Question

complete the following steps for the given function, interval, and value of n.
a. sketch the graph of the function on the given interval.
b. calculate \\(\delta x\\) and the grid points \\(x_0,x_1,\cdots,x_n\\).
c. illustrate the left - and right - riemann sums and determine which riemann sum underestimates and which sum overestimates the area under the curve.
d. calculate the left and right riemann sums.
\\(f(x)=2x^{2}+1\\) on \\(1,6,n = 5\\)
a. sketch the graph of \\(f(x)=2x^{2}+1\\) on the interval \\(1,6\\). choose the correct graph below.
b. \\(\delta x=\\)
list the grid points.
c. illustrate the left riemann sum. (use a comma to separate answers as needed.) choose the correct graph below.

Explanation:

Step1: Recall Riemann - sum concepts

For a decreasing function on an interval $[a,b]$, the left - Riemann sum overestimates the area under the curve and the right - Riemann sum underestimates the area under the curve. For an increasing function on an interval $[a,b]$, the left - Riemann sum underestimates the area under the curve and the right - Riemann sum overestimates the area under the curve. First, find the derivative of $y = f(x)=2x^{2}+1$.

Step2: Differentiate the function

Using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $y^\prime=f^\prime(x)=4x$. On the interval $[1,6]$, since $x\geq1$, $y^\prime = 4x>0$, so the function $y = 2x^{2}+1$ is increasing on the interval $[1,6]$.

Step3: Analyze left and right Riemann sums

For an increasing function on $[1,6]$, the left - Riemann sum will underestimate the area under the curve and the right - Riemann sum will overestimate the area under the curve.

Step4: Sketch the function

The function $y = 2x^{2}+1$ is a parabola opening upwards with vertex at $(0,1)$. When $x = 1$, $y=2\times1^{2}+1=3$; when $x = 6$, $y=2\times6^{2}+1=2\times36 + 1=73$.

Step5: Calculate $\Delta x$

The formula for $\Delta x=\frac{b - a}{n}$, where $a = 1$, $b = 6$ and $n = 5$. So $\Delta x=\frac{6 - 1}{5}=1$.

Step6: List grid points

The grid points $x_{i}=a + i\Delta x$, for $i = 0,1,\cdots,5$. So $x_{0}=1,x_{1}=2,x_{2}=3,x_{3}=4,x_{4}=5,x_{5}=6$.

Answer:

b. $\Delta x = 1$
c. The left - Riemann sum underestimates and the right - Riemann sum overestimates the area under the curve. The grid points are $x_{0}=1,x_{1}=2,x_{2}=3,x_{3}=4,x_{4}=5,x_{5}=6$