# 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 \(f(x) = x^3\).

\(3 x^3\)

\(x^3\)

\(3 x^2\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(4x \, exp(4x)\)

\(4 e^{4x}\)

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

\(4 e^{3x}\)

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

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

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

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

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

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

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

\(e^{6x}\)

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

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

**Question 8**: Define

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?

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

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

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

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

### Integration¶

The next part contains questions on integration.

**Question 9**: Perform the integration

Choose the correct solution:

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

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

\(x^3 + c\)

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

where \(c\) is a constant.

**Question 10**: Perform the integration

Choose the correct solution:

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

\(x log_e(x) + c\)

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

\(log_e(x) + c\)

where \(c\) is a constant.

**Question 11**: Perform the integration

Choose the correct solution:

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

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

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

All of the above

where \(c\) is a constant.

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

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

Choose the correct option:

a) \(9\) and b) \(9\)

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

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

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

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

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

Choose the correct option:

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

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

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

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