Exam (elaborations)
Assignment 1 solutions for calculus 1B.
- Institution
- University Of South Africa (Unisa)
Assignment 1 solutions for calculus 1B for semester 1, 2021.
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
Content preview
Assignment 1.1. Let f be the function
f (x) = x2 ln x8 where x > 1
(a) Use the sign pattern for f 0 (x) to determine the intervals where f rises and where
f falls. (5)
(b) Determine the coordinates of the local extreme point(s). (2)
(c) Find f 00 (x) and determine where the graph of f is concave up and where it is
concave down. (5)
[12]
2. You are designing a poster to contain 50 cm2 of printing with margins of 4 cm each at
the top and bottom and 2 cm at each side. What overall dimensions will minimize
the amount of paper used? [5]
3. Use L’Hôpital’s rule to evaluate:
1 cos3 x
(a) lim (5)
x!0 sin2 x
(b) lim (cos x1 )x (5)
x!1
x ln x
(c) lim 2
x!1 x 1
(5)
(d) lim x2ln x (5)
x!1 x 1
[20]
1 1
4. Find the exact value of tan sin 2
[3]
TOTAL: [40]
SOLUTIONS
1. Let f be the function de…ned by
f (x) = x2 ln x8 where x > 1
Solutions:
(a)
1
, 0
f 0 (x) = (x2 8 ln x)
8
= 2x
x
2(x2 4)
=
x
2(x 2)(x + 2)
= where x > 1
x
We …nd the sign pattern for x > 1 so x in the denominator always has positive values and
may be excluded from the sign pattern. We only look at the sign pattern for x > 1
y=x 2 +
y =x+2 + +
f 0 (x) +
1 2
Thus f rises on (2; 1) and f falls on (1; 2):
(b)
The local extreme point(s) occur at x when f 0 (x) = 0:Thus we must …nd where
2(x 2)(x + 2)
f 0 (x) = = 0 in the interval x > 1
x
This can only happen when x = 2 and the local extreme point is (2; f (2)) = (2; 22 ln 28 ) =
(2; 4 8 ln 2) or (2; 4 ln 256):
(c)
00 8
f (x) = 2 + for x > 1
x2
2x2 + 8
= for x > 1 (*)
x2
We see that y = 2x2 + 8 in (*) is a parabola which have no roots and is always positive,
Similarly y = x2 is always positive for x > 1 .
Thus the second derivative is always positive and so the function is concave up on the interval
(1; 1):
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 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 siphumelelezwane. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R79,00. 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