Differentiation
DerivativesDerivative NotationThe definition of a derivative is the slope of the tangent line.
Derivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Product Rule]:
[tex]\displaystyle (uv)' = u'v + uv'[/tex]
Implicit Differentiation
ApplicationStep 1: DefineLet's define the problem.
We are trying to find the slope of the tangent line.
Curve: [tex]\displaystyle y^2 - 2x^2 = 6 - 2xy[/tex]
Point: (2, 3)
Step 2: DifferentiateLet's find the derivative of the curve using implicit differentiation and basic differentiation techniques listed under "Calculus":
[tex]\displaystyle\begin{aligned}2y \frac{dy}{dx} - 4x & = -2 \bigg( y + x \frac{dy}{dx} \bigg) \\2y \frac{dy}{dx} & = -2y - 2x \frac{dy}{dx} + 4x \\2y \frac{dy}{dx} + 2x \frac{dy}{dx} & = 4x - 2y \\\frac{dy}{dx} \bigg( 2y + 2x \bigg) & = 4x - 2y \\\frac{dy}{dx} & = \frac{4x - 2y}{2y + 2x} \\& = \boxed{ \frac{2x - y}{x + y} } \\\end{aligned}[/tex]
∴ the derivative of the curve is equal to:
[tex]\displaystyle \boxed{ \frac{dy}{dx} = \frac{2x - y}{x + y} }[/tex]
We know by definition that the derivative is the slope of the tangent line. That means we can plug in the point (2, 3) directly into our derivative:
[tex]\displaystyle\begin{aligned}\frac{dy}{dx} \bigg|_{(2 ,\ 3)} & = \frac{2(2) - 3}{2 + 3} \\& = \frac{4 - 3}{5} \\& = \boxed { \frac{1}{5} } \\\end{aligned}[/tex]
∴ the slope of the tangent line is equal to:
[tex]\displaystyle \boxed{ \frac{dy}{dx} \bigg|_{(2 ,\ 3)} = \frac{1}{5} }[/tex]
The slope of the tangent line to the curve [tex]\displaystyle y^2 - 2x^2 = 6 - 2xy[/tex] at the point (2, 3) is equal to 0.20.
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Topic: Calculus I
Unit: Differentiation
AP STATISTICS boxplot: range, mean, IQR, and 3rd quartile, see image below. Really easy for stats people !!
Kristiana collected data on the amount of time in minutes each person in a large sample of customers spent in a local sporting store. The data also included recording the gender for each customer. These data were used to generate the boxplots shown to the right.
a. the range in the amount of time in minutes males in the sample of customers spent in the store is approximately 40 minutes.
b. the mean amount of time in minutes in the sample of customers spent in the store is approximately 20 minutes.
c. the third quartile of the amount of time in minutes males in the sample of customers spent in the store is approximately 45 minutes.
d. the interquartile range of the amount of time in minutes females in the sample customers spent in the store is 15 minutes.
e. approximately half of the males in the sample of customers spent at least as much time in the store as any females in the sample of customers.
1) Briefly explain ONE development that contributed to the patterns of population in Mississippi between 1800 and 1860.
2) Briefly explain ONE effect of the patterns of population in Mississippi reflected in the table.
3) Briefly explain ANOTHER effect of the patterns of population in Mississippi reflected in the table
Before and after becoming a state, Mississippi's population expansion is clearly associated with the increase in cotton output. The number of White people increased from 5,179 in 1800 to 353,901 in 1860, while the number of slaves increased from 3,489 to 436,631.
Mississippi was the main cotton producer in the United States throughout the first half of the 19th century, and owners of sizable estates relied on the work of African slaves. Between 1800 and 1860, the nation saw two significant changes. Rice and tobacco were the previous cash crops; cotton is now the new cash crop. The slave trade expanded to fill the economic gap in the Chesapeake region as the crops changed. Furs from the Great Lakes and the Missouri River; basic foods like maize and wheat from the Midwest; cotton, sugar, and tobacco from the plantations of the Deep South; and just about every other trade commodity you can think of were all transported by the Mississippi River. And in each instance, commerce resulted in a certain culture.
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Given that the econd tudent choen i a girl, what i the probability that the firt tudent wa alo a girl?
Answer:
50% or 1/2
Explanation:
Assuming there is a 50/50 split of boys and girls in a class, the answer is 50%
We know this because the question doesn't specify if the selection is with or without replacement, so we can assume either (so I assume with replacement).
With replacement would mean that the first and second students chosen are independent events and that they do not impact each other. Thus, we are just finding the probability that we choose a girl in a 50/50 split class.
This means our answer is 50%
Which of the following illustrates a situation that
would not be protected by the First Amendment
due to time, place, and manner restrictions?
Constraints of time, place, and manner are imposed by the government and are neutral in terms of content. Regulating the placement or size of signs on public property.
What would not be protected by the First Amendment?Speaking in a way that will soon result in illegal behaviour is not protected by the First Amendment. This type of discourse must be addressed to a particular individual or group. It must be a clear command to take quick, illegal action. There must be a belief that the communication will actually result in illegal behaviour.
Therefore, Anti-business protesters who blocked up busy junctions in Times Square, New York City, during rush hour were detained and charged.
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Directions: Suppose a firm's marginal product of capital and marginal product of labor schedules are as shown in the table below. The firm hires both capital and labor competitively for $5 and $8, respectively. This assignment will be graded out of 6 points with 2 points possible for each question.
Capital MP of Capital Labor MP of Labor
0 0
1 10 1 28
2 9 2 30
3 8 3 24
4 7 4 20
5 6 5 16
6 5 6 12
7 4 7 8
8 3 9 4
1. Suppose the firm is currently using 4 units of capital and 4 units of labor. Is the corresponding output being produced at least cost? How do you know?
2. Consider your answer to the question above. Generally speaking, what would a firm need to do in order to move towards producing at the least cost?
3. What is the least-cost combination of labor and capital?
Answer:
In order to determine whether the firm is currently producing at least cost, we need to compare the marginal cost of using an additional unit of capital or labor to the marginal revenue generated by that unit. Since we do not have information on the marginal revenue in this case, it is impossible to say for sure whether the firm is producing at least cost.
In order to move towards producing at the least cost, a firm would need to adjust the combination of labor and capital it uses in order to minimize the marginal cost of production. This can be achieved by adjusting the mix of labor and capital used in production until the marginal cost of an additional unit of labor is equal to the marginal cost of an additional unit of capital.
In order to determine the least-cost combination of labor and capital, we need to find the point where the marginal cost of using an additional unit of labor is equal to the marginal cost of using an additional unit of capital. From the information given in the table, we can see that this point occurs when the firm is using 3 units of capital and 3 units of labor. At this point, the marginal cost of using an additional unit of labor is 24 and the marginal cost of using an additional unit of capital is 8, which are equal. Therefore, the least-cost combination of labor and capital is 3 units of each.
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How long does it take for a light particle to travel 25 light years?
The speed of light is approximately 186,282 miles per second, and a light year is the distance that light travels in one year, which is approximately 5.88 trillion miles. Therefore, it would take a light particle approximately 25 * 5.88 trillion miles / 186,282 miles per second = 8.46 * 10^14 seconds to travel 25 light years.
This time is so large that it is usually measured in years rather than seconds. To convert the time to years, we would need to divide the time in seconds by the number of seconds in a year, which is approximately 31.5 million seconds. This would give us a time of approximately 8.46 * 10^14 seconds / 31.5 million seconds = 26,847 years.
Therefore, it would take approximately 26,847 years for a light particle to travel 25 light years. Note that this is just an approximate value, as the exact time would depend on various factors such as the exact speed of light and the distance traveled.
principles of science
Answer:
If you drop an apple, it will fall. If you throw an apple in the air, it will follow a specific path while falling down. Newton's laws don't tell us why the apple falls or what causes it to fall, just that it does fall.
Explanation:
The principle of science, the definition, almost, is the following: the test of all knowledge is experiment. Experiment is the sole judge of scientific "truth". But what is the source of knowledge
In 2-3 sentences, describe TWO ways in which colonists “have been driven to madness by injustice.”
Answer:
Two ways in which colonists may have been driven to madness by injustice are:
The lack of representation in the British government, which led to the feeling that their interests and concerns were not being heard or addressed. This could have caused a sense of frustration and anger among colonists, potentially leading to feelings of madness.
The imposition of taxes and other economic policies by the British government without the consent of the colonists, which many colonists saw as unfair and oppressive. This could have caused feelings of resentment and injustice, potentially leading to feelings of madness.
Explanation:
Injustice can cause a range of negative emotions, including frustration, anger, resentment, and hopelessness. These emotions can be intense and overwhelming, and may lead a person to feel like they are losing their sanity or being driven mad. In the case of the colonists, the lack of representation in the British government and the imposition of taxes and other policies without their consent may have caused intense feelings of injustice that could have contributed to feelings of madness.
(A) Identify the assumed protection that is common to both Planned Parenthood v.
Casey (1992) and Roe v. Wade (1973)
(B) Based on the assumed protection identified in part A, explain why the facts of the
case in Planned Parenthood v. Casey and Roe v. Wade demonstrate change over time
in the law.
(C) Explain how the holdings in Planned Parenthood v. Casey and Roe v. Wade
demonstrate change over time in the law.
The assumed protection that is common to both Planned Parenthood v. Casey (1992) and Roe v. Wade (1973) is a woman having the right to an abortion.
(B and C) The reason that facts of the case in Planned Parenthood v. Casey and Roe v. Wade demonstrate change over time in the law and the other is that:
The national political and legal upheaval caused by the Supreme Court's Roe v. Wade decision in 1973 marked the beginning of the abortion discussions. By enacting new legislation that forbade abortions, many states openly disobeyed the Roe decision.
The measure mandated that all abortion facilities in Pennsylvania self-report to the state. The state was sued by Planned Parenthood of Southeastern Pennsylvania on the grounds that the Abortion Control Act contravened the Supreme Court's Roe v. Wade decision. The issue was taken up for review by the Supreme Court, which is today more ideologically conservative than it was when Roe was ruled.
Roe v. Wade was upheld, but the majority of the Pennsylvania law was still valid, according to the Supreme Court in 1992, after great anticipation. Reiterating some of the arguments made in Roe, the Supreme Court first stated that a woman's decision to have an abortion involves significant "liberty interests" and "privacy interests" that the Constitution's Due Process Clause safeguards against state intrusion.
Therefor, Together, these interests provide a "substantive right to private" that is shielded from state meddling in "marriage, procreation, contraception, family relationships, child rearing, and education."
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THIS IS AN EARTH SCIENCE QUESTION
predict how you would use a compass for navigation if you traveled back in time 1 million years
Answer:
with a stick we would use a compass
Decribe one way the pread of Buddhim led Chinee culture to change during the Period 1200-1450
Answer:
One way the spread of Buddhism led Chinese culture to change during the period c. 1200-c. 1450 can be seen through the architecture. Buddhist monasteries, buildings where monks gathered, were erected in major cities across China, signifying its increasing popularity
Explanation:
Why are people attracted to conspiratorial thinking?
There are many reasons why people might be attracted to conspiratorial thinking. One common reason is a desire for control or a sense of power. Believing in a conspiracy can give a person a sense that they are part of a select group of individuals who know the truth about a situation, which can make them feel special or superior to others.
Another reason why people might be attracted to conspiracies is a feeling of uncertainty or fear. In times of uncertainty or crisis, people may seek explanations for events that are outside the realm of their understanding or control. Believing in a conspiracy can provide a sense of certainty or order in a chaotic world.
Additionally, people may be drawn to conspiracies because they offer a simple, easy-to-understand explanation for complex events. In a world that can be overwhelming and confusing, believing in a conspiracy can provide a clear and straightforward narrative that makes sense of otherwise inexplicable occurrences.
Finally, people may be attracted to conspiracies because they provide a sense of belonging or community. Believing in a conspiracy can connect a person to others who share their beliefs, creating a sense of solidarity and shared purpose. This can be particularly appealing to individuals who feel isolated or marginalized in their daily lives.
Evaluating Expressions with Exponents
What is the value of the expression 3x7+3^2
Answer:
30
Explanation:
if you mean 3 times 7+3^2 then its 30
if you meant 3x7+3^2 then its 3x^7+9
Triangle ABC has a side of 8, a side of 6, and a non-included angle of 40 degrees.
Triangle DEF has a side of 16, a side of 12, and a non-included angle of 40 degrees.
What statement is TRUE?
Triangle ABC is congruent to triangle DEF.
Triangle ABC must be similar to triangle DEF.
Triangle ABC must be similar to either triangle DEF or to triangle DFE.
Triangle ABC may or may not be similar to triangle DEF
Identify the following information for each of the three main monotheistic faiths: Christianity, Judaism, and Islam (AP Human Geo)
Answer:
Christianity- Holy book- Bibe, sacred City-Bethlaham Palestine, Founder Jesus, America, Europe sub saharan Africa. No taboos original christanity though is like no drinking but now no. Branches- Protestant, Orthodox, Catholic.Symbol Cross.
Judaism-Holy book Torah
Sacred Site- Jerusalem Palestine
Founder-Abraham
Moet practiced- Palestine
Taboo- Can only cat Kosher foods
Key branches not enough to be
Symbol the jewish Sign
Islam:
Holy book- Quran
Sacred Site-Jerusalam Palestine Sauria Arabia
Founder-Prophet Mohammad
Branches-Mainly Sunni and Shia
Symbols cresent with Little star
Explanation:
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A blind tate tet will be conducted with 9 volunteer to determine whether people can tate a difference between bottled water and tap water. Each participant will tate the water from two different glae and then identify which gla he or he think contain the tap water. Auming that people cannot tate a difference between bottled water and tap water, what i the probability that at leat 8 of the 9 participant will correctly identify the tap water?
The correct answer is 0.01953125 i the probability that at leat 8 of the 9 participant will correctly identify the tap water.
Probability is simply the possibility that something will happen. We may talk about the possibility of one result, or the likelihood of numerous outcomes, when we don't know how an event will turn out. The study of events with a probability distribution is known as statistics. Probability: P(A) = f / N, determines the likelihood that an event will occur. Odds and probability are related, however the odds depend on the probability. Prior to determining the chance of an event, probability must be known. A probability is a numerical representation of the likelihood or possibility that a particular event will occur. Probabilities can be expressed as fractions from 0 to 1, as well as percentages from 0% to 100%.
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