Prepare for your exams
Get points
Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Sell on Docsity

Log in Sign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents

20 Points

For each uploaded document

Answer questions

5 Points

For each given answer (max 1 per day)

All the ways to get free points

Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study

Go to the blog

Calculus II: Practice Problems Answers, Exams of Calculus

Vincennes University (VU)Calculus

Answers to practice problems in calculus ii, including solving for x, finding derivatives, integrals, and solving initial value problems. Topics covered include logarithmic functions, exponential functions, and differential equations.

Typology: Exams

2021/2022

Uploaded on 02/24/2022

anandamayi 🇺🇸

4.2

(9)

250 documents

1 / 3

This page cannot be seen from the preview

Don't miss anything!

Calculus II

Practice Problems 1: Answers

1. Solve for x:

a) 6x



362



x

Answer. Since 36



62, the equation becomes 6x



62



2



x



, so we must have x



2



2



x



which has the

solution x



4



3.

b) ln3x



5

Answer. If we exponentiate both sides we get x



35



243.

c) ln2



x



1



ln2



x



1





ln28

Answer. Since the difference of logarithms is the logarithm of the quotient, we rewrite this as

ln2



x



1

x



1





ln28



which is, after exponentiating, the same as x



1

x



1



8



which gives us x



1



8



x



1



, so that x



9



7.

2. Find the derivative of the given function:

a) y



ln



lnx



Answer. Use the chain rule: dy

dx



1

lnx

d

dx lnx



1

xlnx

b) y



log2



x2



1



Answer. Remember that log2A



lnA



ln2, so y





ln



x2



1



ln2. Then, use the chain rule:

dy

dx



1



ln2



x2



1



2x



2

ln2

x

x2



1



c) y



ex2

x

Answer. Use the quotient rule carefully:

dy

dx



x



2xex2



ex2

x2



2ex2



x



2ex2





ex2



2



x



2



Partial preview of the text

Download Calculus II: Practice Problems Answers and more Exams Calculus in PDF only on Docsity!

Calculus II Practice Problems 1: Answers

Solve for x :

a) 6 x^^ 362 x

Answer. Since 36 62 , the equation becomes 6 x^^ 62

2 x

, so we must have x 2 2 x which has the

solution x 4 3.

b) ln 3 x 5

Answer. If we exponentiate both sides we get x 35 243.

c) ln 2 x 1 ln 2 x 1 ln 2 8

Answer. Since the difference of logarithms is the logarithm of the quotient, we rewrite this as

ln 2

x 1

ln 2 8

which is, after exponentiating, the same as

x 1

which gives us x 1 8 x 1 , so that x 9 7.

Find the derivative of the given function:

a) y ln ln x

Answer. Use the chain rule: dy dx

ln x

d dx

ln x

x ln x

b) y log 2 x^2 1

Answer. Remember that log 2 A ln A ln2, so y ln x^2 1 ln 2. Then, use the chain rule:

dy dx

ln 2 x^2 1

2 x

ln 2

x

x^2 1

c) y

ex

2

x

Answer. Use the quotient rule carefully:

dy dx

x 2 xex

2

ex

2

x^2

2 e

x^2 x 2 ex^2

ex^2 2 x 2

3. Solve: ln x ln x

Answer. By the laws of exponents, this becomes ln x 1 2 ln x. Squaring both sides, we get the equation

4 ln x ln x 2. Thus ln x 0, or ln x 4, giving the solutions x 1 e^4.

Find the integrals:

a)

ln x ^2 1

x

dx

Answer. Let u ln x , so du dx x. Then

ln x 2 1

x

dx u^2 1 du

u^3 3

u C

ln x ^3

ln x C

b) e sin^ x^ cos xdx

Answer. Let u sin x du cos xdx. Then

e sin x^ cos xdx eudu e sin^ x^^ C

c)

xdx

3 x^2 1

Answer. Let u 3 x^2 1 du 6 xdx. Then

xdx

3 x^2 1

du u

ln 3 x^2 1 C

5. Solve the initial value problem x 1 y 2 y y 1 1.

Answer. Separating variables, this becomes

dy y

2 dx

x 1

Integrating both sides,

Calculus II: Practice Problems Answers, Exams of Calculus

Related documents

Partial preview of the text

Download Calculus II: Practice Problems Answers and more Exams Calculus in PDF only on Docsity!

a) 6 x^^ 362 x

Answer. Since 36 62 , the equation becomes 6 x^^ 62

2 x

, so we must have x 2 2  x  which has the

solution x 4 3.

b) ln 3 x 5

Answer. If we exponentiate both sides we get x 35 243.

c) ln 2 x  1  ln 2 x  1  ln 2 8

ln 2

x  1

x  1

x  1

x  1

which gives us x  1 8 x  1  , so that x 9  7.

a) y ln ln x 

ln x

b) y log 2 x^2  1 

Answer. Remember that log 2 A ln A ln2, so y ln x^2  1  ln 2. Then, use the chain rule:

ln 2  x^2  1 

2 x

x^2  1 

c) y

x 2 xex

x^2  x 2 ex^2

ex^2 2  x 2 

3. Solve:  ln x ln  x 

Answer. By the laws of exponents, this becomes  ln x 1  2 ln x. Squaring both sides, we get the equation

4 ln x ln x  2. Thus ln x 0, or ln x 4, giving the solutions x 1 e^4.

a) 

ln x ^2  1

dx

Answer. Let u ln x , so du dx  x. Then

ln x  2  1

dx  u^2  1  du

u  C

ln x ^3

ln x  C

b)  e sin^ x^ cos xdx

Answer. Let u sin x du cos xdx. Then

 e sin x^ cos xdx  eudu e sin^ x^^  C

c) 

3 x^2  1

Answer. Let u 3 x^2  1 du 6 xdx. Then

3 x^2  1

ln 3 x^2  1   C

5. Solve the initial value problem x  1  y  2 y y 1  1.

x  1 

ln y 2 ln x  1  C

which exponentiates to y K x  1  2 , where K eC. The initial values give 1 K 1  1  2 , so K 1  4, and

the solution is y x  1  2 4.

6. If f x  2  x ln x , find f  x .

, so we must have x 2 2 x which has the

solution x 4 3.

c) ln 2 x 1 ln 2 x 1 ln 2 8

x 1

x 1

x 1

x 1

which gives us x 1 8 x 1 , so that x 9 7.

a) y ln ln x

b) y log 2 x^2 1

Answer. Remember that log 2 A ln A ln2, so y ln x^2 1 ln 2. Then, use the chain rule:

ln 2 x^2 1

x^2 1

x^2 x 2 ex^2

ex^2 2 x 2

3. Solve: ln x ln x

Answer. By the laws of exponents, this becomes ln x 1 2 ln x. Squaring both sides, we get the equation

4 ln x ln x 2. Thus ln x 0, or ln x 4, giving the solutions x 1 e^4.

a)

ln x ^2 1

Answer. Let u ln x , so du dx x. Then

ln x 2 1

dx u^2 1 du

u C

ln x ^3

ln x C

b) e sin^ x^ cos xdx

e sin x^ cos xdx eudu e sin^ x^^ C

c)

3 x^2 1

Answer. Let u 3 x^2 1 du 6 xdx. Then

3 x^2 1

ln 3 x^2 1 C

5. Solve the initial value problem x 1 y 2 y y 1 1.

x 1

ln y 2 ln x 1 C

which exponentiates to y K x 1 2 , where K eC. The initial values give 1 K 1 1 2 , so K 1 4, and

the solution is y x 1 2 4.

6. If f x 2 x ln x , find f x .