Question

the following function is positive and negative on the given interval. f(x)=8 - 2x², 0,4 a. sketch the function on the 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. c. use the sketch in part (a) to show which intervals of 0,4 make positive and negative contributions to the net area. a. choose the correct graph.

Explanation:

Step1: Identify function key points

$f(0)=8$, $f(2)=0$, $f(4)=-24$; parabola opens downward.

Step2: Determine correct graph

Graph D matches: starts at (0,8), crosses x-axis at x=2, ends at (4,-24).

Step3: Riemann sum setup

$\Delta x=(4-0)/4=1$; subintervals [0,1],[1,2],[2,3],[3,4].

Step4: Left sum calculation

Left endpoints: 0,1,2,3; $f(0)=8,f(1)=6,f(2)=0,f(3)=-10$; Sum=1*(8+6+0-10)=4.

Step5: Right sum calculation

Right endpoints:1,2,3,4; $f(1)=6,f(2)=0,f(3)=-10,f(4)=-24$; Sum=1*(6+0-10-24)=-28.

Step6: Midpoint sum calculation

Midpoints:0.5,1.5,2.5,3.5; $f(0.5)=7.5,f(1.5)=3.5,f(2.5)=-4.5,f(3.5)=-16.5$; Sum=1*(7.5+3.5-4.5-16.5)=-10.

Step7: Positive/negative intervals

$f(x)=0$ at x=2; positive on [0,2), negative on (2,4].

Answer:

a. D. (Assuming graph D matches the key points)
b. Left Riemann sum: 4, Right Riemann sum: -28, Midpoint Riemann sum: -10
c. Positive on [0,2), negative on (2,4]