the t subintervals of thinner width. finine Eive some value in between. The true
ID: 2884818 • Letter: T
Question
the t subintervals of thinner width. finine Eive some value in between. The true value lies som upper sums and lower sums. In the examples that we looked at, EXERCISES 5.1 Area the distance traveled by the engine using 10 subinterya I with a. left-endpoint values. b. right-endpoint values the graph of the function a. a lower sum with two rectangles of equal width. h. a lower sum with four rectangles of equal width. e, an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. Velocity Time Velocits, (em/sec) (sec) (em/see Time (sec) . f)between -0 and 1 2, f(x)-r, between x 0 and x =1 3. f(x)-1/x between x-1 and x = 5. 4.f(x)-4-r2 between x =-2 and x = 2. 15 Using rectangles each of whose height is given by the value of 10 the function at the midpoint of the rectangle's base (the midpoint rule). estimate the area under the graphs of the following functions, using first two and then four rectangles. 10. ou are 6. fcx)- between 0 andx 1 tidal river watching the incoming tide carry a bottle upstream You record the velocity of the flow every 5 minutes for an hou with the results shown in the accompanying table. About how f upstream did the bottle travel during that hour? Find an estimae using 12 subintervals of length 5 with 7. f(r) at 1 /r between x 1 and x 5. &fx;) = 4-x2 between re-2and x = 2. 9. Distance traveled The accompanying table shows the velocity of a model train engine moving along a track for 10 sec. EstimateExplanation / Answer
4 - x^2 between x = -2 and 2
a) Lower sum with 2 rectangles
So, the endpoints of x are x = -2 , 0 and 2
At x = -2, y = 0
At x = 0, y = 4
At x= 2, y = 0
Between x = -2 and 0, we choose lower y-value of ZERO
Between x = 0 and 2, we choose lower y value of ZERO
So, lower sum = 0 + 0 ----> 0
b) LS with 4 rectangles
So, endpts are = -2 , -1 , 0 , 1 and 2
x = -2, y = 0
x = -1, y = 3
x = 0 , y = 4
x = 1, y = 3
x = 2, y = 0
Choosing lower y-values from each consecutive y-value, we get
0 + 3 + 3 + 0
= 6
To this multiply width of rectangle, i.e 1
6 * 1
6 ---> ANS
c) US with n = 2
Endpts are x = -2 , 0 and 2
x = -2, y = 0
x = 0, y = 4
x = 2, y = 0
Choosing higher values, we get
4 + 4 = 8
To this , multiply width of rectangle ie 2
8 * 2
16 ----> ANS
d) US with n = 4
So, endpts are = -2 , -1 , 0 , 1 and 2
x = -2, y = 0
x = -1, y = 3
x = 0 , y = 4
x = 1, y = 3
x = 2, y = 0
3 + 4 + 4 + 3 --> 14
14 * width
14 * 1
14 ----> ANS
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8)
n = 2
So, endpts are x = -2 , 0 and 2
Midpts are x = -1 and 1
When x = -1, y = 3
When x = 1, y = 3
Adding em up,
3 + 3 = 6
To this multiply width of rectangle, i.e 2
6 * 2
12 ---> ANS
n = 4 rectangles : Endpts are x = -2,-1,0,1,2
Midpts are -1.5,-0.5,0.5 and 1.5
When x = -1.5, y = 1.75
When x = -0.5, y = 3.75
When x = 0.5, y = 3.75
When x = 1.5, y = 1.75
Adding,
1.75 + 3.75 + 3.75 + 1.75
11
To this multiply width of rectangle i.e 1
11 * 1
11 ----> ANS
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10)
Table missing
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