The post What You Need To Know About At-Home AP Exams appeared first on Magoosh Blog | High School.
]]>If you’re enrolled in an AP class, you’re probably wondering how COVID-19 will affect your AP exams. However, the College Board, who makes the tests, has stepped up: you no longer need to decide between prioritizing your health and taking the exam. All spring 2020 tests will now be at-home AP exams administered online.
How do these tests work? What do at-home AP exams mean for you? In this post, we’ll break it down.
There are a few key differences that you’ll need to keep in mind when taking your AP test online. First of all, forget multiple choice; AP students will instead take a 45-minute free-response exam. But if that stresses you out, there’s good news; this year’s AP exams will be open book and open note.
If you’re worried about fairness, you’re not alone—and the AP College Board people are on it! They realize that some AP students have lost more classroom time than others because of differences between school curriculums. To address this, they’ve decided only to include content from your classes that most teachers will have covered before early March.
The College Board is also offering flexibility regarding devices for the at-home AP exam: you can take it on any device you have access to, including computers, tablets, and smartphones. In terms of responses, you can either type and upload your answers, or you can write them by hand and send a photo from your phone—whatever works best for you.
However, CB also realizes that some students may not have access to a device or reliable internet. If that’s the case, they’ll work with you to find a solution—but be sure to contact CB ASAP. If you’re not able to access the form, you can also call their AP customer service line at 888-225-5427 or email them at apstudents@info.collegeboard.org.
The College Board will ensure tests are secure with digital security tools, including software to detect plagiarism.
So far, the College Board has reported widespread support from colleges, who want to make sure you get the AP credit you’ve earned by accepting your AP test scores. Colleges have accepted College Board AP scores from shortened exams for literally decades, so they’re used to working with students who have found themselves in emergency situations.
You can find the AP exam schedule 2020 on the College Board website. They’ve also posted makeup exam dates, so they have you covered just in case.
Every AP test at home will happen at the same time—no matter where you are in the world. This means that if you’re in New York, you’ll sit down for the exam at 2pm your time, while at the exact same moment, a student in Honolulu will be taking the test at 8am their time.
It’s a good idea to prepare for these adjusted times by taking practice tests at the same time of day you’ll be taking the actual at-home AP exam. That way, you’ll be ready for any potential issues (tiredness, hunger) and know how to address them well in advance.
The College Board will put out more info on the at-home AP testing system online as they continue to develop it; expect details in late April.
In the meantime, as of March 25, you can get free AP review live from the test-maker! First, take a look at their daily schedule for 32 courses. Then, go over to the AP YouTube channel to watch the review session.
Finally, don’t forget that Magoosh has tons of AP resources to help you brush up on topics for test day!
Learning a new method of test-taking with at-home AP exams may add an additional layer to your test prep plan. However, there are real benefits to learning the new format–particularly if you’re a junior and may take a digital SAT later this year. By following College Board updates, prepping with official materials (including their review live streams!), and practicing early and often, you’ll maximize your chances of getting the score—and college credits—you want on test day.
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]]>The post APUSH Civil War Quiz – Are You Ready For Test Day? appeared first on Magoosh Blog | High School.
]]>Your AP US History exam is approaching! How are you feeling about it? Undoubtedly, there are some topics you feel stronger about than others. The Civil War is a central topic on the exam, so we’ve devised an APUSH Civil War quiz so you can get a sense of how well you know this material. After you’ve taken the APUSH Civil War quiz, scroll down for additional answer explanations, as well as other APUSH resources.
A – Appomattox Courthouse
Explanation: On April 9, 1865, Robert E. Lee, Confederate commander of all the Southern armies, surrendered to Ulysses S. Grant, commander of the Union armies. He surrendered at Appomattox Courthouse in Virginia, after the Battle of Appomattox Court House, which took place that morning and was one of the last battle of the Civil War.
A – William Sherman
Explanation: Starting November 15, 1864, William Sherman led roughly 60,000 soldiers through Georgia on the “Georgia Campaign.” Their route started in Atlanta and ended in Savannah on December 21. During this march, they destroyed military targets, civilian properties, and transportation means.
A – Jefferson Davis
Explanation: Jefferson Davis was President of the Confederate States of America during the the Civil War. The Confederacy consisted of 11 Southern states that seceded from the Union. The first 7 to secede were: South Carolina, Mississippi, Florida, Alabama, Georgia, Louisiana, and Texas, in defense of their right to own slaves. After the start of the Civil War, Virginia, Arkansas, Tennessee, and North Carolina seceded as well.
A – Kansas
Explanation: “Bleeding Kansas” is another name for The Border War, which was actually a series of very violent altercations that took place in Kansas and Missouri between 1854 and 1861. Ultimately, Kansas was admitted to the Union as a free state, but many historians consider this conflict a significant precursor to the Civil War, also largely fought over the issue of states’ rights concerning slavery. This is an excellent resource on APUSH topics related to Bleeding Kansas.
A – Roger Taney
Explanation: Dred Scott v. Sandford was a Supreme Court case that took place in 1857. Dred Scott, a slave, attempted to sue for his freedom, since he was moved to free territory by his master. The United States Supreme Court, led by Justice Roger B. Taney, ruled against Scott, declaring that black individuals were not considered citizens of the United States. This ruling is often considered among the most upsetting and oppressive in American history. Here is a more thorough overview of APUSH topics related to The Dred Scott decision.
A – 13th Amendment
Explanation: The 13th Amendment to the U.S Constitution, officially ending slavery, was passed by Congress on January 31, 1865 and ratified on December 6, 1865. It declares that “Neither slavery nor involuntary servitude, except as a punishment for crime whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction.” Previous to this, on January 1, 1863, President Lincoln issued the Emancipation Proclamation, in which he stated “that all persons held as slaves are, and henceforth, shall be free.”
A – Gettysburg
Explanation: During the 3-day Battle of Gettysburg, General Robert E. Lee attempted to invade the North by entering Gettysburg, Pennsylvania. From July 1-July 3, 1863, 52,000 men were wounded, killed, or missing in action. The battle was a turning point of the war, leading to Confederate retreat and halting their progress.
A- Virginia
Explanation: Virginia joined the Confederacy reluctantly in 1861, after Abraham Lincoln became president. In 1863, because the state was heavily divided on account of slavery, West Virginia became the 35th state to be admitted to the Union. Read more here on Virginia’s involvement in the Civil War.
For more information on how the Civil War is tested on the APUSH exam, check out the following:
In addition to questions specifically about the Civil War, you should also familiarize yourself with the following topics and themes, which will be addressed throughout the exam:
You’ve probably been discussing this exam all year in your AP US History course, but it can be helpful to review the actual structure of and assessments on the test on your own. This APUSH overview goes over exactly what will be on the test, including question formats and types. It also covers key concepts, events, and strategies.
This is also a great resource covering 9 thematic topics on the APUSH exam, to help you brush up on everything from the British Colonies to the Cold War. Along those lines, it’s also worth studying the specific historical periods that show up on the APUSH exam.
You know the saying: practice makes perfect! You probably take practice tests in your AP US History class as part of your curriculum, but you can always take them at home as well. Here are some tests for more practice:
If you liked this APUSH Civil Rights quiz, check out these other APUSH quizzes!
Let’s face it, working together is always more fun. Try getting together with a few peers from your AP US History course weekly to refresh yourselves on key concepts, review practice test questions, divide and conquer searching for answers, etc. Here are some of our favorite APUSH resources:
AP Exams can be stressful, and there’s a lot of information on the AP US History exam, but we believe in you! The earlier you can start studying, the better, and here are some tips for studying smarter.
Are you looking for more AP US History Exam Prep? We’ve got you covered!
Happy studying and best of luck on test day!
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]]>The post APUSH American Revolution Quiz – Are You Ready For Test Day? appeared first on Magoosh Blog | High School.
]]>And remember, no peeking at the answers below before taking it!
A- Patrick Henry
Explanation: On March 23, 1775, Patrick Henry gave a speech at the Second Virginia Convention. This speech, including the empowering line, “Give me liberty or give me death,” is often credited for passing the resolution to assemble Virginia’s troops for the Revolutionary War.
A- Common Sense
Explanation: Thomas Paine was an activist born in Great Britain who is one of the United States’ founding fathers. Common Sense, one of the pamphlets he authored, was incredibly influential in inspiring the patriots to declare independence from Great Britain in 1776. The text offered a new argument for independence rooted in Enlightenment theories of human rights.
A- The French and Indian War
Explanation: The French and Indian War started in American and moved Europe, becoming part of the Seven Year’s War. It was fought between the French and British over American expansion. The French received help from the Mohawk and Algonquin Indians while the British received help from the Iroquois. The British eventually captured Quebec and Montreal, ending the war.
A- The Townshend Acts
Explanation: The Townshend Acts were a series of taxes on goods imported to the American colonies. These acts were enforced by British troops in America, and were considered unfair by colonists, who had no representation in parliament.
A- Lexington and Concord
Explanation: The Battles of Lexington and Concord commenced on April 19, 1775, and marked the start of the Revolutionary War. Facing rebellion, the British attempted to seize arms being stored in Concord, Massachusetts. Paul Revere and other riders warned of British invasion and assembled troops accordingly. The defensive gunfire that ensued has been come to know as “the shot heard ‘round the world.”
A- Whigs
Explanation: The Whig Party was an American political party from 1834-54. They opposed what they saw as President Jackson’s tyranny, and consisted of former members of the National Republican and Anti-Masonic parties. Among many other platforms, they also advocated for ending the war.
A- Imprison Act.
Explanation: The Quartering Act was passed in 1765 and required colonies to provide food, housing, and other provisions to British forces occupying their territory. The Stamp Act, also passed in 1765, taxed all papers in colonies, including newspapers, pamphlets, legal papers, etc.
A- 1776
Explanation: The Declaration of Independence was signed in 1776 and detailed all of the reasons the American colonies sought independence from Great Britain.
In addition to information on the American Revolution specifically, you’ll also want to brush up on the following themes, which will be tested across the exam:
You’ve probably been discussing this exam all year in your AP US History course, but it can be helpful to review the actual structure of and assessments on the test on your own. This APUSH overview goes over exactly what will be on the test, including question formats and types. It also covers key concepts, events, and strategies.
This is also a great resource covering 9 thematic topics on the APUSH exam, to help you brush up on everything from the British Colonies to the Cold War. Along those lines, it’s also worth studying the specific historical periods that show up on the APUSH exam.
You know the saying: practice makes perfect! You probably take practice tests in your AP US History class as part of your curriculum, but you can always take them at home as well. Here are some tests for more practice:
If you liked the APUSH American Revolution Quiz, check out these others!
Let’s face it, working together is always more fun. Try getting together with a few peers from your AP US History course weekly to refresh yourselves on key concepts, review practice test questions, divide and conquer searching for answers, etc. Here are some of our favorite APUSH resources:
AP Exams can be stressful, and there’s a lot of information on the AP US History exam, but we believe in you! The earlier you can start studying, the better, and here are some tips for studying smarter.
Are you looking for more AP US History Exam Prep? We’ve got you covered!
Happy studying and best of luck on test day!
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]]>The post APUSH Quiz – Are You Ready For Test Day? appeared first on Magoosh Blog | High School.
]]>Remember, no peeking at the answers ahead of time!
A- The 1860’s
Explanation: The Civil War was fought in the United States, between the North and the South, from April 12, 1861-May 9, 1865. It commenced with the Battle of Fort Sumter, when the Confederate Army bombarded Fort Sumter (near Charleston, South Carolina), leading to surrender by the United States Army. The Battle of Palmito Ranch is widely considered the last official battle of the Civil War.
A- Stephen Douglas
Explanation: The Lincoln-Douglas debates, sometimes also known as the “Great Debates of 1858,” were 7 debates between Abraham Lincoln and Stephen Douglas that took place between August and October of 1858. Lincoln was the Republican candidate for US Senator and Douglas was the Democratic incumbent. They debated a range of topics, primarily related to slavery in the United States. While he ultimately lost to Douglas, Abraham Lincoln became a national figure through these debates. More on the Lincoln-Douglas debates here!
A- USS Maine
Explanation: The USS Maine blew up and sank on the night of February 15, 1898. It was sent to protect U.S. interests during Cuba’s revolt against Spain in the Cuban War of Independence. Here’s a fantastic overview of how this incident fits into American imperialism.
A- It acted as a precursor to the US Constitution and established how the national government would function.
Explanation: The Articles of Confederation was a legal document in place from 1781-1789, and was in many ways the first US Constitution. It served as a bridge of sorts between the government put in place by the Continental Congress, and the US Constitution of 1787. This document was an agreement between the 13 original states for how the government would operate. For more information, and a detailed account of each article, you can read the full Articles of Confederation here.
A- John F. Kennedy
Explanation: President Kennedy held office from January 20, 1961-November 22, 1963, when he was assassinated by Lee Harvey Oswald. JFK’s presidency was short, but eventful. The Bay of Pigs took place on April 17, 1961, when a group of 1,400 Cuban exiles attempted to invade the south coast of Cuba, a plan JFK was privy to before his presidency. The Cuban Missile Crisis took place during October of 1962, after the United States discovered Soviet Union nuclear missile sites in Cuba and responded with a naval blockade of Cuba.
A- Great Britain, France, and Russia
Explanation: The Allied Powers of WWI officially joined forces on September 5, 1914 through the Treaty of London. They opposed the Central Powers, consisting of Germany, Austria-Hungary, and Turkey.
A- Many of the European customs that had informed the American lifestyles were found wanting for those pioneers who moved West.
Explanation: Frederick Turner’s Frontier Thesis, published in 1893, stressed the importance of American frontier life on American democracy, and critiqued the limitations of old European customs and mindsets.
A- William Penn
Explanation: After being persecuted for their faith, the Quakers, led by William Penn, founded the colony of Pennsylvania in 1862. Read more about APUSH topics regarding the Quakers here!
Here’s a great overview of all of the themes that will be addressed throughout the APUSH exam:
You’ve probably been discussing this exam all year in your AP US History course, but it can be helpful to review the actual structure of and assessments on the test on your own. This APUSH overview goes over exactly what will be on the test, including question formats and types. It also covers key concepts, events, and strategies.
This is also a great resource covering 9 thematic topics on the APUSH exam, to help you brush up on everything from the British Colonies to the Cold War. Along those lines, it’s also worth studying the specific historical periods that show up on the APUSH exam.
You know the saying: practice makes perfect! You probably take practice tests in your AP US History class as part of your curriculum, but you can always take them at home as well. Here are some tests for more practice:
Let’s face it, working together is always more fun. Try getting together with a few peers from your AP US History course weekly to refresh yourselves on key concepts, review practice test questions, divide and conquer searching for answers, etc. Here are some of our favorite APUSH resources:
AP Exams can be stressful, and there’s a lot of information on the AP US History exam, but we believe in you! The earlier you can start studying, the better, and here are some tips for studying smarter.
Are you looking for more AP US History Exam Prep? We’ve got you covered!
Happy studying and best of luck on test day!
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]]>The post AP Literature Quiz: Are You Ready for Test Day? appeared first on Magoosh Blog | High School.
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First things first: literary terms are super important on the AP Literature exam. You probably already know this—but did you also know that the way that you study them can have a big influence on your score? Starting with lists of words is a good place to start; turning them into flashcards is even better. However, the exam isn’t going to ask you to simply define literary terms. Instead, it’ll ask you to identify them in context. With that in mind, come up with several examples for each key term and turn those into flash cards, instead. When you see a similar construction on test day, you’ll be able to make connections much more easily.
Another way to practice using AP literary terms is to make everything you read into your own AP Literature quiz. How? Highlight every literary device you see used in everything you read for fun. For bonus practice, work with a trusted study partner and exchange materials at the end of the week to see where you agree and disagree (and then figure out who’s right!).
Mastering the material is super important for AP Literature. However, mastering the format of the exam is equally important. For example, how are you going to approach those multiple-choice questions? In this case, nothing beats practicing with official materials.
For most students, the strongest strategy with AP Literature multiple-choice questions is to read the question, come up with your own answer before you look at the printed answer choices, and then see which one matches your prediction best. Why? This will prevent you from getting distracted by wrong answer choices that seem “almost” right. (Tip: no answer choice on the AP exam is “almost” right—there’s only one correct answer, as you’ll see if you take our AP Literature quiz above!)
If timing is an issue for you, practicing with official materials will also help—but there’s another hack. In the last ten minutes of the multiple-choice section, skim through your unanswered questions and eliminate all choices you know are wrong. Then, take your best guess on the remaining choices. Going from five answer choices to four ups your odds of getting the right choice from 20% to 25%. Eliminate one more, and you’re up to a 33% chance of answering correctly. One more and it’s 50%! If you find yourself in a situation where you have to blindly guess on multiple items, choose the same answer choice and plug it in for all of the remaining questions—you’re more likely to get at least a few points than if you chose random letters.
Of course, randomly guessing shouldn’t be your main strategy for on the AP test! That’s why, once you’ve taken the AP Literature quiz, it’s important to keep going with a full-length AP English Literature and Composition Practice Test (official materials are a great place to start).
Many students take practice AP exams, but far fewer know how to use them to their advantage. To maximize how useful this test prep is, plan to spend at least as much time studying your finished exam and your answers as you did taking the test. Review the questions you got wrong. Label what they were testing: vocabulary? Literary devices? Rhetorical devices? Keep a running list of your problem areas and brush up on them between practice exams. Review your correct answers, as well. Did you get them correct by guessing? Should you still go back and review that material? Were you unsure about anything? Thorough review will help you master the test!
Speaking of reviewing your work…are you ready to take a look at the answers and explanations to the AP Literature quiz at the top of this post? Here they are!
Assonance.
Sorry about that (literature joke!). Congratulations on making it through the AP Literature quiz and study advice guide. Ready for next steps? Dive into your practice tests, start making your flashcards, work on mastering your multiple-choice questions, and you’ll be golden by test day (start here: there’s at least two instances of figurative language in that last sentence—can you identify them?). Good luck!
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]]>The post Interval Notation appeared first on Magoosh Blog | High School.
]]>There are a few different types of intervals that commonly arise when studying math, called the open interval and the closed interval, notated respectively as (a, b) and [a, b].
The open interval uses parentheses, and they signify the fact that the interval contains all the real numbers x that are strictly between the numbers a and b, i.e. the interval does NOT actually contain the numbers a and b. Another way of notating an open interval is the set of all x such that a < x < b.
In the case of the closed interval, the square brackets are used to indicate that the endpoints are contained in the interval. Therefore we can notate a closed interval as the set of x so that
There are slightly fancier intervals, called half-open intervals, notated as (a, b] and [a, b), which are the respective sets of all x so that , and .
An interval is called bounded when there is a real positive number M with the property that for any point x inside of the interval, we have that |x| < M.
Supposing as in the setup that a < b, then how many numbers are actually in the interval (a, b)? It turns out that there are uncountably infinite numbers in any interval (a, b) where a < b, no matter how close a and b are together.
It is a fact that actually, there are the same quantity of real numbers in the interval (0, 1) as there are in the entire real numbers, also represented by the interval . This seems counterintuitive, because one interval seems so much more vast than the other, but it is not a contradiction, but rather a beautiful subtly of set theory.
Intervals arise regularly in calculus, and it will be important for you to know the difference between a closed interval and an open interval, since there are some theorems, like the intermediate value theorem, which requires that the interval upon which the function is defined is a closed and bounded interval.
Closed and bounded intervals touch on one of the most important concepts in the broader study of calculus, that of compactness. Many central theories in calculus revolve around compact sets, which in the setting of the real numbers are exactly the closed bounded intervals.
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]]>The post Common Integrals on the AP Calc Exam appeared first on Magoosh Blog | High School.
]]>For a quick review of integration (or, antidifferentiation), you might want to check out the following articles first.
And now, without further ado, here are some of the most common integrals found on the AP Calculus exams!
The following seven integrals (or their close cousins) seem to pop up all the time on the AP Calculus AB and BC exams.
Trigonometric functions are popular on the exam!
You need to recognize when to use the substitution u = kx, for constant k. This substitution generates a factor of 1/k because du = k dx.
For example,
Integrands of the form x f(x) often lend themselves to integration by parts (IBP).
In the following integral, let u = x and dv = sin x dx, and use IBP.
Integrands of the form a/(bx + c) pop up as a result of partial fractions decomposition. (See AP Calculus BC Review: Partial Fractions). While partial fractions is a BC test topic, it’s not rare to see an integral with linear denominator showing up in the AB test as well.
The key is that substituting u = bx + c (and du = b dx) turns the integrand into a constant times 1/u. Let’s see how this works in general. Keep in mind that a, b, and c must be constants in order to use this rule.
The antiderivative of f(x) = ln x is interesting. You have to use a tricky integration by parts.
Let u = ln x, and dv = dx.
By the way, this trick works for other inverse functions too, such as the inverse trig functions, arcsin x, arccos x, and arctan x. For example,
For some trigonometric integrals, you have to rewrite the integrand in an equivalent way. In other words, use a trig identity before integrating. One of the most popular (and useful) techniques is the half-angle identity.
It’s no secret that the AP Calculus exams consist of challenging problems. Perhaps the most challenging integrals are those that require a trigonometric substitution.
The table below summarizes the trigonometric substitutions.
For example, find the integral:
Here, the best substitution would be x = (3/2) sin θ.
Now we’re not out of the woods yet. Use the half-angle identity (see point 6 above). We also get to use the double-angle identity for sine in the second line.
Note, the third line may seem like it comes out of nowhere. But it’s based on the substitution and a right triangle.
If x = (3/2) sin θ, then sin θ = (2x) / 3. Draw a right triangle with angle θ, opposite side 2x, and hypotenuse 3.
By the Pythagorean Theorem, we find the adjacent side is equal to:
That allows us to identify cos θ in the expression (adjacent over hypotenuse).
Finally, θ by itself is equal to arcsin(2x/3).
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]]>The post Interpreting Slope Fields: AP Calculus Exam Review appeared first on Magoosh Blog | High School.
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A slope field shows the direction of flow for solutions to a differential equation.
A slope field is a visual representation of a differential equation of the form dy/dx = f(x, y). At each sample point (x, y), there is a small line segment whose slope equals the value of f(x, y).
That is, each segment on the graph is a representation of the value of dy/dx. (Check out AP Calculus Review: Differential Equations for more about differential equations on the AP Calculus exams.)
Because each segment has slope equal to the derivative value, you can think of the segments as small pieces of tangent lines. Any curve that follows the flow suggested by the directions of the segments is a solution to the differential equation.
Each curve represents a particular solution to a differential equation.
Consider the differential equation dy/dx = x – y. Let’s sketch a slope field for this equation. Although it takes some time to do it, the best way to understand what a slope field does is to construct one from scratch.
First of all, we need to decide on our sample points. For our purposes, I’m going to choose points within the window [-2, 2] × [-2, 2], and we’ll sample points in increments of 1. Just keep in mind that the window could be anything, and increments are generally smaller than 1 in practice.
Now plug in each sample point (x, y) into the (multivariable) function x – y. We will keep track of the work in a table.
x - y | x = -2 | x = -1 | x = 0 | x = 1 | x = 2 |
---|---|---|---|---|---|
y = 2 | -2 - 2 = -4 | -1 - 2 = -3 | 0 - 2 = -2 | 1 - 2 = -1 | 2 - 2 = 0 |
y = 1 | -2 - 1 = -3 | -1 - 1 = -2 | 0 - 1 = -1 | 1 - 1 = 0 | 2 - 1 = 1 |
y = 0 | -2 - 0 = -2 | -1 - 0 = -1 | 0 - 0 = 0 | 1 - 0 = 1 | 2 - 0 = 2 |
y = -1 | -2 - (-1) = -1 | -1 - (-1) = 0 | 0 - (-1) = 1 | 1 - (-1) = 2 | 2 - (-1) = 3 |
y = -2 | -2 - (-2) = 0 | -1 - (-2) = 1 | 0 - (-2) = 2 | 1 - (-2) = 3 | 2 - (-2) = 4 |
Ok, now let’s draw the slope field. Remember, the values in the table above represent slopes — positive slopes mean go up; negative ones mean go down; and zero slopes are horizontal.
Spend some time matching each slope value from the table with its respective segment on the graph.
It’s important to realize that this is just a sketch. A more accurate picture would result from sampling many more points. For example, here is a slope field for dy/dx = x – y generated by a computer algebra system. The viewing window is the same, but now there are 400 sample points (rather than the paltry 25 samples in the first graph).
Slope field for dy/dx = x – y
Now let’s get down to the heart of the matter. What do I need to know about slope fields on the AP Calculus AB or BC exams?
You’ll need to master these basic skills:
We’ve already seen above how to sketch a slope field, so let’s get some practice with the first two skills in the list instead.
The slope field shown above corresponds to which of the following differential equations.
A. dy/dx = y^{2}
B. dy/dx = sin y
C. dy/dx = -sin y
D. dy/dx = sin x
Look for the clues. The segments have the same slopes in any given row (left to right across the graph). Therefore, since the slopes do not change with respect to x, we can assume that dy/dx is a function of y alone. That eliminates choice D.
The horizontal segments occur when y = 0, π, and -π. However, the only point at which y^{2} equals 0 is y = 0 (not π or -π). That narrows it down to a choice between B and C.
Finally, notice that slopes are positive when 0 < y < π and negative when -π < y < 0. This pattern corresponds to the values of sin y. (The signs are opposite for -sin y, ruling out choice C).
The correct choice is B.
Suppose y = f(x) is a particular solution to the differential equation dy/dx = x – y such that f(0) = 0. Use the slope field shown earlier to estimate the value of f(2).
A. -2.5
B. -1.3
C. 0.2
D. 1.1
Because f(0) = 0, the solution curve must begin at (0, 0). Then sketch the curve carefully following the directions of the segments. It helps to imagine that the segments are showing the currents in a river. Your solution should be like a raft carried along by the currents.
Then find the approximate value of f(2) on your solution curve.
The best choice from among the answers is: D. 1.1.
Often if you see a slope field problem in the Free Response section of the exam, one part of the problem might be to use Euler’s Method to estimate a value of a solution curve.
While the slope field itself can be used to estimate solutions, Euler’s Method is much more precise and does not rely on the visual representation. Check out this review article for practice using the method: AP Calculus BC Review: Euler’s Method.
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]]>The post What is Logarithmic Differentiation? AP Calc Review appeared first on Magoosh Blog | High School.
]]>First of all, let’s review what a logarithm is. More specifically, we need to understand how the logarithm function can be used to break down complicated expressions.
You might want to check out the following article before getting started: AP Calculus Review: Properties of Exponents and Logartithms. However, the most important properties for us will be the product, quotient, and power properties for logarithms. Here, we focus on a particular logarithm: the natural logarithm, ln x, though the properties remain true in any base.
In other words, logarithms change…
I like to think of the logarithm as a powerful acid that can dissolve a complicated algebraic expression.
Let me illustrate the point with an example.
Notice how the original expression involves a huge fraction with roots and powers all over the place. After applying the properties of logarithms, the resulting expression mostly has only plus and minus. (Of course, there is a trade-off — there are now three natural logs in the simplified expression.)
Now let’s get down to business! How can we exploit these logarithmic simplification rules to help find derivatives?
The most straightforward case is when the function already has a logarithm involved.
Find the derivative of .
First simplify using the properties of logarithms (see work above). Then you can take the derivative of each term. But be careful — the final term requires a Product Rule!
In the above example, there was already a logarithm in the function. But what if we want to use logarithmic differentiation when our function has no logarithm?
Suppose f(x) is a function with a lot of products, quotients, and/or powers. Then you might use the method of logarithmic differentiation to find f ‘(x).
Use logarithmic differentiation to find the derivative of:
Let’s follow the steps outlined above. The first two steps are routine.
On the other hand, Step 3 requires us to break down the logarithmic expression using the properties. The work in this step depends on the function. In our case, there is a product of two factors, so we’ll start with the product property. The power property helps to break down the radical. Finally, don’t forget the cancellation rule: ln(e^{x}) = x.
Next, in Steps 4 and 5 apply the derivative and work out the right-hand side.
Finally, in Step 6, we solve for the unknown derivative by multiplying both sides by y. Don’t forget to substitute back the original function f(x) in place of y.
A very famous question in Calculus class is: What is the derivative of x^{x} ?
So what is the derivative of x^{x} ?
Well, it turns out that only logarithmic differentiation can decide this one for us!
In fact any time there is a function raised to a function power (that is, neither the exponent nor the base is constant), then you will have to use logarithms to break it down before you can take a derivative.
Let’s see how it works in the simplest case: x^{x}.
First, write y = x^{x}.
Then, apply the logarithm to both sides: ln y = ln x^{x}.
Break down the right-hand side of the equation using the algebraic properties of logarithms. In this case, only the power property plays a role.
ln y = x ln x
Now you can take derivatives of the functions on both sides. But be careful… the function on the right requires a Product Rule.
(1/y)(dy/dx) = (1) ln x + x(1/x) = ln x + 1
Finally, multiply both sides by the original function (y = x^{x}) to isolate dy/dx.
dy/dx = x^{x}(ln x + 1)
And there you have it! The derivative of x^{x} turns out to be trickier than you might have thought at first, but it’s not impossible.
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]]>In this review article, we’ll explore the methods and applications of linear approximation. We’ll also take a look at plenty of examples along the way to better prepare you for the AP Calculus exams.
By definition the linear approximation for a function f(x) at a point x = a is simply the equation of the tangent line to the curve at that point. And that means that derivatives are key! (Check out How to Find the Slope of a Line Tangent to a Curve or Is the Derivative of a Function the Tangent Line? for some background material.)
Given a point x = a and a function f that is differentiable at a, the linear approximation L(x) for f at x = a is:
L(x) = f(a) + f '(a)(x – a)
The main idea behind linearization is that the function L(x) does a pretty good job approximating values of f(x), at least when x is near a.
In other words, L(x) ≈ f(x) whenever x ≈ a.
Find the linear approximation of the parabola f(x) = x^{2} at the point x = 1.
A. x^{2} + 1
B. 2x + 1
C. 2x – 1
D. 2x – 2
C.
Note that f '(x) = 2x in this case. Using the formula above with a = 1, we have:
L(x) = f(1) + f '(1)(x – 1)
L(x) = 1^{2} + 2(1)(x – 1) = 2x – 1
Clearly, the graph of the parabola f(x) = x^{2} is not a straight line. However, near any particular point, say x = 1, the tangent line does a pretty good job following the direction of the curve.
How good is this approximation? Well, at x = 1, it’s exact! L(1) = 2(1) – 1 = 1, which is the same as f(1) = 1^{2} = 1.
But the further away you get from 1, the worse the approximation becomes.
x | f(x) = x^{2} | L(x) = 2x - 1 |
---|---|---|
1.1 | 1.21 | 1.2 |
1.2 | 1.44 | 1.4 |
1.5 | 2.25 | 2 |
2 | 4 | 3 |
The formula for linear approximation can also be expressed in terms of differentials. Basically, a differential is a quantity that approximates a (small) change in one variable due to a (small) change in another. The differential of x is dx, and the differential of y is dy.
Based upon the formula dy/dx = f '(x), we may identify:
dy = f '(x) dx
The related formula allows one to approximate near a particular fixed point:
f(x + dx) ≈ y + dy
Suppose g(5) = 30 and g '(5) = -3. Estimate the value of g(7).
A. 24
B. 27
C. 28
D. 33
A.
In this example, we do not know the expression for the function g. Fortunately, we don’t need to know!
First, observe that the change in x is dx = 7 – 5 = 2.
Next, estimate the change in y using the differential formula.
dy = g '(x) dx = g '(5) · 2 = (-3)(2) = -6.
Finally, put it all together:
g(5 + 2) ≈ y + dy = g(5) + (-6) = 30 + (-6) = 24
Approximate using differentials. Express your answer as a decimal rounded to the nearest hundred-thousandth.
1.03333.
Here, we should realize that even though the cube root of 1.1 is not easy to compute without a calculator, the cube root of 1 is trivial. So let’s use a = 1 as our basis for estimation.
Consider the function . Find its derivative (we’ll need it for the approximation formula).
Then, using the differential, , we can estimate the required quantity.
Sometimes we are interested in the exact change of a function’s values over some interval. Suppose x changes from x_{1} to x_{2}. Then the exact change in f(x) on that interval is:
Δy = f(x_{2}) – f(x_{1})
We also use the “Delta” notation for change in x. In fact, Δx and dx typically mean the same thing:
Δx = dx = x_{2} – x_{1}
However, while Δy measures the exact change in the function’s value, dy only estimates the change based on a derivative value.
Let f(x) = cos(3x), and let L(x) be the linear approximation to f at x = π/6. Which expression represents the absolute error in using L to approximate f at x = π/12?
A. π/6 – √2/2
B. π/4 – √2/2
C. √2/2 – π/6
D. √2/2 – π/4
B.
Absolute error is the absolute difference between the approximate and exact values, that is, E = | f(a) – L(a) |.
Equivalently, E = | Δy – dy |.
Let’s compute dy ( = f '(x) dx ). Here, the change in x is negative. dx = π/12 – π/6 = -π/12. Note that by the Chain Rule, we obtain: f '(x) = -3 sin(3x). Putting it all together,
dy = -3 sin(3 π/6 ) (-π/12) = -3 sin(π/2) (-π/12) = 3π/12 = π/4
Ok, next we compute the exact change.
Δy = f(π/12) – f(π/6) = cos(π/4) – cos(π/2) = √2/2
Lastly, we take the absolute difference to compute the error,
E = | √2/2 – π/4 | = π/4 – √2/2.
Linear approximations also serve to find zeros of functions. In fact Newton’s Method (see AP Calculus Review: Newton’s Method for details) is nothing more than repeated linear approximations to target on to the nearest root of the function.
The method is simple. Given a function f, suppose that a zero for f is located near x = a. Just linearize f at x = a, producing a linear function L(x). Then the solution to L(x) = 0 should be fairly close to the true zero of the original function f.
Estimate the zero of the function using a tangent line approximation at x = -1.
A. -1.48
B. -1.53
C. -1.62
D. -1.71
D.
Remember, The purpose of this question is to estimate the zero. First of all, the tangent line approximation is nothing more than a linearization. We’ll need to know the derivative:
Then find the expression for L(x). Note that g(-1) ≈ 2.13 and g '(-1) = 3.
L(x) = 2.13 + 3(x – (-1)) = 5.13 + 3x.
Finally to find the zero, set L(x) = 0 and solve for x
0 = 5.13 + 3x → x = -5.13/3 = -1.71
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