Science:Math Exam Resources/Courses/MATH105/April 2013/Question 01 (d)

MATH105 April 2013

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[hide] Question 01 (d)
Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Fill in the blanks with right, left, or midpoint; an interval; and a value of $n$. $\sum _{k=0}^{3}f(1.5+k)\cdot 1$ is a _________ Riemann sum for $\displaystyle f$ on the interval [____ , ____] with $\displaystyle n=$ ______.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint

The left Riemann sum of a function ƒ over the interval [a,b] with n intervals is given by $\sum _{k=0}^{n-1}f(a+k\Delta x)\cdot \Delta x$ with $\Delta x={\frac {b-a}{n}}$ The midpoint Riemann sum of a function ƒ over the interval [a,b] with n intervals is given by $\sum _{k=0}^{n-1}f(a+(k+1/2)\Delta x)\cdot \Delta x$ with $\Delta x={\frac {b-a}{n}}$ The right Riemann sum of a function ƒ over the interval [a,b] with n intervals is given by $\sum _{k=1}^{n}f(a+k\Delta x)\cdot \Delta x$ or $\sum _{k=0}^{n-1}f(a+(k+1)\Delta x)\cdot \Delta x$ with $\Delta x={\frac {b-a}{n}}$

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] Solution 1
There are multiple correct answers to this problem. The following is but one of three possible interpretations: This is a left Riemann sum from 1.5 to 5.5 with $n=4$ rectangles. If we look at our left endpoint formula in the hint, we realize that ${\begin{aligned}a&=1.5\\n&=4\\\Delta {x}&={\frac {b-a}{n}}=1.\end{aligned}}$ This last relation tells us that b=n+a=5.5.

[show] Solution 2

There are multiple correct answers to this problem. The following is but one of three possible interpretations: This is a midpoint Riemann sum from 1 to 5 with $n=4$ rectangles. We can recognize this by writing $\sum _{k=0}^{3}f(1.5+k)=\sum _{k=0}^{3}f(1+(k+1/2))$ which is in the form of our midpoint formula in the hint if we take ${\begin{aligned}a&=1\\n&=4\\\Delta {x}&={\frac {b-a}{n}}=1.\end{aligned}}$ This last relation tells us that b=n+a=5.

[show] Solution 3

There are multiple correct answers to this problem. The following is but one of three possible interpretations: This is a right endpoint Riemann sum from 0.5 to 4.5 with $n=4$ rectangles. We can recognize this by writing $\sum _{k=0}^{3}f(1.5+k)=\sum _{k=0}^{3}f(0.5+(k+1))$ which is in the form of our right endpoint formula in the hint if we take ${\begin{aligned}a&=0.5\\n&=4\\\Delta {x}&={\frac {b-a}{n}}=1.\end{aligned}}$ This last relation tells us that b=n+a=4.5.

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