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Given $h(x) = 3 \, ext{cos} \, 2x$, find the value of $h \left( \frac{\pi}{6} \right)$ - Scottish Highers Maths - Question 3 - 2018

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Given--$h(x)-=-3-\,--ext{cos}-\,-2x$,-find-the-value-of-$h-\left(-\frac{\pi}{6}-\right)$-Scottish Highers Maths-Question 3-2018.png

Given $h(x) = 3 \, ext{cos} \, 2x$, find the value of $h \left( \frac{\pi}{6} \right)$

Worked Solution & Example Answer:Given $h(x) = 3 \, ext{cos} \, 2x$, find the value of $h \left( \frac{\pi}{6} \right)$ - Scottish Highers Maths - Question 3 - 2018

Step 1

Differentiate the function

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Answer

To find the value of h(π6)h \left( \frac{\pi}{6} \right), we first need to differentiate the function. The derivative of h(x)=3cos(2x)h(x) = 3 \text{cos}(2x) is given by:

h′(x)=−3⋅2sin(2x)=−6sin(2x)h'(x) = -3 \cdot 2 \text{sin}(2x) = -6 \text{sin}(2x)

Step 2

Evaluate the derivative at $x = \frac{\pi}{6}$

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Answer

Next, we evaluate h′(π6)h'\left( \frac{\pi}{6} \right):

h′(π6)=−6sin(2⋅π6)=−6sin(π3)h'\left( \frac{\pi}{6} \right) = -6 \text{sin}\left( 2 \cdot \frac{\pi}{6} \right) = -6 \text{sin}\left( \frac{\pi}{3} \right)

Using the sine value, extsin(π3)=32 ext{sin}\left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}, we continue:

h′(π6)=−6⋅32=−33h'\left( \frac{\pi}{6} \right) = -6 \cdot \frac{\sqrt{3}}{2} = -3\sqrt{3}

Step 3

Final Result

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Answer

Thus, the value of h′(π6)h' \left( \frac{\pi}{6} \right) is −33-3\sqrt{3}.

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