section 6.3 estimating distance traveled
TRANSCRIPT
Section 6.3 Estimating Distance Traveled
Distance Traveled Suppose a man is driving a car and we know his velocity. Suppose further that
we have a graph of his velocity curve from time t = a to time t = b.
Question: How do we determine the distance the man traveled during that total time span?
Answer: The distance traveled is the area under the velocity curve.
Estimating Area/Distance with Rectangles
Suppose v(t), the velocity, is a function defined on [a, b]. We can divide the interval [a, b] into n
subintervals of equal width �x = (b� a)/n. We let x0 = a, x1, x2, . . . , xn = b be the endpoints of these
subintervals.
The subintervals will look like .
We can estimate the area under the velocity curve (distance) by dividing the area into n rectangles
where the width of each rectangle is �x = (b� a)/n and the heights are given by
v(x0 = a), v(x1), v(x2), . . . , v(xn = b). In other words, the heights are the velocity function evaluated
at the endpoints. If we add up the areas of the rectangles, then we will have an estimate for the area
under the curve ( the distance).
Note: We are finding the distance on the interval [a, b], using n rectangles, each with a width of �x,
which is given by
�x =(b� a)
n
Left-Hand Sum: The left-hand sum, Ln, is what we calculate when we use only the left endpoints to
estimate the area, and is given by
Ln = �x (v(x0 = a) + v(x1) + v(x2) + . . .+ v(xn�1))
Right-Hand Sum: The right-hand sum, Rn, is what we calculate when we use only the right endpoints
to estimate the area, and is given by
Rn = �x (v(x1) + v(x2) + . . .+ v(xn = b))
I"
-
E¥xD.Ix
. . x xn -
- b
-
- -
--
-- -
-
-
- w - w u w
-
-
w w w -
Theorem: Let f be a continuous function on an interval [a, b]
If f is increasing on I: The left-hand sum will be and the
right-hand sum will be
If f is decreasing on I: The left-hand sum will be and the
right-hand sum will be
1. Finding the Left-Endpoints and Right-Endpoints: Suppose you have a velocity function
v(x), and you want to estimate the distance using a left-hand sum and a right-hand sum. If you
want to estimate the distance on the interval [5, 10] using n = 3 rectangles, find the left-endpoints
and the right-endpoints.
2 Spring 2019, Maya Johnson
-lower estimate
upper estimate
- uppertower
a a nut )( Left-hand ) ( Left - hand )
←Upper
Estimateput)
¥÷÷.
a ab
a
tnad" " I Right-hand ?
Lowerof Estimate "
-
44[ 4,10 ] - -
-
On :{4104gn =3 , DX -_b-£ = 10-4=26-3--2 c- width
Left - End pts : Xo =4, × ,
= Xotbx = 4+2=6 ,×z=XitDX=6+2=8
Xo=4,Xi=6,xz=8
-
Right - End pts : X , =XotBx= 4+2=6 , Xz-
- Xittsx = 6+2=8 , ×z=XztDX=8+2=10
Xi=6,Xz=8,Xz=l
2. The speed of a runner increased steadily during the first twelve seconds of a race. Her speed at
two-second intervals is given in the table. Find lower and upper estimates for the distance that
she traveled during these twelve seconds using a left-hand sum and a right-hand sum with n = 6.
t(s) 0 2 4 6 8 10 12
v(ft/s) 0 6.7 9.2 14.1 17.5 19.4 20.2
3 Spring 2019, Maya Johnson
-
- - - -
- - -
[ O,123 , n = 6 DX = b = l2 = 2
a
Upper Estimate ( Rought - he d)
Right - end pts : X ,= 2 , Xz = 4 , X 3=6 , Xy = 8 ,
X s= to I X 6=12
R yI DX ( v ( 27 t V (4) t VC 6) t VC 8 It V I lost VC 12 ) )
6. 7 t 9 a2 t 14 .
I t 17 . 5 t 19 . 4 t 20 . 2)::m.=mn¥¥÷÷F¥Lower Estimate ( Left - hand ) :
Left - end points : Xo = o, × ,
= 2,
x 2=4 ,X
z-
-6
, X 4=8 , Xs = to
( 6= DX ( v to ) t V C 22 t v (4) t V (6) t V (8) t v Clo ) )
= 2 ( O t 6.7 t 9 .2 t 14 .
I t He 5 t 19 .
4)= 2 ( 66
. 9) = l33n µ
" "
. tf '
3. Speedometer readings for a motorcycle at 12-second intervals are given in the table.
t(s) 0 12 24 36 48 60
v(ft/s) 32 27 24 22 25 28
(a) Estimate the distance traveled by the motorcycle during this time period using a left-hand
sum with n = 5.
(b) Estimate the distance traveled by the motorcycle during this time period using a right-hand
sum with n = 5.
(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
4 Spring 2019, Maya Johnson
-DX
-
- - 49.603,DX = but = 6051=13
Left - end pt : Xo = O,
X i = 12, Xz -
- 24 , Xz = 36, X 4=48
↳ z DX (V ( o ) t Vl 12 ) tv I 24 ) t VI 36 ) t VC 48 ) )
= 12 ( 32 t 27 t 24 t 22 t 25 ) = 12 ( I 30 )
=l5
-
Right - end pts : X , = 12 , Xz = 24, X 3=36 , Xy = 48 , Xs = GO
Rs = DX ( v 42 ) t VC 24 ) t VC 365 tv ( 48 ) t VC Go ) )
= 12 ( 27 t 24 t 22 t 25 t 28 ) = 12 ( I 26 )
I l5l
Neither,
since the function is not strictly
decreasing increasing on the interval [ o,
60 ].
4. A model rocket has upward velocity v(t) = 35t2 ft/s, t seconds after launch. Use the interval [0, 8]
with n = 4 and equal subintervals to compute the following approximations of the distance the
rocket traveled. (Round answers to two decimal places.)
(a) Left-hand sum
(b) Right-hand sum
(c) average of the two sums
5 Spring 2019, Maya Johnson
-
rectangles
-
on [ o,
83 with n -
-4
,the width is DX = =3
Left - endpoints :
Xo -
- O, X , = 2
,Xz = 4
, X 3= 6
L 4= DX ( V I o ) t v (2) t V (4) t VC 6 ) )
± 2 ( o t 35125 t 351457351632 )
= 2 ( 1960 ) =392
Right - endpoints :
X i= 2
,X z = 4
,X z
= 6,
X 4= 8
R 4 = DX ( VL 22 t Vl 4) t VC 6) t V 18 ) )
= 2 (35125+35145+35165+35185)
= 2 ( 4200 ) =84
Average = 3920ftt
= 6160ft
Note:AvesageofanytwowumbsspsqisPtI
5. An object has a velocity v(t) =5
t+65 ft/s. Use the interval [1, 9] with n = 4 and equal subintervals
to compute the following approximations of the distance the object traveled. (Round final answers
to two decimal places.)
(a) Left-hand sum
(b) Right-hand sum
(c) average of the two sums
6 Spring 2019, Maya Johnson
- retakes
On [ I, 9) with h = 4
,the width Ax = 9-41=3
Left - end points : Xo -
-I
, X ,=3
, X 2=5 ,x 3=7
Ly = DX ( v Ll ) t V (3) t Vl 5) t VCD )
= 2 ( ⇐+65 ) t (Est 65) t (Est 65 ) t (E t 65J
= 2 ( 268. 38 ) --536.76ft
Right - end points : X,
=3,
Xz = 5,
X 3=7 ,X 4
= 9
Ry = DX ( v ( 3) t VC 5) t VL 7 ) t VC 9 ) )
= 2 ( (Est 6 5) t (Est 65) t +65 ) t ( It 65 ) )= 2 ( 263 .
94 ) = 527.87ft
Average = 536e76ftz5ft = 532.32ft