Answers

MATH 101 Webwork 7 Homework Questions and Answers PDF from 2020

83 views 0 purchase

Course
MATH 101 (MATH101)

Institution
University Of British Columbia (UBC )

This is a pdf copy of the webwork 6 questions and answers for math 101 at ubc. Note that the exact numbers will be different for you but this provides insight into solving your problems nonetheless. Plus there is always the chance that the same questions are asked of you:) You can use this to help ...

[Show more]

Preview 2 out of 6 pages

View example

Uploaded on May 25, 2020
Number of pages 6
Written in 2019/2020
Type Answers
Person Unknown

math 101 differential and integral calculus
first year get ahead engineering ubc
webwork 7 questions and answers pdf
simpsons rule

CA$14.67

Added

Add to cart Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

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

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller new2282307660. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for CA$14.67. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

72841 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling