Mathematics for Management -- Supplementary Electronic Materials

Quiz: Derivatives of Trigonometic Functions

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1. Determine the derivative \(f'(x)\) of \(f(x) = \sin(x) - \cos(x)\).

\( f'(x) = \sin(x) + \cos(x) \) \( f'(x) = 2 \sin(x) \) \( f'(x) = \sin(x) - \cos(x) \) \( f'(x) = \cos(x) - \sin(x) \)

2. Determine the derivative \(f'(x)\) of \(f(x) = 5 \sin(x) - 2 \cos(x)\).

\( f'(x) = 5 \cos(x) - 2 \sin(x) \) \( f'(x) = 5 \cos(x) + 2 \sin(x) \) \( f'(x) = 2 \cos(x) + 5 \sin(x) \) \( f'(x) = 2 \cos(x) - 5 \sin(x) \)

3. Determine the derivative \(f'(x)\) of \(f(x) = \frac{3x}{\cos(x)}\).

\( f'(x) = -\frac{3}{\sin(x)} \) \( f'(x) = \frac{3 \cos(x) - 3 x \sin(x)}{\cos^2(x)} \) \( f'(x) = \frac{3 \cos(x) + 3 x \sin(x)}{\cos^2(x)} \) \( f'(x) = \frac{3}{\sin(x)} \)

4. Determine the derivative \(f'(x)\) of \(f(x) = \cos(x) \cdot \cos(x)\).

\( f'(x) = -\sin^2(x) \) \( f'(x) = 2 \sin(x) \cos(x) \) \( f'(x) = -2 \sin(x) \cos(x) \) \( f'(x) = -2\sin(x) \)

5. Determine the derivative \(f'(x)\) of \(f(x) = \frac{\tan(x)}{2x-3}\).

\( f'(x) = \frac{(2x-3) \sec(x) \tan(x) - 2 \tan(x)}{(2x-3)^2} \) \( f'(x) = \frac{(2x-3) \sec^2(x) - 2 \tan(x)}{(2x-3)^2} \) \( f'(x) = \frac{\sec^2(x) - 2 \tan(x)}{(2x-3)^2} \) \( f'(x) = \frac{(2x-3) \csc^2(x) - 2 \tan(x)}{(2x-3)^2} \)

6. Determine the derivative \(f'(x)\) of \(f(x) = \frac{2}{\sin(x)} + \frac{1}{\cot(x)}\).

\( f'(x) = 2 \cos(x) - \csc^2(x) \) \( f'(x) = 2 \csc(x) \cot(x) - \csc^2(x) \) \( f'(x) = -2 \csc(x) \cot(x) + \sec^2(x) \) \( f'(x) = 2 \csc(x) \cot(x) - \sec^2(x) \)

7. Determine the value of the slope of the tangent line to the curve \(y = 7 \cos(x)\) at the point with \(x = \frac{\pi}{4}\).

\( -\frac{7}{2} \) \( -\frac{7 \sqrt{2}}{2} \) \( -\frac{7 \sqrt{3}}{2} \) \( \frac{7 \sqrt{2}}{2} \)

8. The equation of line tangent to the curve \(y = x \cdot \sin(x)\) at the point \(x = \left( \frac{\pi}{2} , \frac{\pi}{2} \right)\).

\( y = x - \pi \) \( y = \frac{\pi}{2} \) \( y = \pi - x \) \( y = x \)

9. Determine the value of the following limit (if it exists): \[ \lim_{x \to 0} \frac{1 - \cos(x)}{x} \, . \]

\( 1 \) \( 0 \) \( \infty \) does not exist

10. Determine the value of the following limit (if it exists): \[ \lim_{x \to 0} \frac{\sin(x)}{x^2 + 3x} \, . \]

\( 1 \) \( \frac{1}{3} \) \( 3 \) \( \infty \)

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Solution: 1a; 2b; 3c; 4c; 5b; 6c; 7b; 8d; 9b; 10b

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.