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## Section 1.2 Exercises

a. −1 b. Decreasing

a. 3/4 b. Increasing

a. 4/3 b. Increasing

a. 0 b. Horizontal

y = −6 x + 9 y = −6 x + 9

y = 1 3 x + 4 y = 1 3 x + 4

y = 1 2 x y = 1 2 x

y = 3 5 x − 3 y = 3 5 x − 3

a. ( m = 2 , b = −3 ) ( m = 2 , b = −3 ) b.

a. ( m = −6 , b = 0 ) ( m = −6 , b = 0 ) b.

a. ( m = 0 , b = −6 ) ( m = 0 , b = −6 ) b.

a. ( m = − 2 3 , b = 2 ) ( m = − 2 3 , b = 2 ) b.

a. 2 b. 5 2 , −1 ; 5 2 , −1 ; c. −5 d. Both ends rise e. Neither

a. 2 b. ± 2 ± 2 c. −1 d. Both ends rise e. Even

a. 3 b. 0, ± 3 ± 3 c. 0 d. Left end rises, right end falls e. Odd

a. 13 , −3 , 5 13 , −3 , 5 b.

a. −3 2 , −1 2 , 4 −3 2 , −1 2 , 4 b.

True; n = 3 n = 3

False; f ( x ) = x b , f ( x ) = x b , where b b is a real-valued constant, is a power function

a. V ( t ) = −2733 t + 20500 V ( t ) = −2733 t + 20500 b. ( 0 , 20 , 500 ) ( 0 , 20 , 500 ) means that the initial purchase price of the equipment is $20,500; ( 7.5 , 0 ) ( 7.5 , 0 ) means that in 7.5 years the computer equipment has no value. c.$6835 d. In approximately 6.4 years

a. C = 0.75 x + 125 C = 0.75 x + 125 b. $245 c. 167 cupcakes a. V ( t ) = −1500 t + 26,000 V ( t ) = −1500 t + 26,000 b. In 4 years, the value of the car is$20,000.

96% of the total capacity

## Section 1.3 Exercises

4 π 3 rad 4 π 3 rad

− π 3 − π 3

11 π 6 rad 11 π 6 rad

210 ° 210 °

−540 ° −540 °

− 2 2 − 2 2

3 − 1 2 2 3 − 1 2 2

a. b = 5.7 b = 5.7 b. sin A = 4 7 , cos A = 5.7 7 , tan A = 4 5.7 , csc A = 7 4 , sec A = 7 5.7 , cot A = 5.7 4 sin A = 4 7 , cos A = 5.7 7 , tan A = 4 5.7 , csc A = 7 4 , sec A = 7 5.7 , cot A = 5.7 4

a. c = 151.7 c = 151.7 b. sin A = 0.5623 , cos A = 0.8273 , tan A = 0.6797 , csc A = 1.778 , sec A = 1.209 , cot A = 1.471 sin A = 0.5623 , cos A = 0.8273 , tan A = 0.6797 , csc A = 1.778 , sec A = 1.209 , cot A = 1.471

a. c = 85 c = 85 b. sin A = 84 85 , cos A = 13 85 , tan A = 84 13 , csc A = 85 84 , sec A = 85 13 , cot A = 13 84 sin A = 84 85 , cos A = 13 85 , tan A = 84 13 , csc A = 85 84 , sec A = 85 13 , cot A = 13 84

a. y = 24 25 y = 24 25 b. sin θ = 24 25 , cos θ = 7 25 , tan θ = 24 7 , csc θ = 25 24 , sec θ = 25 7 , cot θ = 7 24 sin θ = 24 25 , cos θ = 7 25 , tan θ = 24 7 , csc θ = 25 24 , sec θ = 25 7 , cot θ = 7 24

a. x = − 2 3 x = − 2 3 b. sin θ = 7 3 , cos θ = − 2 3 , tan θ = − 14 2 , csc θ = 3 7 7 , sec θ = −3 2 2 , cot θ = − 14 7 sin θ = 7 3 , cos θ = − 2 3 , tan θ = − 14 2 , csc θ = 3 7 7 , sec θ = −3 2 2 , cot θ = − 14 7

sec 2 x sec 2 x

sin 2 x sin 2 x

sec 2 θ sec 2 θ

1 sin t ( = csc t ) 1 sin t ( = csc t )

{ π 6 , 5 π 6 } { π 6 , 5 π 6 }

{ π 4 , 3 π 4 , 5 π 4 , 7 π 4 } { π 4 , 3 π 4 , 5 π 4 , 7 π 4 }

{ 2 π 3 , 5 π 3 } { 2 π 3 , 5 π 3 }

{ 0 , π , π 3 , 5 π 3 } { 0 , π , π 3 , 5 π 3 }

y = 4 sin ( π 4 x ) y = 4 sin ( π 4 x )

y = cos ( 2 π x ) y = cos ( 2 π x )

a. 1 b. 2 π 2 π c. π 4 π 4 units to the right

a. 1 2 1 2 b. 8 π 8 π c. No phase shift

a. 3 b. 2 2 c. 2 π 2 π units to the left

Approximately 42 in.

≈ 30.9 in 2 ≈ 30.9 in 2

a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods

a. Amplitude = 10 ; period = 24 10 ; period = 24 b. 47.4 ° F 47.4 ° F c. 14 hours later, or 2 p.m. d.

## Section 1.4 Exercises

Not one-to-one

a. f −1 ( x ) = x + 4 f −1 ( x ) = x + 4 b. Domain : x ≥ −4 , range : y ≥ 0 : x ≥ −4 , range : y ≥ 0

a. f −1 ( x ) = x − 1 3 f −1 ( x ) = x − 1 3 b. Domain: all real numbers, range: all real numbers

a. f −1 ( x ) = x 2 + 1 , f −1 ( x ) = x 2 + 1 , b. Domain: x ≥ 0 , x ≥ 0 , range: y ≥ 1 y ≥ 1

These are inverses.

These are not inverses.

− π 6 − π 6

a. x = f −1 ( V ) = 0.04 − V 500 x = f −1 ( V ) = 0.04 − V 500 b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V . c. 0.1 cm; 0.14 cm; 0.17 cm

a. $31,250,$66,667, $107,143 b. ( p = 85 C C + 75 ) ( p = 85 C C + 75 ) c. 34 ppb a. ~ 92 ° ~ 92 ° b. ~ 42 ° ~ 42 ° c. ~ 27 ° ~ 27 ° x ≈ 6.69 , 8.51 ; x ≈ 6.69 , 8.51 ; so, the temperature occurs on June 21 and August 15 ~ 1.5 sec ~ 1.5 sec tan −1 ( tan ( 2.1 ) ) ≈ − 1.0416 ; tan −1 ( tan ( 2.1 ) ) ≈ − 1.0416 ; the expression does not equal 2.1 since 2.1 > 1.57 = π 2 2.1 > 1.57 = π 2 —in other words, it is not in the restricted domain of tan x . cos −1 ( cos ( 2.1 ) ) = 2.1 , tan x . cos −1 ( cos ( 2.1 ) ) = 2.1 , since 2.1 is in the restricted domain of cos x . cos x . ## Section 1.5 Exercises a. 125 b. 2.24 c. 9.74 a. 0.01 b. 10,000 c. 46.42 Domain: all real numbers, range: ( 2 , ∞ ) , y = 2 ( 2 , ∞ ) , y = 2 Domain: all real numbers, range: ( 0 , ∞ ) , y = 0 ( 0 , ∞ ) , y = 0 Domain: all real numbers, range: ( − ∞ , 1 ) , y = 1 ( − ∞ , 1 ) , y = 1 Domain: all real numbers, range: ( −1 , ∞ ) , y = −1 ( −1 , ∞ ) , y = −1 8 1 / 3 = 2 8 1 / 3 = 2 5 2 = 25 5 2 = 25 e −3 = 1 e 3 e −3 = 1 e 3 e 0 = 1 e 0 = 1 log 4 ( 1 16 ) = −2 log 4 ( 1 16 ) = −2 log 9 1 = 0 log 9 1 = 0 log 64 4 = 1 3 log 64 4 = 1 3 log 9 150 = y log 9 150 = y log 4 0.125 = − 3 2 log 4 0.125 = − 3 2 Domain: ( 1 , ∞ ) , ( 1 , ∞ ) , range: ( − ∞ , ∞ ) , x = 1 ( − ∞ , ∞ ) , x = 1 Domain: ( 0 , ∞ ) , ( 0 , ∞ ) , range: ( − ∞ , ∞ ) , x = 0 ( − ∞ , ∞ ) , x = 0 Domain: ( −1 , ∞ ) , ( −1 , ∞ ) , range: ( − ∞ , ∞ ) , x = −1 ( − ∞ , ∞ ) , x = −1 2 + 3 log 3 a − log 3 b 2 + 3 log 3 a − log 3 b 3 2 + 1 2 log 5 x + 3 2 log 5 y 3 2 + 1 2 log 5 x + 3 2 log 5 y − 3 2 + ln 6 − 3 2 + ln 6 ln 15 3 ln 15 3 log 7.21 log 7.21 2 3 + log 11 3 log 7 2 3 + log 11 3 log 7 x = 1 25 x = 1 25 x = 4 x = 4 x = 3 x = 3 1 + 5 1 + 5 ( log 82 log 7 ≈ 2.2646 ) ( log 82 log 7 ≈ 2.2646 ) ( log 211 log 0.5 ≈ − 7.7211 ) ( log 211 log 0.5 ≈ − 7.7211 ) ( log 0.452 log 0.2 ≈ 0.4934 ) ( log 0.452 log 0.2 ≈ 0.4934 ) ~ 17 , 491 ~ 17 , 491 Approximately$131,653 is accumulated in 5 years.

i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid

a. ~ 333 ~ 333 million b. 94 years from 2013, or in 2107

a. k ≈ 0.0578 k ≈ 0.0578 b. ≈ 92 ≈ 92 hours

The San Francisco earthquake was 10 3.4 or ≈ 2512 10 3.4 or ≈ 2512 times more intense than the Japanese earthquake.

## Review Exercises

Domain: x > 5 , x > 5 , range: all real numbers

Domain: x > 2 x > 2 and x < − 4 , x < − 4 , range: all real numbers

Degree of 3, y y -intercept: 0, zeros: 0, 3 − 1 , −1 − 3 3 − 1 , −1 − 3

cos 2 x - sin 2 x = cos 2 x = 1 - 2 sin 2 x = 2 cos 2 x - 1 cos 2 x - sin 2 x = cos 2 x = 1 - 2 sin 2 x = 2 cos 2 x - 1

0 , ± 2 π 0 , ± 2 π

One-to-one; yes, the function has an inverse; inverse: f −1 ( x ) = 1 y f −1 ( x ) = 1 y

x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7 x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7

a. C ( x ) = 300 + 7 x C ( x ) = 300 + 7 x b. 100 shirts

The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

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## Exercises: Calculus (OpenStax)

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These are homework exercises to accompany OpenStax's "Calculus" Textmap.

• 1.E: Functions and Graphs (Exercises) These are homework exercises to accompany Chapter 1 of OpenStax's "Calculus" Textmap.
• 2.E: Limits (Exercises) These are homework exercises to accompany Chapter 2 of OpenStax's "Calculus" Textmap.
• 3.E: Derivatives (Exercises) These are homework exercises to accompany Chapter 3 of OpenStax's "Calculus" Textmap.
• 4.E: Applications of Derivatives (Exercises) These are homework exercises to accompany Chapter 4 of OpenStax's "Calculus" Textmap.
• 5.E: Integration (Exercises) These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
• 6.E: Applications of Integration (Exercises) These are homework exercises to accompany Chapter 6 of OpenStax's "Calculus" Textmap.
• 7.E: Techniques of Integration (Exercises) These are homework exercises to accompany Chapter 7 of OpenStax's "Calculus" Textmap.
• 8.E: Differential Equations (Exercises) These are homework exercises to accompany Chapter 8 of OpenStax's "Calculus" Textmap.
• 9.E: Sequences and Series (Exercises) These are homework exercises to accompany Chapter 9 of OpenStax's "Calculus" Textmap.
• 10.E: Power Series (Exercises) These are homework exercises to accompany Chapter 10 of OpenStax's "Calculus" Textmap.
• 11.E: Parametric Equations and Polar Coordinates (Exercises) These are homework exercises to accompany Chapter 11 of OpenStax's "Calculus" Textmap.
• 12.E: Vectors in Space (Exercises) These are homework exercises to accompany Chapter 12 of OpenStax's "Calculus" Textmap.
• 13.E: Vector-Valued Functions (Exercises) These are homework exercises to accompany Chapter 13 of OpenStax's "Calculus" Textmap.
• 14.E: Differentiation of Functions of Several Variables (Exercise) These are homework exercises to accompany Chapter 14 of OpenStax's "Calculus" Textmap.
• 15.E: Multiple Integration (Exercises) These are homework exercises to accompany Chapter 15 of OpenStax's "Calculus" Textmap.
• 16.E: Vector Calculus (Exercises) These are homework exercises to accompany Chapter 16 of OpenStax's "Calculus" Textmap.
• 17.E: Second-Order Differential Equations (Exercises) These are homework exercises to accompany Chapter 17 of OpenStax's "Calculus" Textmap.

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## Unit 1: Limits and continuity

Unit 2: derivatives: definition and basic rules, unit 3: derivatives: chain rule and other advanced topics, unit 4: applications of derivatives, unit 5: analyzing functions, unit 6: integrals, unit 7: differential equations, unit 8: applications of integrals.

## Math 104: Calculus I – Homework

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Boost your assignment and exam grades with these best AI homework helpers.

• March 17, 2024

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## AIMath is the most efficient math AI solver for math learning

AIMath is a free online math AI solver designed for people struggling with complex...

## AIMath is the most efficient math AI solver for math learning

O ften perceived as an enigmatic language, mathematics has been revered and feared. For some, it's a thrilling puzzle waiting to be solved; for others, it's a daunting maze of symbols and theorems.

Recognizing the difficulties students encounter in grasping intricate mathematical concepts, AIMath emerges as a supportive solution.

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Let’s delve into the realm of AIMath, exploring its potential to reshape the future of math education.

## AIMath overview: What is it?

AIMath represents a revolutionary advance in educational technology, constituting an invaluable aid for students and educators in the field of mathematics.

It is an artificial intelligence-powered system designed to assist with a wide range of mathematical problems.

This AI-driven platform, often called a math AI solver, distinguishes itself by providing users with step-by-step guidance and explanations-not just answers.

Its technology is tailored to enhance and facilitate learning and is capable of addressing topics from basic arithmetic to complex calculus equations.

## How can AIMath enhance your math learning experience?

Now that we understand the essence of AIMath, the next question arises: How can we harness its potential?

## Application in self-study and exam prep

Utilizing AIMath proves highly beneficial for self-study and exam preparation.

This online math AI solver is a dynamic addition to students’ study routines, seamlessly complementing traditional methods and textbooks.

Its adaptability and swift feedback on practice problems make it an indispensable resource, empowering students to deepen their comprehension and achieve improved academic performance.

## Support for curriculum and homework

Moreover, AIMath is a valuable support system for formal curriculums and homework assignments.

The broad spectrum of solutions and detailed explanations provided align seamlessly with educational systems’ diverse requirements.

By offering reliable and immediate assistance, AIMath becomes a crucial ally for students seeking to successfully complete assignments and grasp new concepts introduced in their classes.

## Who is AIMath designed for?

AIMath, designed for a diverse user base, is particularly tailored to:

• Students: The sophisticated technologies embedded within AIMath have a proven track record of improving student test scores. By delivering comprehensive explanations and fostering independent problem-solving skills, this math AI homework solver significantly boosts academic performance among its users. The AI not only equips students with the correct answers but also ensures that they understand the rationale behind each step, resulting in deeper learning and better retention of mathematical concepts.
• Educators: AIMath also serves as a formidable resource for educators. By integrating this AI math homework solver into their teaching methodologies, teachers can offer supplemental support to students outside the classroom. The math AI solver’s in-depth explanations and real-time assistance enable teachers to direct their students towards an additional layer of instruction, reinforcing lesson plans and improving educational outcomes.

## Key features of AIMath

With so many math AI solvers on the market, what sets AIMath apart from its competitors?

## Comprehensive coverage of mathematical topics

AIMath distinguishes itself by its ability to address a wide spectrum of mathematical subjects.

These include arithmetic, algebra, geometry, trigonometry, calculus, and even advanced fields such as discrete math and linear algebra.

By incorporating a comprehensive range of topics, AIMath ensures that users with varying degrees of proficiency can find the help they need, whether they are tackling primary school-level calculations or university-level problems.

## Global accessibility through multi-language support

One of the significant advantages of AIMath is its multi-language support, catering to a global audience.

By providing solutions in numerous languages, this math AI solver bridges the communication gap and ensures that language barriers do not hinder students’ quest for mathematical understanding.

## Real-time problem analysis and solutions

AIMath’s efficiency as an AI math helper derives from its real-time problem analysis capabilities.

This feature enables the system to produce accurate solutions promptly, which is especially beneficial for students requiring immediate assistance with homework or exam preparation.

The state-of-the-art algorithms employed by AIMath allow it to interpret and solve even the most intricate math problems within seconds, drastically reducing students’ time on problem-solving.

## Universal access and 24/7 availability

AIMath’s universal access and 24/7 availability highlight its commitment to constantly supporting learners of all ages and from all locales.

With just an internet connection, users can access the robust features of this AI math homework solver at any time and from anywhere in the world.

## How do you solve your homework problems with AIMath?

Let's walk through the step-by-step guide for using AIMath, the AI-powered math solver:

• Visit the AIMath website.
• Type your math problem into the provided input box.
• Click the button to submit your math problem.
• AIMath will process your query using its AI algorithms. Within seconds, you will receive an accurate answer and a detailed explanation.

## The evolving role of AIMath in bridging education gaps

As we look to the future, what benefits can we anticipate from the continued use of AIMath?

## Addressing geographic and socio-economic barriers

One crucial role of AIMath in education is its remarkable potential to mitigate geographic and socioeconomic disparities in learning.

With AIMath, students in remote or underserved areas gain improved access to quality math education.

It democratizes learning by making high-quality math assistance available to anyone with internet access, regardless of their location or socio-economic status.

## Achievement of educational equity

The ongoing evolution of AIMath actively contributes to the attainment of educational equity.

By ensuring that all students have equal opportunities to understand and excel in mathematics, this math AI solver lays the groundwork for a more equitable educational landscape.

Education is a fundamental right, and the increasing prominence of AI math solvers positions it as a key player in realizing this ideal on a global scale.

## Final thoughts

As an AI math solver, AIMath provides precision and insightful guidance through the complexities of mathematical landscapes.

It effectively addresses comprehension gaps and fosters a deep appreciation for the elegance embedded in numerical intricacies.

Trustworthy and helpful, AIMath stands as a reliable AI Math companion for students, teachers, and anyone eager to learn mathematics.

Have any thoughts on this? Drop us a line below in the comments, or carry the discussion to our  Twitter  or  Facebook .

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The post AIMath is the most efficient math AI solver for math learning appeared first on KnowTechie .

March 12, 2024

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## The numbers do not add up for mathematics homework, according to a new study

by University of South Australia

Supporting kids with math homework is a common afterschool activity. But beyond the basics, new curricula and teaching strategies are making it harder for parents to help and it's taking a serious toll on children's confidence and learning.

In a study from the University of South Australia, researchers have found that mathematics homework can sometimes cause more harm than good.

Exploring how homework policies and practices affect families, researchers found that mathematics homework could inadvertently affect a child when it often:

• was too difficult for a child to complete independently, and/or with the support of a parent
• required significant support from parents and seeped into family time
• resulted in a negative experience for the child and their parent, leading to negative associations with mathematics and potentially students' disengagement from the subject
• generated feelings of despair, stress, and negativity among parents who were unable to help
• made students feel inadequate when they struggled with the work.

UniSA researcher Associate Professor Lisa O'Keeffe says such negativity around mathematics has broad implications.

"Homework has long been accepted as a practice that reinforces children's learning and improves academic success," Assoc. Prof. O'Keeffe says.

"But when it is too complex for a student to complete even with parent support, it raises the question as to why it was set as a homework task in the first place. We know that parents play a key role in supporting their children with schooling and homework. When children need help, their parents are often the first people they turn to.

"But many parents are unsure of the current mathematics strategies and approaches that their children are learning as these have changed a lot since they were at school. Like many things, mathematics teaching has evolved over time. But when parents realize that their tried-and-true methods are different to those which their children are learning, it can be hard to adapt, and this can add undue pressure. When children see their parents struggle with mathematics homework, or where mathematics homework becomes a shared site of frustration for families, it can lead to negativity across generations.

"For example, we might hear adults saying things like, 'I wasn't very good at math, so my child won't be either.' Negative interactions with mathematics, and negative discourses like these can lead to reduced confidence, reduced self-efficacy, and can negatively affect children's resilience, persistence, and ultimately their inclination to continue with mathematics."

Any decline in STEM subjects such as math can have long-term impacts for Australia's future. Statistics show that fewer than 10% of students are studying a higher level of math, with math capabilities declining more than 25 points (15-year-olds in 2022 scored at a level that would have been expected of 14-year-olds, 20 years earlier).

Co-researcher, UniSA's Dr. Sarah McDonald, says the research also identified gendered biases.

"Our research showed that it was overwhelmingly mothers who were responsible for managing children's homework. And they often experience frustration or despair when they were unable to understand the math problems," Dr. McDonald says. "When mothers find math hard, there is concern that this may demonstrate to their children , especially their girls, that this is not an area in which they would naturally excel.

"The last thing teachers want to do is disadvantage girls in developing potentially strong mathematical identities. We need a greater understanding of homework policies and expectations.

"The experiences of the families in our study do not support the often-quoted claim by researchers that that homework has potential non-academic benefits such as fostering independence, creating positive character traits, developing good organizational skills, or virtues such as self-discipline and responsibility."

Provided by University of South Australia

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## Doing more but learning less: Addressing the risks of AI in research

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