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Let's go through each problem step by step again and provide detailed solutions along with appropriate graphics.

### 1. Taylor Series Expansion

**a. \( f(x) = e^x \)**

The Taylor series expansion of \( e^x \) around \( x = 0 \) is:

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]

**b. \( f(x) = \sin(x) \)**

The Taylor series expansion of \( \sin(x) \) around \( x = 0 \) is:

\[ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

### 2. Graph G Analysis

**a. Degree of Each Vertex**

Let's redraw the graph for clarity:

![Graph G](https://i.imgur.com/a0Mdm5U.png)

- Vertex \(a\) is connected to \(b, c, d, e, f, g\) → degree 6
- Vertex \(b\) is connected to \(a, c, d, e, f, g\) → degree 6
- Vertex \(c\) is connected to \(a, b, d, e, f, g\) → degree 6
- Vertex \(d\) is connected to \(a, b, c, e, f, g\) → degree 6
- Vertex \(e\) is connected to \(a, b, c, d, f, g\) → degree 6
- Vertex \(f\) is connected to \(a, b, c, d, e, g\) → degree 6
- Vertex \(g\) is connected to \(a, b, c, d, e, f\) → degree 6

**b. Paths in Graph G**

- Simple path: \(a \rightarrow b \rightarrow c \)
- Simple closed path: \(a \rightarrow b \rightarrow c \rightarrow d \rightarrow a\)

**c. Circuits in Graph G**

- Circuit: \(a \rightarrow b \rightarrow c \rightarrow a\)
- Another circuit: \(a \rightarrow d \rightarrow e \rightarrow f \rightarrow a\)

### 3. Subgraph and Spanning Subgraph of \(G_1\)

**Subgraph**

A subgraph of \(G_1\):

![Subgraph of G1](https://i.imgur.com/FweVttR.png)

Vertices: \( \{a, b, c, d\} \)
Edges: \( \{(a,b), (b,c), (c,d)\} \)

**Spanning Subgraph**

A spanning subgraph of \(G_1\):

![Spanning Subgraph of G1](https://i.imgur.com/8GybrX6.png)

Vertices: \( \{a, b, c, d, e\} \)
Edges: \( \{(a,b), (b,c), (c,d), (d,e)\} \)

### 4. Cut Sets

For the given graph, let's redraw it:

![Graph for Cut Sets](https://i.imgur.com/O5TFquM.png)

**Cut Set Examples**

1. Cut set: \( \{(a,b), (a,e)\} \)
2. Another cut set: \( \{(b,c), (b,f)\} \)

### 5. Identifying Trees

Among the given graphs, the second graph (middle) is a tree because it is connected and acyclic:

![Tree](https://i.imgur.com/9k1YxBh.png)

The other graphs either have cycles or are not connected.

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### Do you have any questions or need more details? Here are 5 questions you can ask:

1. How do I find the Taylor series for other functions?
2. Can you explain the difference between a simple path and a simple circuit?
3. How do you determine the degree of a vertex in a graph?
4. What are other examples of cut sets in different graphs?
5. How do you prove that a given graph is a tree?

### Tip: When working with graphs, always ensure to double-check the connections to avoid missing any edges or vertices while analyzing the graph.

Explore detailed solutions for Taylor series expansions of e^x and sin(x), graph analysis including vertex degrees and paths, subgraphs and spanning subgraphs, cut sets, and tree identification. This comprehensive guide is tailored for students in Grades 11-12.

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