To solve this, we need to calculate the limit of the function $f(x) = \frac{7}{2}x + 4$ over the interval $[0, 4]$ and verify the result using geometry.

Step 1: Define the function and the interval

The function is $f(x) = \frac{7}{2}x + 4$. The interval is $[0, 4]$.

Step 2: Evaluate the function at the endpoints

At $x = 0$: $f(0) = \frac{7}{2}(0) + 4 = 4$

At $x = 4$: $f(4) = \frac{7}{2}(4) + 4 = 14 + 4 = 18$

Step 3: Geometric interpretation

The graph of $f(x) = \frac{7}{2}x + 4$ is a straight line with a slope of $\frac{7}{2}$ and a y-intercept at 4. Over the interval $[0, 4]$, the line rises from $f(0) = 4$ to $f(4) = 18$.

To calculate the area under the curve geometrically, the area forms a trapezoid. The trapezoid's parallel sides are the values of $f(x)$ at the endpoints ($4$ and $18$), and the base length is the interval length ($4$).

Step 4: Calculate the area of the trapezoid

The formula for the area of a trapezoid is: $\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$ Here:

• $\text{Base}_1 = 4$ (value of $f(0)$),
• $\text{Base}_2 = 18$ (value of $f(4)$),
• $\text{Height} = 4$ (interval length).

$\text{Area} = \frac{1}{2} \times (4 + 18) \times 4 = \frac{1}{2} \times 22 \times 4 = 44$

Final Answer:

The limit of the function $f(x)$ over the interval $[0, 4]$, calculated geometrically, is: $\boxed{44}$

Do you have any questions or would you like further clarification on any part of the solution?

Here are 5 related questions for deeper exploration:

1. What is the derivative of $f(x) = \frac{7}{2}x + 4$ and how does it relate to the slope?
2. How does the geometric area under a curve relate to the integral of the function?
3. Can this approach be extended to non-linear functions? How would the calculation differ?
4. What is the significance of the line equation $f(x) = mx + b$ in geometry?
5. How does this problem illustrate the relationship between algebra and geometry?

Tip: Always visualize a problem graphically when possible—it can make abstract concepts clearer!