1. **State the problem:** We need to find the difference between the upper and lower sums for the function $f(x) = 3^x$ over the interval $[0,2]$ using four subintervals. 2. **Formula and explanation:** The interval $[0,2]$ is divided into $n=4$ equal subintervals, each of width $$\Delta x = \frac{2-0}{4} = 0.5.$$ For an increasing function like $3^x$, the lower sum uses the left endpoints and the upper sum uses the right endpoints of each subinterval. 3. **Calculate the subinterval points:** $$x_0=0, x_1=0.5, x_2=1, x_3=1.5, x_4=2.$$ 4. **Evaluate the function at these points:** $$f(x_0) = 3^0 = 1,$$ $$f(x_1) = 3^{0.5} = \sqrt{3} \approx 1.732,$$ $$f(x_2) = 3^1 = 3,$$ $$f(x_3) = 3^{1.5} = 3 \times \sqrt{3} \approx 5.196,$$ $$f(x_4) = 3^2 = 9.$$ 5. **Calculate the lower sum $L_4$ using left endpoints:** $$L_4 = \Delta x \times \left(f(x_0) + f(x_1) + f(x_2) + f(x_3)\right) = 0.5 \times (1 + 1.732 + 3 + 5.196) = 0.5 \times 10.928 = 5.464.$$ 6. **Calculate the upper sum $U_4$ using right endpoints:** $$U_4 = \Delta x \times \left(f(x_1) + f(x_2) + f(x_3) + f(x_4)\right) = 0.5 \times (1.732 + 3 + 5.196 + 9) = 0.5 \times 18.928 = 9.464.$$ 7. **Find the difference between upper and lower sums:** $$U_4 - L_4 = 9.464 - 5.464 = 4.$$ **Final answer:** The difference between the upper and lower sums is **4**.