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xyy=12x13Q: Letf(x)be a function that isdecreasingfromx=0 tox=5. Which Riemann sumapproximation ofż50f(x)dxis the largest–left, right, or midpoint?Q: Give an example of a functionf(x), an interval[a,b], and a numbernsuch that themidpoint Riemann sum off(x)over[a,b]usingnintervals islarger thanboth the left andright Riemann sums off(x)over[a,b]usingnintervals.In Questions6through10, we practice using sigma notation. There are many ways to write a given sum insigma notation. You can practice finding several, and deciding which looks the clearest.Q: Express the following sums in sigma notation:(a) 3+4+5+6+7(b) 6+8+10+12+14(c) 7+9+11+13+15(d) 1+3+5+7+9+11+13+153
INTEGRATION1.1 DEFINITION OF THEINTEGRALQ: Express the following sums in sigma notation:(a)13+19+127+181(b)23+29+227+281(c)´23+29´227+281(d)23´29+227´281Q: Express the following sums in sigma notation:(a)13+13+527+781+9243(b)15+111+129+183+1245(c) 1000+200+30+4+12+350+71000Q: Evaluate the following sums. You might want to use the formulas from Theorems1.1.5 and 1.1.6 in the CLP–II text.(a)100ÿi=035i(b)100ÿi=5035i(c)10ÿi=1i2´3i+5(d)bÿn=11en+en3, wherebis some integer greater than 1.Q: Evaluate the following sums. You might want to use the formulas fromTheorem 1.1.6 in the CLP–II text.(a)100ÿi=50(i´50) +50ÿi=0i(b)100ÿi=10(i´5)3(c)11ÿn=1(´1)n(d)11ÿn=2(´1)2n+1Questions11through15are meant to give you practice interpreting the formulas in Definition 1.1.11 of theCLP–II text. The formulas might look complicated at first, but if you understand what each piece means,they are easy to learn.4
INTEGRATION1.1 DEFINITION OF THEINTEGRALQ: In the picture below, draw in the rectangles whose (signed) area is beingcomputed by the midpoint Riemann sum4ÿi=1b´a4¨fa+i´12b´a4.xybay=f(x)Q(˚):4ÿk=1f(1+k)¨1 is a left Riemann sum for a functionf(x)on the interval[a,b]withnsubintervals. Find the values ofa,bandn.Q: Draw a picture illustrating the area given by the following Riemann sum.3ÿi=12¨(5+2i)2Q: Draw a picture illustrating the area given by the following Riemann sum.5ÿi=1π20¨tanπ(i´1)20Q(˚): Fill in the blanks with right, left, or midpoint; an interval; and a value of n.3řk=0f(1.5+k)¨1 is aRiemann sum forfon the interval[,]withn=.Q: Evaluate the following integral by interpreting it as a signed area, and usinggeometry:ż50xdx5
INTEGRATION1.1 DEFINITION OF THEINTEGRALQ: Evaluate the following integral by interpreting it as a signed area, and usinggeometry:ż5´2xdx§§Stage 2Q(˚): Use sigma notation to write the midpoint Riemann sum forf(x) =x8on[5, 15]withn=50. Do not evaluate the Riemann sum.Q(˚): Estimateż5´1x3dxusing three approximating rectangles and left hand end points.