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Quiz 2: Differentiation and integration

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Quiz

This quiz tests materials in the section called “Differentiation and Integration”. If you are struggling with the questions go back and re-read the relevant material.

Differentiation

The first part contains questions on differentiation

Question 1: Differentiate the function \(f(x) = x^3\).

  1. \(3 x^3\)

  2. \(x^3\)

  3. \(3 x^2\)

  4. \(\frac{x^4}{4}\)

Question 2: Differentiate the function \(f(x) = x^{-7}\).

  1. \(-7 x^{-8}\)

  2. \(\frac{x^{-9}}{9}\)

  3. \(\frac{-8}{x^8}\)

  4. \(\frac{7 \times 8 x}{x^8}\)

Question 3: Differentiate the function \(f(x) = \sqrt{1+x^2}\).

  1. \(\frac{2x}{(1 + x^2)^{1/2}}\)

  2. \(\frac{x (1 + x)^{-1/2}}{2}\)

  3. \(\{x (1+x^2)\}^{-1/2}\)

  4. \(\frac{x}{\sqrt{1 + x^2}}\)

Question 4: Differentiate the function \(f(x) = \frac{1}{log_e(x)}\).

  1. \(\frac{x}{log_e(x)}\)

  2. \(\frac{-1}{x(log_e(x))^2}\)

  3. \(\frac{-x}{(log_e(x))^2}\)

  4. \(\frac{-1}{(x log_e(x))^2}\)

Question 5: Differentiate the function \(f(x) = exp(4x)\).

  1. \(4x \, exp(4x)\)

  2. \(4 e^{4x}\)

  3. \(16 \, exp(3x)\)

  4. \(4 e^{3x}\)

Question 6: Differentiate the function \(f(x) = \frac{(1 + x)}{x^2}\).

  1. \(-\frac{2(1+x)}{x^3}\)

  2. \(\frac{1}{x^2}\)

  3. \(-\frac{(2+x)}{x^3}\)

  4. \(\frac{(2+x)}{x^3}\)

Question 7: Differentiate the function \(f(x) = e^{3x^2}\).

  1. \(3x^2 exp(3 x^2)\)

  2. \(e^{6x}\)

  3. \(3x^2 \times 6x \times e^{3 x^2}\)

  4. \(6x e^{3x^2}\)

Question 8: Define

\[L = -\frac{(x - \mu)^2}{s} - \log(s)\]

Find \(\frac{\partial L}{\partial s}\), \(\frac{\partial L}{\partial \mu}\), \(\frac{\partial^2 L}{\partial s^2}\), \(\frac{\partial^2 L}{\partial \mu^2}\), \(\frac{\partial^2 L}{\partial s \partial \mu}\), \(\frac{\partial^2 L}{\partial \mu \partial s}\).

Which of the following are correct?

  1. \(\frac{\partial^2 L}{\partial s^2} = \frac{2(x-\mu)^2}{s^3} + \frac{1}{s^2}, \ \ \qquad \frac{\partial^2 L}{\partial \mu^2} = \frac{-2}{s^2}, \qquad \frac{\partial^2 L}{\partial s \partial \mu} = \frac{-2(x-\mu)}{s^2}\)

  2. \(\frac{\partial^2 L}{\partial s^2} = \frac{-2(x-\mu)^2}{s^3} - \frac{1}{s^2}, \qquad \frac{\partial^2 L}{\partial \mu^2} = \frac{2}{s}, \qquad \frac{\partial^2 L}{\partial s \partial \mu} = \frac{2(x-\mu)}{s^2}\)

  3. \(\frac{\partial^2 L}{\partial s^2} = \frac{-2(x-\mu)^2}{s^3} + \frac{1}{s^2}, \qquad \frac{\partial^2 L}{\partial \mu^2} = \frac{-2}{s}, \qquad \frac{\partial^2 L}{\partial s \partial \mu} = \frac{-2(x-\mu)}{s^2}\)

  4. \(\frac{\partial^2 L}{\partial s^2} = \frac{-2(x-\mu)^2}{s^3} - \frac{1}{s^2}, \qquad \frac{\partial^2 L}{\partial \mu^2} = \frac{-2}{s}, \qquad \frac{\partial^2 L}{\partial s \partial \mu} = \frac{-2(x-\mu)^2}{s^2}\)

Integration

The next part contains questions on integration.

Question 9: Perform the integration

\[\int 3 x^4 dx\]

Choose the correct solution:

  1. \(\frac{3 x^4}{5} + c\)

  2. \(\frac{3 x^5}{5} + c\)

  3. \(x^3 + c\)

  4. \(\frac{3 x^3}{4} + c\)

where \(c\) is a constant.

Question 10: Perform the integration

\[\int \frac{1}{x} dx\]

Choose the correct solution:

  1. \(-\frac{1}{x^2} + c\)

  2. \(x log_e(x) + c\)

  3. \(log_e\left( \frac{1}{x} \right) + x\)

  4. \(log_e(x) + c\)

where \(c\) is a constant.

Question 11: Perform the integration

\[\int \frac{x}{x^2-4} dx\]

Choose the correct solution:

  1. \(\frac{1}{2} log(x^2+4) + c\)

  2. \(log( \sqrt{x^2+4}) + c\)

  3. \(\frac{1}{2} log((x-2)(x+2)) + c\)

  4. All of the above

where \(c\) is a constant.

Question 12: Find the area under the curve for the following function:

\[y = x^2\]

a) between \(x = 0\) and \(x = 3\), and b) between \(x = -3\) and \(x = -0\)

Choose the correct option:

  1. a) \(9\) and b) \(9\)

  2. a) \(9\) and b) \(0\)

  3. a) \(9\) and b) \(-9\)

  4. a) \(3\) and b) \(-3\)

Question 13: Find the area under the curve for the following function:

\[y = x^3 – 3x\]

a) between \(x = 0\) and \(x = 3\), and b) between \(x = -2\) and \(x = 2\)

Choose the correct option:

  1. a) \(\frac{3^3}{4}\) and b) \(0\)

  2. a) \(\frac{9}{4}\) and b) \(4\)

  3. a) \(6.75\) and b) \(-8\)

  4. a) \(6.75\) and b) \(10\)

Quiz 1: Powers and logs Quiz 3: Matrices