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We are given that \( G(x) = 4f(x) - g(x) \), where \( f(x) \) and \( g(x) \) are shown in the graph, and we need to find \( G'(1) \), which is the derivative of \( G(x) \) at \( x = 1 \).

### Step-by-step solution:

1. **Derivative of G(x):**
   \[
   G'(x) = \frac{d}{dx} [4f(x) - g(x)] = 4f'(x) - g'(x)
   \]
   So, to find \( G'(1) \), we need to calculate \( f'(1) \) and \( g'(1) \), the derivatives of \( f(x) \) and \( g(x) \) at \( x = 1 \).

2. **Determine slopes from the graph:**
   The derivative at a point is the slope of the tangent line at that point. From the graph, we can estimate the slopes of \( f(x) \) and \( g(x) \) at \( x = 1 \).

   - The function \( f(x) \) appears to be a straight line with a positive slope in the segment where \( x = 1 \). We can calculate the slope by observing the rise over run between two points on this line. 
   
     For \( f(x) \):
     - Between \( (0, 2) \) and \( (2, 4) \), the slope is:
       \[
       f'(x) = \frac{4 - 2}{2 - 0} = \frac{2}{2} = 1
       \]
       So, \( f'(1) = 1 \).

   - The function \( g(x) \) also appears linear in the segment where \( x = 1 \). From the graph, we observe:
   
     For \( g(x) \):
     - Between \( (0, 0) \) and \( (2, 4) \), the slope is:
       \[
       g'(x) = \frac{4 - 0}{2 - 0} = \frac{4}{2} = 2
       \]
       So, \( g'(1) = 2 \).

3. **Calculate \( G'(1) \):**
   Now that we have \( f'(1) = 1 \) and \( g'(1) = 2 \), we can substitute into the expression for \( G'(x) \):
   \[
   G'(1) = 4f'(1) - g'(1) = 4(1) - 2 = 4 - 2 = 2
   \]

### Final Answer:
\[
G'(1) = 2
\]

Would you like further details on any step, or have any questions about this solution?

Here are five related questions to explore:

1. How do you find the derivative of a function from a graph if the function is not linear?
2. What is the geometric interpretation of the derivative at a point?
3. How do we differentiate a product of functions?
4. Can the second derivative give us more information about the graph's behavior at \( x = 1 \)?
5. How do you compute the derivative of a composite function?

**Tip:** When interpreting graphs, remember that the steeper the line, the larger the magnitude of the derivative.

Let G = 4f - g, where the graphs of f and g are shown in the figure. Find G'(1).

Learn how to find the derivative of the function G(x) = 4f(x) - g(x) using the slopes of linear functions f(x) and g(x) from a graph. This solution includes calculating f'(1) and g'(1) and combining them to find G'(1). Suitable for high school students in grades 10-12.

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