Exam (elaborations)
Calculus I (MAT 1320) Final Exam Questions and Solutions
- Course
- Institution
Questions and solutions to Calculus I Final Exams (Summer 2020) at the University of Ottawa.
[Show more]
Seller
Follow
lizysteven
Content preview
lOMoARcPSD|23478024MAT1320 Final Exam (With Solutions)
Calculus I (University of Ottawa)
Studocu is not sponsored or endorsed by any college or university
Downloaded by Lizy Steven (lizysteven832@gmail.com)
, lOMoARcPSD|23478024
MAT1320X Solution to Final Examination Summer 2020
Solution to the Final Examination
MAT1320X, Summer 2020
Part I. Multiple-Choice Questions
3 10 = 30 points
In all questions, (A) is the right answer.
1.1. The domain of the function f (x) = 1 ln( x e) is
(A) e < x 2e; (B) e x 2e; (C) x > 2e; (D) x < e.
Solution. 1 ln(x e) 0, ln(x e) 1, 0 < x e e, e < x 2e.
1.2. The domain of the function f (x) = 1 ln(e x ) is
(A) 0 x < e; (B) 0 x e; (C) x < 0; (D) x e.
Solution. 1 ln(e x) 0, ln(e x) 1, 0 < e x e, 0 x < e.
1.3. The domain of the function f (x) = 1 ln(e x ) is
(A) e < x 0; (B) e x < 0; (C) x < e; (D) x 0.
Solution. 1 ln(e + x) 0, ln(e + x) 1, 0 < e + x e. e < x 0.
2.1. Some values of functions f (x) and g(x), and their derivatives f '(x) and g'(x) are given in the
following table:
x f (x) f '(x) g(x) g'(x)
1 3 1 2 3
2 1 2 3 5
3 2 4 1 7
Let z = h(x) = (f g)(x), what is h(1) + h'(1)?
(A) 7; (B) 14; (C) 8; (D) 22.
Solution. h(1) = f (g(1)) = f (2) = 1. h'(1) = f '(g(1))g'(1) = f '(2)g'(1) = 2 3 = 6. h(1) + h'(1) = 7.
2.2. Some values of functions f (x) and g(x), and their derivatives f '(x) and g'(x) are given in the
following table:
1
Downloaded by Lizy Steven (lizysteven832@gmail.com)
, lOMoARcPSD|23478024
MAT1320X Solution to Final Examination Summer 2020
x f (x) f '(x) g(x) g'(x)
1 1 1 3 3
2 3 2 1 5
3 2 4 2 7
Let z = h(x) = (f g)(x), what is h(1) + h'(1)?
(A) 14; (B) 7; (C) 8; (D) 20.
Solution. h(1) = f (g(1)) = f (3) = 2. h'(1) = f '(g(1))g'(1) = f '(3)g'(1) = 4 3 = 12. h(1) + h'(1) =
14.
2.3. Some values of functions f (x) and g(x), and their derivatives f '(x) and g'(x) are given in the
following table:
x f (x) f '(x) g(x) g'(x)
1 3 1 1 3
2 1 2 3 5
3 2 4 2 7
Let z = h(x) = (f g)(x), what is h(2) + h'(2)?
(A) 22; (B) 14; (C) 7; (D) 8.
Solution. h(2) = f (g(2)) = f (3) = 2. h'(2) = f '(g(2))g'(2) = f '(3)g'(2) = 4 5 = 20. h(2) + h'(2) =
22.
2
e( x )
3.1. The derivative of the function f (x) = at x = 2is
x
7 4 4 4 7 4 3 4
(A) e ; (B) e ; (C) e ; (D) e .
4 3 3 4
2 2 2
2 x 2e x e x e x (2 x 2 1) 7
Solution. By the quotient rule, f '(x) = 2
2
. When x =2, f '(2) = e4 .
x x 4
sin 2 x
3.2. The derivative of the function f (x) = at x = is
x 4
4( 2) 4 2 4( 2) 4 2
(A) ; (B) ; (C) ; (D) .
2
2
2
2
2
Downloaded by Lizy Steven (lizysteven832@gmail.com)
, lOMoARcPSD|23478024
MAT1320X Solution to Final Examination Summer 2020
2 x sin x cos x sin 2 x
Solution. (B) By the quotient rule, f '(x) = 2
. When x = ,
x 4
1 1
2 2 2 4( 2)
f ' .
4
2
2
4
(ln x ) 2
3.3. The derivative of the function f (x) = at x = e is
x
(A) e2; (B) e; (C) e−1; (D) e2.
ln x
2 x (ln x ) 2
x ln x (2 ln x )
Solution. (E) Use the quotient rule. f '(x) = 2
. When x = e, f '(e)
x x2
= e−2.
(11 3x ) e x 1
2
4.1. Let f (x) = . Then f '(1) =
( x 1) 3 2 x
9 5 7 5
(A) ; (B) ; (C) ; (D) .
2 2 3 4
Solution. Taking the logarithm on both sides,
2 1
ln f (x) = ln(11 3x ) ( x 2 1) ln( x 1) ln(3 2 x) .
3 2
Then take the derivative with respect to x on both sides:
f '( x ) 2 1 1
2x .
f ( x) 11 3x x 1 3 2x
2 1 1
f '(x) = f (x) 2x .
11 3x x 1 3 2x
(11 3) e0 2 1 9
When x = 1, f (1) = 2 , f '(1) = 2 2 1 .
2 1 8 2 2
(3x 11)1/ 3 e x 1
2
4.2. Let f (x) = . Then f '(1) =
( x 3) x 5
3
Downloaded by Lizy Steven (lizysteven832@gmail.com)
, lOMoARcPSD|23478024
MAT1320X Solution to Final Examination Summer 2020
5 7 5 9
(A) ; (B) ; (C) ; (D) .
4 3 2 2
Solution. Taking the logarithm on both sides,
1 1
ln f (x) = ln(3x 11) ( x 2 1) ln( x 3) ln( x 5) .
3 2
Then take the derivative with respect to x on both sides:
f '( x ) 1 1 1
2x .
f ( x ) 3x 11 x 3 2( x 5)
1 1 1
f '(x) = f (x) 2x .
3x 11 x 3 2( x 5)
( 3 11)1/ 3 e0 1 11 1 1 5
When x = 1, f (1) = , f '(1) = 2 .
22 2 28 2 8 4
(2 x 5) e x 4
2
4.3. Let f (x) = . Then f '(2) =
( x 4) 3x 7
7 9 5 5
(A) ; (B) ; (C) ; (D) .
3 2 2 4
Solution. Taking the logarithm on both sides,
2 1
ln f (x) = ln(2 x 5) ( x 2 4) ln( x 4) ln(3x 7) .
3 2
Then take the derivative with respect to x on both sides:
f '( x ) 4 1 3
2x .
f ( x ) 3(2 x 5) x 4 2(3x 7)
4 1 3
f '(x) = f (x) 2x .
3(2 x 5) x 4 2(3x 7)
( 4 5) e0 1 14 1 3 2 7
When x = 2, f (2) = , f '(1) = 4 3 .
2 1 2 23 2 2 3 3
4
Downloaded by Lizy Steven (lizysteven832@gmail.com)
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 lizysteven. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.89. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
60904 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now