Nikita Sharma 2019W2 MATH 101 209
Assignment Homework 7 due 9/12/14 at 9pm
and, for i = 0, . . . , 8,
1. (1 point) Given the following graph of the function iπ
y = f (x) and n = 6, answer the following questions about the xi = i∆x = ,
48
area under the curve from x = 0 to x = 6.
so that
iπ
f (xi ) = f (i∆x) = sec2 .
48
Therefore,
π/48 2 2 π 2 2π 2 3π
T8 = sec (0)+2 sec +2 sec +2 sec +2 s
2 48 48 48
R π/6
Similarly, we estimate the integral 0 sec2 x dx using the
Midpoint Rule with 8 subdivisions, namely
x0 + x1 x0 + x1 x1 + x2 x2 + x3
M8 = ∆x f +f +f +f +
2 2 2 2
1. Use the Trapezoidal Rule to estimate the area. In our case, we obtain
Answer: T6 =
π 2 π 2 3π 2 5π 2 7π 2
2. Use Simpson’s Rule to estimate the area. T8 = sec +sec +sec +sec +sec
48 96 96 96 96
Answer: S6 = Correct Answers:
• 0.577899396
Note: You can click on the graph to enlarge the image. • 0.577075882
Correct Answers: 3. (1 point)
• 1/2*(2*1+2*3+2*1+2*0.5+2)+1*6
Approximate the following integral using the indicated meth-
• 1/3*(4*1+2*3+4*1+2*0.5+2)+1*6 ods. Leave your answers in ”calculator-ready” form.
Z 1
2
2. (1 point) e−3x dx
Calculate the integral approximations T8 (with the Trapezoid 0
Rule) and M8 (with the Midpoint Rule) for (a) Trapezoidal Rule with 4 subintervals
R π/6
0 sec2 x dx.
Leave your answers in calculator-ready form.
T8 = (b) Midpoint Rule with 4 subintervals
M8 = (c) Simpson’s Rule with 4 subintervals
2
(d) With f (x) = e−3x , It can be shown by direct computation
Solution: that | f (4) (x)| ≤ 108 on the interval [0, 1]. Using this information
Solution: and the error formula:
2
We have f (x) = sec x. We first estimate the integral M(b − a)5
E(n) ≤ ,
180n4
R π/6
0 sec2 x dx using the Trapezoid Rule with 8 subdivisions,
namely what is the least value of n so that the Simpson’s Rule approx-
imation for the given integral is guaranteed to be accurate to
∆x
T8 = f (x0 )+2 f (x1 )+2 f (x2 )+2 f (x3 )+2 f (x4 )+2 f (x5 )+2 f (x6within
)+2 f (x 0.00001?
7 )+ f (x8 . answer must be a whole number.)
(Your
)
2
Since n = 8, we have n=
π/8 − 0 π Correct Answers:
∆x = = • 0.502817651253163
8 48
1
, • 0.505095217095571 The left Rule with n = 6 is
• 0.504213520516726 h i
• 16 L6 = ∆x f (x0 ) + f (x1 ) + f (x2 ) + f (x3 ) + f (x4 ) + f (x5 )
h i
= 2 f (0) + f (2) + f (4) + f (6) + f (8) + f (10)
h i
≈ 2 8 + 7.9 + 7.6 + 7.0 + 6.2 + 5.2
≈ 83.8
4. (1 point) Use six rectangles to find an estimate of each type
for the area under the given graph of f from x = 0 to x = 12.
2. Because the function is concave down, the Left Sum is
an overestimate. (This is also apparent from the picture).
3. The function together with the rectangles for the Right
Rule is plotted below. We use the picture to estimate the values
of the function at the required values of x.
1. Take the sample points from the left-endpoints.
Answer: L6 =
The Right Sum with n = 6 is
2. Is your estimate L6 an underestimate or overestimate of h i
the true area? R6 = ∆x f (x1 ) + f (x2 ) + f (x3 ) + f (x4 ) + f (x5 ) + f (x6 )
• Choose one h i
= 2 f (2) + f (4) + f (6) + f (8) + f (10) + f (12)
• Underestimate h i
• Overestimate ≈ 2 7.9 + 7.6 + 7.0 + 6.2 + 5.2 + 4.0
3. Take the sample points from the right-endpoints. ≈ 75.8
Answer: R6 =
4. Is your estimate R6 an underestimate or overestimate of
the true area? 4. Because the function is concave down, the Right Sum is
• Choose one an underestimate. (This is also apparent from the picture).
• Underestimate
• Overestimate 5. The function together with the rectangles for the mid-
point Rule is plotted below. We use the picture to estimate the
5. Take the sample points from the midpoints.
values of the function at the required values of x.
Answer: M6 =
Note: You can click on the graph to enlarge the image.
Solution:
SOLUTION
12
We have a = 0, b = 12 and n = 6. So ∆x = b−a n = 6 = 2, x +x
The midpoints are x̄k = k−12 k , k = 1, 2, . . . , 6.
and xk = a + k∆x so
The Midpoint Sum with n = 6 is
x0 = 0, x1 = 2 , x2 = 4, x3 = 6, x4 = 8, x5 = 10 and x6 = 12. h i
M6 = ∆x f (x̄1 ) + f (x̄2 ) + f (x̄3 ) + f (x̄4 ) + f (x̄5 ) + f (x̄6 )
1. The function together with the rectangles for the Left h i
= 2 f (1) + f (3) + f (5) + f (7) + f (9) + f (11)
Rule is plotted below. We use the picture to estimate the values h i
of the function at the required values of x. ≈ 2 8.0 + 7.8 + 7.3 + 6.6 + 5.8 + 4.6
≈ 80.1
Correct Answers:
• 83.7778
• Overestimate
• 75.7778
• Underestimate
2
Assignment Homework 7 due 9/12/14 at 9pm
and, for i = 0, . . . , 8,
1. (1 point) Given the following graph of the function iπ
y = f (x) and n = 6, answer the following questions about the xi = i∆x = ,
48
area under the curve from x = 0 to x = 6.
so that
iπ
f (xi ) = f (i∆x) = sec2 .
48
Therefore,
π/48 2 2 π 2 2π 2 3π
T8 = sec (0)+2 sec +2 sec +2 sec +2 s
2 48 48 48
R π/6
Similarly, we estimate the integral 0 sec2 x dx using the
Midpoint Rule with 8 subdivisions, namely
x0 + x1 x0 + x1 x1 + x2 x2 + x3
M8 = ∆x f +f +f +f +
2 2 2 2
1. Use the Trapezoidal Rule to estimate the area. In our case, we obtain
Answer: T6 =
π 2 π 2 3π 2 5π 2 7π 2
2. Use Simpson’s Rule to estimate the area. T8 = sec +sec +sec +sec +sec
48 96 96 96 96
Answer: S6 = Correct Answers:
• 0.577899396
Note: You can click on the graph to enlarge the image. • 0.577075882
Correct Answers: 3. (1 point)
• 1/2*(2*1+2*3+2*1+2*0.5+2)+1*6
Approximate the following integral using the indicated meth-
• 1/3*(4*1+2*3+4*1+2*0.5+2)+1*6 ods. Leave your answers in ”calculator-ready” form.
Z 1
2
2. (1 point) e−3x dx
Calculate the integral approximations T8 (with the Trapezoid 0
Rule) and M8 (with the Midpoint Rule) for (a) Trapezoidal Rule with 4 subintervals
R π/6
0 sec2 x dx.
Leave your answers in calculator-ready form.
T8 = (b) Midpoint Rule with 4 subintervals
M8 = (c) Simpson’s Rule with 4 subintervals
2
(d) With f (x) = e−3x , It can be shown by direct computation
Solution: that | f (4) (x)| ≤ 108 on the interval [0, 1]. Using this information
Solution: and the error formula:
2
We have f (x) = sec x. We first estimate the integral M(b − a)5
E(n) ≤ ,
180n4
R π/6
0 sec2 x dx using the Trapezoid Rule with 8 subdivisions,
namely what is the least value of n so that the Simpson’s Rule approx-
imation for the given integral is guaranteed to be accurate to
∆x
T8 = f (x0 )+2 f (x1 )+2 f (x2 )+2 f (x3 )+2 f (x4 )+2 f (x5 )+2 f (x6within
)+2 f (x 0.00001?
7 )+ f (x8 . answer must be a whole number.)
(Your
)
2
Since n = 8, we have n=
π/8 − 0 π Correct Answers:
∆x = = • 0.502817651253163
8 48
1
, • 0.505095217095571 The left Rule with n = 6 is
• 0.504213520516726 h i
• 16 L6 = ∆x f (x0 ) + f (x1 ) + f (x2 ) + f (x3 ) + f (x4 ) + f (x5 )
h i
= 2 f (0) + f (2) + f (4) + f (6) + f (8) + f (10)
h i
≈ 2 8 + 7.9 + 7.6 + 7.0 + 6.2 + 5.2
≈ 83.8
4. (1 point) Use six rectangles to find an estimate of each type
for the area under the given graph of f from x = 0 to x = 12.
2. Because the function is concave down, the Left Sum is
an overestimate. (This is also apparent from the picture).
3. The function together with the rectangles for the Right
Rule is plotted below. We use the picture to estimate the values
of the function at the required values of x.
1. Take the sample points from the left-endpoints.
Answer: L6 =
The Right Sum with n = 6 is
2. Is your estimate L6 an underestimate or overestimate of h i
the true area? R6 = ∆x f (x1 ) + f (x2 ) + f (x3 ) + f (x4 ) + f (x5 ) + f (x6 )
• Choose one h i
= 2 f (2) + f (4) + f (6) + f (8) + f (10) + f (12)
• Underestimate h i
• Overestimate ≈ 2 7.9 + 7.6 + 7.0 + 6.2 + 5.2 + 4.0
3. Take the sample points from the right-endpoints. ≈ 75.8
Answer: R6 =
4. Is your estimate R6 an underestimate or overestimate of
the true area? 4. Because the function is concave down, the Right Sum is
• Choose one an underestimate. (This is also apparent from the picture).
• Underestimate
• Overestimate 5. The function together with the rectangles for the mid-
point Rule is plotted below. We use the picture to estimate the
5. Take the sample points from the midpoints.
values of the function at the required values of x.
Answer: M6 =
Note: You can click on the graph to enlarge the image.
Solution:
SOLUTION
12
We have a = 0, b = 12 and n = 6. So ∆x = b−a n = 6 = 2, x +x
The midpoints are x̄k = k−12 k , k = 1, 2, . . . , 6.
and xk = a + k∆x so
The Midpoint Sum with n = 6 is
x0 = 0, x1 = 2 , x2 = 4, x3 = 6, x4 = 8, x5 = 10 and x6 = 12. h i
M6 = ∆x f (x̄1 ) + f (x̄2 ) + f (x̄3 ) + f (x̄4 ) + f (x̄5 ) + f (x̄6 )
1. The function together with the rectangles for the Left h i
= 2 f (1) + f (3) + f (5) + f (7) + f (9) + f (11)
Rule is plotted below. We use the picture to estimate the values h i
of the function at the required values of x. ≈ 2 8.0 + 7.8 + 7.3 + 6.6 + 5.8 + 4.6
≈ 80.1
Correct Answers:
• 83.7778
• Overestimate
• 75.7778
• Underestimate
2
