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The image contains questions from a mathematics exam paper, covering topics in calculus, differentiation, integration, partial derivatives, Maclaurin's series, Taylor's series, and more. I'll summarize the questions:

### 1. Definitions and Theorems:
(a) Define and give examples of:
- Function
- Differentiation
- Integration
- Partial derivative
- Ordinary derivative

(b) State and prove:
- Mean Value Theorem

(c) State:
- Rolle’s Theorem

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### 2. Derivatives and Series:
(a) State, without proof:
- Taylor's series in two variables
- Maclaurin's series in two variables

(b) Complete the following table with the first derivative \( f'(x) \):

| S/N | \( f(x) \) | \( f'(x) \) |
|-----|------------|-------------|
| 1   | \( \cos x \) | |
| 2   | \( \ln x \) | |
| 3   | \( e^x \) | |
| 4   | \( \sin 2x \) | |
| 5   | \( \tan x \) | |
| 6   | \( \cos 2x \) | |

(c) Using appropriate symbols, write an expression for the total derivative with respect to a small increment on a function.

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### 3. Series and Derivatives:
(a) Prove that \( f(x) = \sin 3x \) leads to the given Maclaurin series expansion.

(b) State, without proof:
- Maclaurin’s Series Theorem

(c) Distinguish between:
- Taylor's and Maclaurin's series

(d) If \( y = \sin^2 x \), find \( \frac{dy}{dx} \).

---

### 4. Partial Derivatives:
(a) Given \( f(x, y) = 2xy - y^2 + x^2 \), find the following:
- \( \frac{\partial f}{\partial x} \)
- \( \frac{\partial^2 f}{\partial x^2} \)
- \( \frac{\partial f}{\partial x \partial y} \)
- \( \frac{\partial^2 f}{\partial y \partial x} \)
- \( \frac{\partial^2 f}{\partial y^2} \)

(b) If \( y = \cos^2 x \), find \( \frac{dy}{dx} \).

(c) If \( y = \tan^{-1} x \), find \( \frac{dy}{dx} \).

(d) If \( f(x, y) = x \sin 3y \), find:
- \( \frac{\partial f}{\partial x} \)
- \( \frac{\partial f}{\partial y} \)

Show that \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \).

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### 5. Maclaurin Series and Integration:
(a) Generate Maclaurin’s series for the following functions:
- \( f(x) = \sin x \)
- \( f(x) = \cos x \)

(b) Distinguish between:
- Maximum and minimum value of a function
- Differentiation and integration of a function

(c) Discuss with examples:
- The term "multiple integration"

(d) Mention:
- The three basic types of integration studied
- One application of these types of integration

(e) Evaluate the multiple integrals:
1. \( \int_0^1 \int_0^x (x + y)^2 \, dy \, dx \)
2. \( \int_0^1 \int_1^2 \int_2^3 (x + y + z) \, dx \, dy \, dz \)

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If you'd like to work on any of these specific questions, feel free to let me know, and I can assist further.

Would you like detailed solutions for any of these problems? 

### Here are five related questions to expand your understanding:
1. How do you apply the Mean Value Theorem to a given function?
2. What are the differences between ordinary and partial derivatives?
3. How do you determine higher-order partial derivatives for multivariable functions?
4. What is the role of Maclaurin's series in approximating functions near \( x = 0 \)?
5. How is multiple integration applied in calculating volumes and areas?

**Tip:** Remember, partial derivatives focus on how a function changes with respect to one variable, while keeping the others constant.

Mathematics exam paper covering topics such as differentiation, integration, partial derivatives, Taylor's and Maclaurin's series, and multiple integration.

This exam paper covers essential topics in calculus, including differentiation, integration, partial derivatives, Taylor's series, and Maclaurin's series. It also includes questions on multiple integration with examples. Suitable for university students studying advanced mathematics.

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