Difference between revisions of "009B Sample Midterm 2, Problem 1"

From Grad Wiki
Jump to navigation Jump to search
Line 16: Line 16:
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1:    
 
!Step 1:    
 +
|-
 +
|Let <math>f(x)=\frac{1}{x^2}</math>
 
|-
 
|-
 
|Since our interval is <math>[1,5]</math> and we are using 4 rectangles, each rectangle has width 1. So, the left-endpoint Riemann sum is  
 
|Since our interval is <math>[1,5]</math> and we are using 4 rectangles, each rectangle has width 1. So, the left-endpoint Riemann sum is  
 
|-
 
|-
 
|<math>1(f(1)+f(2)+f(3)+f(4))</math>.   
 
|<math>1(f(1)+f(2)+f(3)+f(4))</math>.   
 +
|-
 +
|
 
|}
 
|}
  
Line 29: Line 33:
 
|<math>1(f(1)+f(2)+f(3)+f(4))=\bigg(1+\frac{1}{4}+\frac{1}{9}+{1}{16}\bigg)=\frac{205}{144}</math>.   
 
|<math>1(f(1)+f(2)+f(3)+f(4))=\bigg(1+\frac{1}{4}+\frac{1}{9}+{1}{16}\bigg)=\frac{205}{144}</math>.   
 
|-
 
|-
−
|
+
|The left-endpoint Riemann sum overestimates the area of <math>S</math>.
−
|-
 
−
|
 
 
|}
 
|}
  
Line 38: Line 40:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|
+
|Let <math>n</math> be the number of rectangles used in the left-endpoint Riemann sum for <math>f(x)=\frac{1}{x^2}</math>.
 
|-
 
|-
−
|
+
|The width of each rectangle is <math>\Delta x=\frac{5-1}{n}=\frac{4}{n}</math>.
 
|-
 
|-
 
|
 
|
Line 50: Line 52:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
+
|So, the left-endpoint Riemann sum is
 
|-
 
|-
−
|
+
|<math>\Delta x \bigg(f(1)+f\bigg(1+\frac{4}{n}\bigg)+f\bigg(1+2\frac{4}{n}\bigg)+\ldots +f\bigg(1+(n-1)\frac{4}{n}\bigg)\bigg)</math>.
 
|-
 
|-
−
|  
+
|Now, we let <math>n</math> go to infinity to get a limit. 
 
|-
 
|-
−
|
+
|So, the area of <math>S</math> is equal to <math>\lim_{n\to\infty} \frac{4}{n}\sum_{i=0}^{n-1}f\bigg(1+i\frac{4}{n}\bigg)</math>.
 
|}
 
|}
  
Line 62: Line 64:
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)''' <math>\frac{205}{144}</math>  
+
|'''(a)''' Left-endpoint Riemann sum: <math>\frac{205}{144}</math>, The left-endpoint Riemann sum overestimates the area of <math>S</math>.
 
|-
 
|-
−
|'''(b)'''  
+
|'''(b)''' Using left-endpoint Riemann sums: <math>\lim_{n\to\infty} \frac{4}{n}\sum_{i=0}^{n-1}f\bigg(1+i\frac{4}{n}\bigg)</math>
 
|}
 
|}
 
[[009B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:34, 31 January 2016

Consider the region bounded by and the -axis.

a) Use four rectangles and a Riemann sum to approximate the area of the region . Sketch the region and the rectangles and indicate your rectangles overestimate or underestimate the area of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .
b) Find an expression for the area of the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} as a limit. Do not evaluate the limit.


Foundations:  
Link to Riemann sums page

Solution:

(a)

Step 1:  
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{1}{x^2}}
Since our interval is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [1,5]} and we are using 4 rectangles, each rectangle has width 1. So, the left-endpoint Riemann sum is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1(f(1)+f(2)+f(3)+f(4))} .
Step 2:  
Thus, the left-endpoint Riemann sum is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1(f(1)+f(2)+f(3)+f(4))=\bigg(1+\frac{1}{4}+\frac{1}{9}+{1}{16}\bigg)=\frac{205}{144}} .
The left-endpoint Riemann sum overestimates the area of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .

(b)

Step 1:  
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} be the number of rectangles used in the left-endpoint Riemann sum for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{1}{x^2}} .
The width of each rectangle is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x=\frac{5-1}{n}=\frac{4}{n}} .
Step 2:  
So, the left-endpoint Riemann sum is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x \bigg(f(1)+f\bigg(1+\frac{4}{n}\bigg)+f\bigg(1+2\frac{4}{n}\bigg)+\ldots +f\bigg(1+(n-1)\frac{4}{n}\bigg)\bigg)} .
Now, we let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} go to infinity to get a limit.
So, the area of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty} \frac{4}{n}\sum_{i=0}^{n-1}f\bigg(1+i\frac{4}{n}\bigg)} .
Final Answer:  
(a) Left-endpoint Riemann sum: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{205}{144}} , The left-endpoint Riemann sum overestimates the area of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .
(b) Using left-endpoint Riemann sums: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty} \frac{4}{n}\sum_{i=0}^{n-1}f\bigg(1+i\frac{4}{n}\bigg)}

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009B_Sample_Midterm_2,_Problem_1&oldid=1081"