤ π/4, x ∈ R.

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Answer

The average value of a function $f(x)$ over the interval $[a, b]$ is given by the formula:
[ \text{Average} = \frac{1}{b-a} \int_a^b f(x) , dx ]
In this case:

$f(x) = h(x) = \cos(2x)$
$a = 0$ , $b = \frac{\pi}{4}$ .

Set up the integral:

[ \text{Average} = \frac{1}{\frac{\pi}{4} - 0} \int_0^{\frac{\pi}{4}} \cos(2x) , dx ]

This simplifies to:
[ = \frac{4}{\pi} \int_0^{\frac{\pi}{4}} \cos(2x) , dx ]
Compute the integral:

The integral of $\cos(2x)$ is:
[ \int \cos(2x) , dx = \frac{1}{2} \sin(2x) + C ]
Evaluating the definite integral:
[ \int_0^{\frac{\pi}{4}} \cos(2x) , dx = \left[ \frac{1}{2} \sin(2x) \right]_0^{\frac{\pi}{4}} = \frac{1}{2} \left( \sin\left(\frac{\pi}{2}\right) - \sin(0) \right) = \frac{1}{2} (1 - 0) = \frac{1}{2} ]
Final calculation:

Plugging back into the average formula:
[ \text{Average} = \frac{4}{\pi} \cdot \frac{1}{2} = \frac{2}{\pi} ]
Therefore, the average value of $h(x)$ over the specified interval is $\frac{2}{\pi}$ .

Find the angle that this tangent makes with the positive sense of the x-axis.

Find the average value of h(x) over the interval 0 ≤ x ≤ π/4, x ∈ R.

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