Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.5 Trapezoidal Rule.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 3 Quick Review Solutions
Transcript of Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 5.5 Trapezoidal Rule.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
5.5Trapezoidal Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 2
Quick Review
4 2
3
Tell whether the curve is concave up or concave down on the given interval.1. cos on [-1,0]2. 3 6 on [8,17]
3. sin on [48 ,50 ]2
4. on [-5,5]5. 1/ on [4, 8]6. csc
x
y xy x x
xy
y ey xy x
on 0,7. sin - cos on [1,2]y x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 3
Quick Review Solutions
4 2
3
concavTell whet
e dowher the curve is concave up or concave down on the given interval.
1. cos on [-1,0] 2. 3 6 on [8,17]
3. sin on [48
nconca
,50
ve up
concave do] 2
4.
wn
y xy x x
xy
y e
on [-5,5] 5. 1/ on [4, 8] 6. c
concave upconcave up
concasc on 0, 7. sin - cos on [1,2
ve upconca] ve d own
x
y xy xy x x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 4
What you’ll learn about
Trapezoidal Approximations Other Algorithms Error Analysis
… and whySome definite integrals are best found bynumerical approximations, and rectangles are notalways the most efficient figures to use.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 5
Trapezoidal Approximations
0 1 11 2
0
1 2 1
0 1 2 1
0 1 1 1 1
( ) ...2 2 2
...2 2
2 2 ... 2 ,2
where ( ), ( ), ..., ( ), ( ).
b n n
a
n
n
n n
n n n
y y y yy yf x dx h h h
y yh y y y
h y y y y y
y f a y f x y f x y f b
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 6
The Trapezoidal Rule
0 1 2 1
To approximate ( ) , use
2 2 ... 2 ,2
where [ , ] is partitioned into n subintervals of equal length( - ) / .
LRAM RRAMEquivalently, ,2
where LRAM and RRAM are the Rienamm
b
a
n n
n n
n n
f x dxhT y y y y y
a bh b a n
T
sums using the leftand right endpoints, respectively, for for the partition.f
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 7
Simpson’s Rule
0 1 2 3 2 1
To approximate ( ) , use
4 2 4 ... 2 4 ,3
where [ , ] is partitioned into an even number subintervalsof equal length ( - ) / .
b
a
n n n
f x dxhS y y y y y y y
a b nh b a n
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 8
Error Bounds
( 4 )
2 4
If and represent the approximations to ( ) given by the Trapezoidal Rule and Simpson's Rule, respectively, then the errors
and satisfy
and 12 180n
b
a
T s
T sf f
T S f x dx
E Eb a b aE h M E h M
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 9
Quick Quiz Sections 5.4 and 5.5
You may use a graphing calculator to solve the following problems1. The function is continuous on the closed interval [1,7] and hasvalues that are given below:
f
x 1 4 6 7
f(x) 10 30 40 20
7
1
Using the subintervals [1,4], [4,6], and [6,7], what is the trapezoidalapproximation of ( ) ?(A) 110(B) 130(C) 160(D) 190(E) 210
f x dx
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 10
Quick Quiz Sections 5.4 and 5.5
Quick Quiz Sections 5.4 and 5.5
You may use a graphing calculator to solve the following problems1. The function is continuous on the closed interval [1,7] and hasvalues that are given below:
f
x 1 4 6 7
f(x) 10 30 40 20
7
1
Using the subintervals [1,4], [4,6], and [6,7], what is the trapezoidalapproximation of ( ) ?(A)
(C) 160
110(B) 130
(D) 190(E) 210
f x dx
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 11
Quick Quiz Sections 5.4 and 5.5
32. Let ( ) be an antiderivative of sin . If (1) 0, then (8)(A) 0.00(B) 0.021(C) 0.373(D) 0.632(E) 0.968
F x x F F
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 12
Quick Quiz Sections 5.4 and 5.5
32. Let ( ) be an antiderivative of sin . If (1) 0, then (8)(A) 0.00(B) 0.0
(
21(C) 0.373D) 0.632
(E) 0.968
F x x F F
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 13
Quick Quiz Sections 5.4 and 5.5
2 23
-23. Let ( ) . At what value of is ( ) a minimum?(A) For no value of (B) 1/2(C) 3/2(D) 2 (E) 3
x x tf x e dt x f xx
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 14
Quick Quiz Sections 5.4 and 5.5
2 23
-23. Let ( ) . At what value of is ( ) a minimum?(A) For no value of (B) 1/2(C) (D) 2 (E
3/2
) 3
x x tf x e dt x f xx
