To solve the problem of finding the upper sum for $f(x) = 17 - 2x^2$ on $[1, 2]$ with $n = 4$ subintervals:

Step 1: Define the Subintervals

The interval $[1, 2]$ is divided into $n = 4$ equal subintervals. The length of each subinterval is: $\Delta x = \frac{2 - 1}{4} = 0.25$

The subintervals are: [ [1, 1.25], [1.25, 1.5], [1.5, 1.75], [1.75, 2] ]

Step 2: Compute the Maximum Value on Each Subinterval

The function $f(x) = 17 - 2x^2$ decreases as $x$ increases. Therefore, the maximum value on each subinterval occurs at the left endpoint.

1. Subinterval $[1, 1.25]$: $f(1) = 17 - 2(1)^2 = 15$
2. Subinterval $[1.25, 1.5]$: $f(1.25) = 17 - 2(1.25)^2 = 17 - 2(1.5625) = 17 - 3.125 = 13.875$
3. Subinterval $[1.5, 1.75]$: $f(1.5) = 17 - 2(1.5)^2 = 17 - 2(2.25) = 17 - 4.5 = 12.5$
4. Subinterval $[1.75, 2]$: $f(1.75) = 17 - 2(1.75)^2 = 17 - 2(3.0625) = 17 - 6.125 = 10.875$

Step 3: Compute the Upper Sum

The upper sum is calculated as: $\text{Upper Sum} = \sum (\text{Maximum value on subinterval} \times \Delta x)$

Substituting the values: $\text{Upper Sum} = (15)(0.25) + (13.875)(0.25) + (12.5)(0.25) + (10.875)(0.25)$

Calculate each term: $(15)(0.25) = 3.75,\quad (13.875)(0.25) = 3.46875,\quad (12.5)(0.25) = 3.125,\quad (10.875)(0.25) = 2.71875$

Add them together: $\text{Upper Sum} = 3.75 + 3.46875 + 3.125 + 2.71875 = 13.0625$

Final Answer:

The upper sum is approximately: $\boxed{13.06}$

Let me know if you need further explanation or assistance!

Here are five follow-up questions to deepen understanding:

1. What is the difference between upper sums and lower sums?
2. How would the result change if we used $n = 8$ subintervals?
3. Can this method be applied to non-decreasing functions?
4. How does this relate to finding the exact area under the curve?
5. What are the limitations of using upper sums in approximations?

Tip: Always sketch the function to visualize how the upper sum or lower sum captures the approximation!