Question

the following function is negative on the given interval. f(x)= - 3 - x^3, 3,7 a. sketch the function on the given interval. b. approximate the net area bounded by the graph of f and the x - axis on the interval using a left, right, and midpoint riemann sum with n = 4. a. choose the correct graph.

Explanation:

Step1: Analyze function behavior

\( f(x) = -3 - x^3 \), derivative \( f'(x) = -3x^2 < 0 \) (strictly decreasing). At \( x=3 \), \( f(3) = -30 \); \( x=7 \), \( f(7) = -346 \). All values negative on [3,7].

Step2: Determine correct graph

Graph must show decreasing curve with negative y-values from (3,-30) to (7,-346).

Step3: Calculate Δx for Riemann sums

\( \Delta x = \frac{7-3}{4} = 1 \). Subintervals: [3,4],[4,5],[5,6],[6,7].

Step4: Left Riemann sum

Left endpoints: 3,4,5,6. Sum: \( 1[(-30)+(-67)+(-128)+(-219)] = -444 \).

Step5: Right Riemann sum

Right endpoints:4,5,6,7. Sum: \( 1[(-67)+(-128)+(-219)+(-346)] = -760 \).

Step6: Midpoint Riemann sum

Midpoints:3.5,4.5,5.5,6.5. Sum: \( 1[(-45.875)+(-94.125)+(-169.375)+(-277.625)] = -587 \).

Answer:

a. (Assuming correct graph shows decreasing negative curve: e.g., Option B if it matches the described behavior)
b. Left sum: -444, Right sum: -760, Midpoint sum: -587