graph the function f(x)=x² - 1 over the interval 0, 2. partition the interval into four subintervals of…

graph the function f(x)=x² - 1 over the interval 0, 2. partition the interval into four subintervals of equal length. then add to your sketch the rectangles associated with the riemann sum ∑ₖ₌₁⁴ f(cₖ)δxₖ, given that cₖ is the (a) left - hand endpoint, (b) right - hand endpoint, (c) midpoint of the kth subinterval. (a) choose the correct graph below. (b) choose the correct graph below.

graph the function f(x)=x² - 1 over the interval 0, 2. partition the interval into four subintervals of equal length. then add to your sketch the rectangles associated with the riemann sum ∑ₖ₌₁⁴ f(cₖ)δxₖ, given that cₖ is the (a) left - hand endpoint, (b) right - hand endpoint, (c) midpoint of the kth subinterval. (a) choose the correct graph below. (b) choose the correct graph below.

Answer

Explanation:

Step1: Calculate sub - interval width

The interval is $[0,2]$ and $n = 4$. The width $\Delta x=\frac{b - a}{n}=\frac{2-0}{4}=0.5$. The sub - intervals are $[0,0.5]$, $[0.5,1]$, $[1,1.5]$, $[1.5,2]$.

Step2: Analyze left - hand endpoints (a)

For left - hand endpoints: $c_1 = 0$, $c_2=0.5$, $c_3 = 1$, $c_4=1.5$. The rectangles will have heights $f(c_1)=0^{2}-1=-1$, $f(c_2)=(0.5)^{2}-1=- 0.75$, $f(c_3)=1^{2}-1 = 0$, $f(c_4)=(1.5)^{2}-1=1.25$. The rectangles for the left - hand endpoints will have the left - most point of each sub - interval touching the curve.

Step3: Analyze right - hand endpoints (b)

For right - hand endpoints: $c_1 = 0.5$, $c_2 = 1$, $c_3=1.5$, $c_4 = 2$. The rectangles will have heights $f(c_1)=(0.5)^{2}-1=-0.75$, $f(c_2)=1^{2}-1 = 0$, $f(c_3)=(1.5)^{2}-1=1.25$, $f(c_4)=2^{2}-1 = 3$. The rectangles for the right - hand endpoints will have the right - most point of each sub - interval touching the curve.

Step4: Analyze mid - points (c)

The mid - points of the sub - intervals are $c_1=0.25$, $c_2 = 0.75$, $c_3=1.25$, $c_4=1.75$. The heights are $f(c_1)=(0.25)^{2}-1=-0.9375$, $f(c_2)=(0.75)^{2}-1=-0.4375$, $f(c_3)=(1.25)^{2}-1 = 0.5625$, $f(c_4)=(1.75)^{2}-1=2.0625$.

(a) For left - hand endpoints, the rectangles start from the left side of each sub - interval. The correct graph is A. (b) For right - hand endpoints, the rectangles start from the right side of each sub - interval. The correct graph is B.

Answer:

(a) A. (b) B.