Giả sử doanh số (tính bằng số sản phẩm) của một sản phẩm mới (trong vòng một số năm nhất định) tuân theo quy luật logistic được mô hình hoá bằng hàm số \(y=f\left( t \right)=\frac{5\,000}{1+5{{\text{e}}^{-t}}},\,t\ge 0\), trong đó thời gian \(t\) (năm) được tính kể từ khi phát hành sản phẩm mới. Khi đó, đạo hàm \(f\text{ }\!\!'\!\!\text{ }\left( t \right)\) sẽ biểu thị tốc độ bán hàng. Sau khi phát hành bao nhiêu năm thì tốc độ bán hàng là lớn nhất? (làm tròn kết quả tới chữ số hàng phần mười)
Hãy suy nghĩ và trả lời câu hỏi trước khi xem đáp án
Ta có: \(f\text{ }\!\!'\!\!\text{ }\left( t \right)=\frac{-5\,000\left( 1+5{{\text{e}}^{-t}} \right)}{{{\left( 1+5{{\text{e}}^{-t}} \right)}^{2}}}=\frac{25\,000{{\text{e}}^{-t}}}{{{\left( 1+5{{\text{e}}^{-t}} \right)}^{2}}}\)
Tốc độ bán hàng là lớn nhất khi \(f\text{ }\!\!'\!\!\text{ }\left( t \right)\) lớn nhất.
Đặt \(h\left( t \right)=\frac{25\,000{{\text{e}}^{-t}}}{{{\left( 1+5{{\text{e}}^{-t}} \right)}^{2}}}\).
\(\begin{align} & h\text{ }\!\!'\!\!\text{ }\left( t \right)=\frac{-25\,000{{\text{e}}^{-t}}{{\left( 1+5{{\text{e}}^{-t}} \right)}^{2}}-2.\left( -5{{\text{e}}^{-t}} \right).\left( 1+5{{\text{e}}^{-t}} \right).25\,000{{\text{e}}^{-t}}}{{{\left( 1+5{{\text{e}}^{-t}} \right)}^{4}}} \\ & =\frac{-25\,000{{\text{e}}^{-t}}\left( 1+5{{\text{e}}^{-t}} \right)\left( 1+5{{\text{e}}^{-t}}-10{{\text{e}}^{-t}} \right)}{{{\left( 1+5{{\text{e}}^{-t}} \right)}^{4}}}=\frac{-25\,000{{\text{e}}^{-t}}\left( 1-5{{\text{e}}^{-t}} \right)}{{{\left( 1+5{{\text{e}}^{-t}} \right)}^{3}}} \\ \end{align}\)
\(h\text{ }\!\!'\!\!\text{ }\left( t \right)=0\Leftrightarrow \frac{-25\,000{{\text{e}}^{-t}}\left( 1-5{{\text{e}}^{-t}} \right)}{{{\left( 1+5{{\text{e}}^{-t}} \right)}^{3}}}=0\Leftrightarrow 1-5{{\text{e}}^{-t}}=0\Leftrightarrow {{\text{e}}^{-t}}=\frac{1}{5}\Leftrightarrow t=\text{ln}5\) (thỏa mãn)
Lập bảng biến thiên với \(t\in \left[ 0;+\infty \right)\):
Vậy sau khi phát hành khoảng \(\text{ln}5\approx 1,6\) năm thì thì tốc độ bán hàng là lớn nhất.
Bộ Đề Kiểm Tra Giữa Học Kì I - Toán 12 - Cánh Diều là tài liệu học tập hữu ích giúp học sinh ôn tập và hệ thống lại kiến thức đã học. Với các dạng bài trắc nghiệm và tự luận phong phú, đề thi giúp học sinh rèn luyện khả năng tính toán, tư duy logic, đồng thời nâng cao kỹ năng giải quyết vấn đề một cách hiệu quả. Đây là công cụ hỗ trợ đắc lực để chuẩn bị cho các kỳ thi quan trọng.
Câu hỏi liên quan
Read the following announcement and choose the correct option that best fits each of the numbered blanks.
School Announcement: Environmental Clean-Up Day
Dear Students and Staff,
We are excited to announce our upcoming Environmental Clean-Up Day! (13) __________ us on Saturday, August 10th, from 9 AM to 12 PM as we work together to clean up our local park. This event is a great opportunity to help protect our environment and earn community service hours.
✔ All necessary supplies (14) __________.
Please sign (15) __________ in the main office by August 5th. Let's make a difference together!
Thank you,
Choose the correct answer to fill in the blank in each of the following questions.
The team worked overtime to meet the deadline, __________ demonstrated their commitment to the project.
Read the following passage and choose the correct answer to each of the questions.
1. In recent years, there has been a significant shift towards more eco-friendly alternatives to plastic products. Among these changes is the introduction of paper straws, which have become increasingly popular in cafes, restaurants, and homes. This shift is largely driven by growing awareness of the environmental damage caused by plastic pollution, particularly in our oceans.
2. Plastic straws have long been criticised for their contribution to pollution. They are lightweight, making them easily blown into waterways and oceans, where they can harm marine life. Unlike plastic, which takes hundreds of years to decompose, paper straws break down much more quickly. This means they are less likely to accumulate in the environment, making them a seemingly better option.
3. However, the production of paper straws also has an environmental impact. The manufacturing process requires cutting down trees, which can contribute to deforestation if not managed sustainably. Moreover, many paper straws have a thin plastic lining to prevent them from getting soggy quickly. This plastic lining can still cause environmental issues and reduce the benefits of switching from plastic to paper.
4. There are also practical concerns. Many people find paper straws less durable than plastic ones. They can become soft and unusable in a short amount of time, especially when used in hot beverages. This inconvenience can lead to increased waste if people use more straws as replacements.
5. On the other hand, some companies are working to address these issues. They are developing more durable and completely biodegradable paper straws, as well as investing in sustainable forestry practices to ensure that the production of paper products does not contribute to deforestation.
6. In conclusion, while paper straws do offer a more environmentally friendly alternative to plastic straws, they are not without their own set of problems. The key to making a positive impact lies in continuing to innovate and improve these products, as well as encouraging responsible production and disposal practices. As consumers, being aware of these issues can help us make more informed choices and support companies that prioritise sustainability.
(35) The word soggy in paragraph 3 is OPPOSITE in meaning to __________.
Read the following passage and choose the correct answer to each of the questions.
1. In recent years, there has been a significant shift towards more eco-friendly alternatives to plastic products. Among these changes is the introduction of paper straws, which have become increasingly popular in cafes, restaurants, and homes. This shift is largely driven by growing awareness of the environmental damage caused by plastic pollution, particularly in our oceans.
2. Plastic straws have long been criticised for their contribution to pollution. They are lightweight, making them easily blown into waterways and oceans, where they can harm marine life. Unlike plastic, which takes hundreds of years to decompose, paper straws break down much more quickly. This means they are less likely to accumulate in the environment, making them a seemingly better option.
3. However, the production of paper straws also has an environmental impact. The manufacturing process requires cutting down trees, which can contribute to deforestation if not managed sustainably. Moreover, many paper straws have a thin plastic lining to prevent them from getting soggy quickly. This plastic lining can still cause environmental issues and reduce the benefits of switching from plastic to paper.
4. There are also practical concerns. Many people find paper straws less durable than plastic ones. They can become soft and unusable in a short amount of time, especially when used in hot beverages. This inconvenience can lead to increased waste if people use more straws as replacements.
5. On the other hand, some companies are working to address these issues. They are developing more durable and completely biodegradable paper straws, as well as investing in sustainable forestry practices to ensure that the production of paper products does not contribute to deforestation.
6. In conclusion, while paper straws do offer a more environmentally friendly alternative to plastic straws, they are not without their own set of problems. The key to making a positive impact lies in continuing to innovate and improve these products, as well as encouraging responsible production and disposal practices. As consumers, being aware of these issues can help us make more informed choices and support companies that prioritise sustainability.
(32) Which of the following could be the best title for the passage?
Choose the correct answer to fill in the blank in each of the following questions.
Before making any changes, the school will __________ a survey to get feedback from students.
Read the following passage and choose the correct option that best fits each of the numbered blanks.
Carpooling, the practice of sharing a ride with others, is an effective way to reduce the number of vehicles on the road. One of the most significant advantages of carpooling is its positive impact on the environment. (22) __________ cars on the road, the fewer harmful greenhouse gases are emitted. This reduction in vehicle emissions can lead to improved air quality, particularly in urban areas (23) __________ pollution levels are typically higher.
Carpooling also offers economic benefits. Sharing a ride can significantly lower transportation costs for individuals by splitting (24) __________ such as fuel, tolls, and parking fees. This makes commuting more affordable and can ease the financial burden of car ownership.
Additionally, carpooling fosters a (25) __________ of community. It provides an opportunity for social interaction and networking (26) __________ passengers can engage in conversations and build relationships during their shared journeys. This can enhance the overall commuting experience, making it more enjoyable and less stressful.
Choose the correct answer to fill in the blank in each of the following questions.
She was knitting a colourful scarf while her brother __________ gifts for their friends.
Read the following passage and choose the correct option that best fits each of the numbered blanks.
Urbanisation, which involves the movement of people from rural to urban areas, (16) __________. This has been happening quickly in recent years because people are looking for better jobs, healthcare, and education. Urbanisation has many benefits. Cities often have more job opportunities, which can help people earn more money. They also usually have better transportation, sanitation, and public services, (17) __________ for residents.
However, there are also challenges. Rapid growth can lead to overcrowding, which makes it hard to find enough housing and puts pressure on public services. (18) __________ in some areas. Additionally, more people in cities can lead to more pollution and use up natural resources more quickly. Urban areas might also struggle with traffic congestion, higher crime rates, and (19) __________.
In summary, while urbanisation can bring economic growth and better living conditions, (20) __________ to avoid problems like overcrowding and pollution. Sustainable development, which considers (21) __________, is key to making sure cities grow in a way that benefits everyone.
Read the following passage and choose the correct option that best fits each of the numbered blanks.
Urbanisation, which involves the movement of people from rural to urban areas, (16) __________. This has been happening quickly in recent years because people are looking for better jobs, healthcare, and education. Urbanisation has many benefits. Cities often have more job opportunities, which can help people earn more money. They also usually have better transportation, sanitation, and public services, (17) __________ for residents.
However, there are also challenges. Rapid growth can lead to overcrowding, which makes it hard to find enough housing and puts pressure on public services. (18) __________ in some areas. Additionally, more people in cities can lead to more pollution and use up natural resources more quickly. Urban areas might also struggle with traffic congestion, higher crime rates, and (19) __________.
In summary, while urbanisation can bring economic growth and better living conditions, (20) __________ to avoid problems like overcrowding and pollution. Sustainable development, which considers (21) __________, is key to making sure cities grow in a way that benefits everyone.
Read the following passage and choose the correct option that best fits each of the numbered blanks.
Carpooling, the practice of sharing a ride with others, is an effective way to reduce the number of vehicles on the road. One of the most significant advantages of carpooling is its positive impact on the environment. (22) __________ cars on the road, the fewer harmful greenhouse gases are emitted. This reduction in vehicle emissions can lead to improved air quality, particularly in urban areas (23) __________ pollution levels are typically higher.
Carpooling also offers economic benefits. Sharing a ride can significantly lower transportation costs for individuals by splitting (24) __________ such as fuel, tolls, and parking fees. This makes commuting more affordable and can ease the financial burden of car ownership.
Additionally, carpooling fosters a (25) __________ of community. It provides an opportunity for social interaction and networking (26) __________ passengers can engage in conversations and build relationships during their shared journeys. This can enhance the overall commuting experience, making it more enjoyable and less stressful.
Read the following passage and choose the correct answer to each of the questions.
1. Have you ever checked to see what the weather would be like for the week? You can thank meteorologists for the weather forecast! A meteorologist is a scientist who studies the atmosphere to predict the weather. Their work involves observing weather patterns, analysing data from satellites and weather stations, and using computer models to forecast future weather conditions. Meteorologists play a crucial role in helping people prepare for various everyday conditions as well as providing timely warnings about extreme weather events like hurricanes and tornadoes.
2. To become a meteorologist, one typically needs a degree in meteorology or a related field like atmospheric science. Many meteorologists also study physics, mathematics, and computer science, as these subjects are essential for understanding and predicting weather patterns. Some meteorologists work in research, studying climate change and developing new forecasting models. Others work in broadcasting, delivering weather reports on television or radio.
3. Meteorologists often work in offices, but they may also spend time in the field, gathering data and monitoring weather conditions firsthand. The job can be challenging, as weather is unpredictable, and forecasts are not always accurate. However, it is also rewarding, as accurate weather predictions can save lives and property by providing timely warnings of severe weather.
4. The work of a meteorologist is never dull, as they must constantly keep up with new developments in weather technology and science. Whether they are tracking a major storm or predicting a sunny day, meteorologists help people plan their activities and stay safe.
(31) The word tracking in paragraph 4 is closest in meaning to __________.
Read the following passage and choose the correct answer to each of the questions.
1. In recent years, there has been a significant shift towards more eco-friendly alternatives to plastic products. Among these changes is the introduction of paper straws, which have become increasingly popular in cafes, restaurants, and homes. This shift is largely driven by growing awareness of the environmental damage caused by plastic pollution, particularly in our oceans.
2. Plastic straws have long been criticised for their contribution to pollution. They are lightweight, making them easily blown into waterways and oceans, where they can harm marine life. Unlike plastic, which takes hundreds of years to decompose, paper straws break down much more quickly. This means they are less likely to accumulate in the environment, making them a seemingly better option.
3. However, the production of paper straws also has an environmental impact. The manufacturing process requires cutting down trees, which can contribute to deforestation if not managed sustainably. Moreover, many paper straws have a thin plastic lining to prevent them from getting soggy quickly. This plastic lining can still cause environmental issues and reduce the benefits of switching from plastic to paper.
4. There are also practical concerns. Many people find paper straws less durable than plastic ones. They can become soft and unusable in a short amount of time, especially when used in hot beverages. This inconvenience can lead to increased waste if people use more straws as replacements.
5. On the other hand, some companies are working to address these issues. They are developing more durable and completely biodegradable paper straws, as well as investing in sustainable forestry practices to ensure that the production of paper products does not contribute to deforestation.
6. In conclusion, while paper straws do offer a more environmentally friendly alternative to plastic straws, they are not without their own set of problems. The key to making a positive impact lies in continuing to innovate and improve these products, as well as encouraging responsible production and disposal practices. As consumers, being aware of these issues can help us make more informed choices and support companies that prioritise sustainability.
(33) What is the main reason for the growing popularity of paper straws?
Choose the word whose underlined part differs from that of the other three in pronunciation in each of the following questions.
Read the following announcement and choose the correct option that best fits each of the numbered blanks.
School Announcement: Environmental Clean-Up Day
Dear Students and Staff,
We are excited to announce our upcoming Environmental Clean-Up Day! (13) __________ us on Saturday, August 10th, from 9 AM to 12 PM as we work together to clean up our local park. This event is a great opportunity to help protect our environment and earn community service hours.
✔ All necessary supplies (14) __________.
Please sign (15) __________ in the main office by August 5th. Let's make a difference together!
Thank you,
Read the following announcement and choose the correct option that best fits each of the numbered blanks.
School Announcement: Environmental Clean-Up Day
Dear Students and Staff,
We are excited to announce our upcoming Environmental Clean-Up Day! (13) __________ us on Saturday, August 10th, from 9 AM to 12 PM as we work together to clean up our local park. This event is a great opportunity to help protect our environment and earn community service hours.
✔ All necessary supplies (14) __________.
Please sign (15) __________ in the main office by August 5th. Let's make a difference together!
Thank you,
Read the following passage and choose the correct option that best fits each of the numbered blanks.
Carpooling, the practice of sharing a ride with others, is an effective way to reduce the number of vehicles on the road. One of the most significant advantages of carpooling is its positive impact on the environment. (22) __________ cars on the road, the fewer harmful greenhouse gases are emitted. This reduction in vehicle emissions can lead to improved air quality, particularly in urban areas (23) __________ pollution levels are typically higher.
Carpooling also offers economic benefits. Sharing a ride can significantly lower transportation costs for individuals by splitting (24) __________ such as fuel, tolls, and parking fees. This makes commuting more affordable and can ease the financial burden of car ownership.
Additionally, carpooling fosters a (25) __________ of community. It provides an opportunity for social interaction and networking (26) __________ passengers can engage in conversations and build relationships during their shared journeys. This can enhance the overall commuting experience, making it more enjoyable and less stressful.
Read the following passage and choose the correct option that best fits each of the numbered blanks.
Urbanisation, which involves the movement of people from rural to urban areas, (16) __________. This has been happening quickly in recent years because people are looking for better jobs, healthcare, and education. Urbanisation has many benefits. Cities often have more job opportunities, which can help people earn more money. They also usually have better transportation, sanitation, and public services, (17) __________ for residents.
However, there are also challenges. Rapid growth can lead to overcrowding, which makes it hard to find enough housing and puts pressure on public services. (18) __________ in some areas. Additionally, more people in cities can lead to more pollution and use up natural resources more quickly. Urban areas might also struggle with traffic congestion, higher crime rates, and (19) __________.
In summary, while urbanisation can bring economic growth and better living conditions, (20) __________ to avoid problems like overcrowding and pollution. Sustainable development, which considers (21) __________, is key to making sure cities grow in a way that benefits everyone.
Choose the correct answer to fill in the blank in each of the following questions.
The company __________ a significant increase in employee satisfaction since the new policy was introduced.
Read the following advertisement and choose the correct option that best fits each of the numbered blanks.
Exciting Job Opportunity at Our Community Centre!
The Sunshine Community Center is looking for enthusiastic students to join our dynamic team.
The job includes:
- Organising community events and activities
- Assisting in (10) __________ tasks
- Helping with social media updates
✔ No (11) __________ prior experience needed!
Contact Mr. Brown at brown.james@communitycenter.org by May 15 to apply.
Don't miss (12) __________!
Choose the word that differs from the other three in the position of stress in each of the following questions.
