Other
MAT1613 ASSIGNMENT 1 2021
- Institution
- University Of South Africa (Unisa)
This document contains MAT1613 ASSIGNMENT 1 2021 solutions. All workings are shown clearly step by step and explanations are also provided.
[Show more]
1 review
By: mariskamiskyjordaan • 3 year ago
Content preview
MAT1613ASSIGNMENT 1 2021
QUESTION 1
𝐴 = 4𝜋𝑟 2 (𝑎𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑠𝑝ℎ𝑒𝑟𝑒)
4
𝑉 = 𝜋𝑟 3 (𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑠𝑝ℎ𝑒𝑟𝑒)
3
𝑑𝐴 𝑑𝐴 𝑑𝑟 𝑑𝑉
= × ×
𝑑𝑡 𝑑𝑟 𝑑𝑉 𝑑𝑡
4
𝑉 = 𝜋𝑟 3
3
𝑑𝑉
= 4𝜋𝑟 2
𝑑𝑟
𝑑𝑟 1
=
𝑑𝑉 4𝜋𝑟 2
𝐴 = 4𝜋𝑟 2
𝑑𝐴
= 8𝜋𝑟
𝑑𝑟
𝑑𝐴 1 𝑑𝑉
= (8𝜋𝑟) × ( 2
)×
𝑑𝑡 4𝜋𝑟 𝑑𝑡
𝑑𝐴 2 𝑑𝑉 𝑑𝑉
= × ∴ 𝑟 = 10, = 15 𝑐𝑚3 ⁄𝑠𝑒𝑐
𝑑𝑡 𝑟 𝑑𝑡 𝑑𝑡
𝑑𝐴 2
= × 15
𝑑𝑡 (10)
𝑑𝐴
= 3 𝑐𝑚2 ⁄𝑠𝑒𝑐
𝑑𝑡
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 = 3 𝑐𝑚2 ⁄𝑠𝑒𝑐
QUESTION 2
−𝑥
𝑓(𝑥) = 𝑥𝑒
a).
,𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 𝑓(0)
= (0)𝑒 −(0)
=0
b).
𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒:
lim 𝑓(𝑥) = lim 𝑥𝑒 −𝑥
𝑥↔∞ 𝑥↔∞
𝑥
lim 𝑓(𝑥) = lim ( 𝑥 )
𝑥↔∞ 𝑥↔∞ 𝑒
𝑈𝑠𝑖𝑛𝑔 𝐿′ℎ𝑜𝑠𝑝𝑖𝑡𝑎𝑙
𝑑
[𝑥]
lim 𝑓(𝑥) = lim 𝑑𝑥
𝑥↔∞ 𝑥↔∞ 𝑑
[𝑒 𝑥 ]
𝑑𝑥
1
= lim
𝑥↔∞ 𝑒 𝑥
1
=
𝑒∞
1
=
∞
=0
𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒: 𝑦 = 0
𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒:
𝑥
𝑓(𝑥) = 𝑥𝑒 −𝑥 =
𝑒𝑥
∴ 𝑒 𝑥 > 0 𝑎𝑙𝑤𝑎𝑦𝑠 , 𝑚𝑒𝑎𝑛𝑖𝑛𝑔 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
c).
i).
𝑓(𝑥) = 𝑥𝑒 −𝑥
𝐿𝑒𝑡: 𝑢 = 𝑥 𝑎𝑛𝑑 𝑣 = 𝑒 −𝑥
, 𝑑𝑣 𝑑𝑢
= −𝑒 −𝑥 𝑎𝑛𝑑 =1
𝑑𝑥 𝑑𝑥
𝑈𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑟𝑢𝑙𝑒:
𝑑𝑣 𝑑𝑢
𝑓 ′ (𝑥) = 𝑢 +𝑣
𝑑𝑥 𝑑𝑥
= (𝑥)(−𝑒 −𝑥 ) + (𝑒 −𝑥 )(1)
= −𝑥𝑒 −𝑥 + 𝑒 −𝑥
= 𝑒 −𝑥 − 𝑥𝑒 −𝑥
= 𝑒 −𝑥 (1 − 𝑥)
(1 − 𝑥)
=
𝑒𝑥
∴ 𝑓(𝑥) 𝑟𝑖𝑠𝑒𝑠 𝑤ℎ𝑒𝑛 𝑓 ′ (𝑥) > 0
𝑓 ′ (𝑥) > 0
(1 − 𝑥)
>0 ∴ 𝑒 𝑥 𝑖𝑠 𝑎𝑙𝑤𝑎𝑦𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑧𝑒𝑟𝑜
𝑒𝑥
(1 − 𝑥) > 0
𝑥<1
𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑤ℎ𝑒𝑛 𝑓(𝑥) 𝑟𝑖𝑠𝑒𝑠 = (−∞, 1)
∴ 𝑓(𝑥) 𝑓𝑎𝑙𝑙𝑠 𝑤ℎ𝑒𝑛 𝑓 ′ (𝑥) < 0
𝑓 ′ (𝑥) < 0
(1 − 𝑥)
<0
𝑒𝑥
(1 − 𝑥) < 0
𝑥>1
𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑤ℎ𝑒𝑛 𝑓(𝑥) 𝑓𝑎𝑙𝑙𝑠 = (1, ∞)
ii).
𝑙𝑜𝑐𝑎𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑎:
𝑓 ′ (𝑥) = 0
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 EFT, 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 this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jctutor0814378595. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R80,00. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
50843 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy summaries for 14 years now