Second Derivatives of Trigonometric Functions Worksheet
Grade 12 Calculus worksheet focusing on finding and interpreting second derivatives of trigonometric functions.
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Second Derivatives of Trigonometric Functions
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Read each question carefully and show all your work. Simplify your answers where possible.
1. Find the second derivative of the function $f(x) = \sin(2x)$.
2. Determine the second derivative of $g(x) = \cos(x^2)$.
3. For the function $h(x) = x + \sin(x)$, which of the following intervals is the function concave up?
$(2n\pi, (2n+1)\pi)$
$((2n+1)\pi, (2n+2)\pi)$
$(\frac{\pi}{2} + 2n\pi, \frac{3\pi}{2} + 2n\pi)$
Nowhere
4. A point of inflection occurs where the second derivative changes sign, indicating a change in .
5. Find the x-coordinates of the points of inflection for $f(x) = \sin(x)$ on the interval $[0, 2\pi]$. The points are .
6. If $f''(x) > 0$ on an interval, then the function $f(x)$ is concave down on that interval.
True
False
7. The second derivative of $f(x) = \tan(x)$ is $2\sec^2(x)\tan(x)$.
True
False
8. The position of a particle is given by $s(t) = 3\cos(t) + 4\sin(t)$. Find the acceleration of the particle at $t = \frac{\pi}{2}$.