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Flashcard 149625149

Tags
#calculus #elasticity #has-images #mathematics
Question

In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output)[1] at point a is defined as[2] using functions and their derivatives

Ef(a) = \frac{a}{f(a)}f'(a)

meaning in words: [...]

Answer
It is thus the ratio of the relative (percentage) change in the function's output \(f(a)\) with respect to the relative change in its input \(a\), for infinitesimal changes from a point (a, f(a)).


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the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output)[1] at point a is defined as[2] using functions and their derivatives meaning in words: <span>It is thus the ratio of the relative (percentage) change in the function's output \(f(a)\) with respect to the relative change in its input \(a\), for infinitesimal changes from a point . <span><body><html>

Original toplevel document

Elasticity of a function - Wikipedia, the free encyclopedia
er:filter:minify-css:7:3904d24a08aa08f6a68dc338f9be277e */ Elasticity of a function From Wikipedia, the free encyclopedia Jump to: navigation, search <span>In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output)[1] at point a is defined as[2] or equivalently It is thus the ratio of the relative (percentage) change in the function's output with respect to the relative change in its input , for infinitesimal changes from a point . Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. The elasticity of a function is a constant if and only if the function has the form for a constant . The elasticity at a point is the limit of the arc elasticity between two points as







#mathematics #polynomials #precalculus

Rewrite \(5^2-2^2\) as a product.


We have

\[5^2-2^2 = (5-2) \times (5+2) = 3\times 7.\]

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Difference Of Squares | Brilliant Math &amp; Science Wiki
ction contains examples and problems to boost understanding in the usage of the difference of squares identity: \(a^2-b^2=(a+b)(a-b)\). Here are the examples to learn the usage of the identity. <span>Rewrite \(5^2-2^2\) as a product. We have \[5^2-2^2 = (5-2) \times (5+2) = 3\times 7. \ _\square\] Calculate \(299\times 301\). You can brute force the answer to this problem by using a calculator, but we have a sweeter way. We can apply the difference of two squares identity. At fir




Flashcard 3199204461836

Tags
#Gastroenterology #small_bowel_pathologies
Question
Which bacreria causes Whipple’s disease?
Answer
Whipple’s disease is a rare infectious bacterial disease caused by Tropheryma whipplei


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#Gastroenterology #small_bowel_pathologies
  • In Whipple’s disease, 87% are males, usually white and middle- aged.
  • It presents with arthritis and arthralgia, progressing over years to weight loss and diarrhoea with abdominal pain, systemic symptoms of fever and weight loss.
  • Peripheral lymphadenopathy and involvement of the heart, lung, joints and brain occur, simulating many neurological conditions

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Flashcard 3199210753292

Tags
#Gastroenterology #small_bowel_pathologies
Question
  • In Whipple’s disease, 87% are [...].
  • It presents with arthritis and arthralgia, progressing over years to weight loss and diarrhoea with abdominal pain, systemic symptoms of fever and weight loss.
  • Peripheral lymphadenopathy and involvement of the heart, lung, joints and brain occur, simulating many neurological conditions
Answer
males, usually white and middle- aged


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In Whipple’s disease, 87% are males, usually white and middle- aged. It presents with arthritis and arthralgia, progressing over years to weight loss and diarrhoea with abdominal pain, systemic symptoms of fever and weight loss. Peripheral lymphadenopat

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Flashcard 3199212326156

Tags
#Gastroenterology #small_bowel_pathologies
Question
  • In Whipple’s disease, 87% are males, usually white and middle- aged.
  • It presents with [...], progressing over years to weight loss and diarrhoea with abdominal pain, systemic symptoms of fever and weight loss.
  • Peripheral lymphadenopathy and involvement of the heart, lung, joints and brain occur, simulating many neurological conditions
Answer
arthritis and arthralgia


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In Whipple’s disease, 87% are males, usually white and middle- aged. It presents with arthritis and arthralgia, progressing over years to weight loss and diarrhoea with abdominal pain, systemic symptoms of fever and weight loss. Peripheral lymphadenopathy and involvement of the heart, lung, join

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Flashcard 3199213899020

Tags
#Gastroenterology #small_bowel_pathologies
Question
  • In Whipple’s disease, 87% are males, usually white and middle- aged.
  • It presents with arthritis and arthralgia, progressing over years to weight loss and diarrhoea with abdominal pain, systemic symptoms of fever and weight loss.
  • Peripheral lymphadenopathy and involvement of the [...] occur, simulating many neurological conditions
Answer
heart, lung, joints and brain


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arthritis and arthralgia, progressing over years to weight loss and diarrhoea with abdominal pain, systemic symptoms of fever and weight loss. Peripheral lymphadenopathy and involvement of the <span>heart, lung, joints and brain occur, simulating many neurological conditions <span>

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Flashcard 3199217044748

Tags
#Gastroenterology #small_bowel_pathologies
Question
How diagnosis of Whipple’s disease is made?
Answer
  • Diagnosis is made by small bowel biopsy.
  • Periodic acid–Schiff (PAS)-positive macrophages are present but are nonspecific.
  • On electron microscopy, the characteristic trilaminar cell wall of T. whipplei can be seen within macrophages.
  • T. whipplei antibodies can be identified by immunohistochemistry.
  • A confirmatory PCR-based assay is available


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Flashcard 3199219404044

Tags
#Gastroenterology #small_bowel_pathologies
Question
What is the treatment of Whipple’s disease?
Answer
  • Treatment is with antibiotics which cross the blood-brain barrier,
  • such as 160 mg Trimethoprim and 800 mg sulpha- methoxazole (co-trimoxazole) daily for 1 year.
  • This is preceded by a 2-week course of streptomycin and penicillin or ceftriaxone.
  • Treatment periods of less than a year are associated with relapse in about 40%


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#Gastroenterology #IBD
in 10% of cases of IBD causing colitis a definitive diagnosis of either UC or CD is not possible and the diagnosis is termed colitis of undetermined type and etiology (CUTE).

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Flashcard 3199224909068

Tags
#Gastroenterology #IBD
Question
in 10% of cases of IBD causing colitis a definitive diagnosis of either UC or CD is not possible and the diagnosis is termed [...]
Answer
colitis of undetermined type and etiology (CUTE).


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in 10% of cases of IBD causing colitis a definitive diagnosis of either UC or CD is not possible and the diagnosis is termed colitis of undetermined type and etiology (CUTE).

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#Gastroenterology #IBD
Jewish people are more prone to inflammatory bowel disease than any other ethnic group.

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Flashcard 3199228841228

Tags
#Gastroenterology #IBD
Question
[...] people are more prone to inflammatory bowel disease than any other ethnic group.
Answer
Jewish


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Jewish people are more prone to inflammatory bowel disease than any other ethnic group.

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Flashcard 3199230414092

Tags
#Gastroenterology #IBD
Question
Although the aetiology of IBD is unknown, it is increasingly clear that IBD represents the interaction between several co-factors, what are they?
Answer
  • Genetic susceptibility
  • The environment
  • The intestinal microbiota
  • Host immune response


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#Gastroenterology #IBD
Crohn's Disease and Ulcerative colitis are complex polygenic diseases and having a positive family history is the largest independent risk factor.

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Flashcard 3199235132684

Tags
#Gastroenterology #IBD
Question
Crohn's Disease and Ulcerative colitis are [...] and having a positive family history is the largest independent risk factor.
Answer
complex polygenic diseases


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Crohn's Disease and Ulcerative colitis are complex polygenic diseases and having a positive family history is the largest independent risk factor.

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ORIE_LIGNE=( #next lines reorient all the line element on ’topbeam’ #along a vector lying along y global axis #with origin at node ’masseN

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An identity equation is an equation that is always true for any value substituted into the variable. \(_\square\)

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Solving Identity Equations | Brilliant Math &amp; Science Wiki
ave an account? Log in here. Quiz Solving Identity Equations Relevant For... Algebra > Polynomial Arithmetic Ram Mohith , Beakal Tiliksew , Mahindra Jain , and 1 other Jimin Khim contributed <span>An identity equation is an equation that is always true for any value substituted into the variable. \(_\square\) For example, \(2(x+1)=2x+2\) is an identity equation. One way of checking is by simplifying the equation. \[\begin{align} 2(x+1)&=2x+2\\ 2x+2&=2x+2\\ 2&=2. \end{align}\] \(2




Flashcard 3199243783436

Tags
#mathematics #polynomials #precalculus
Question
Since the two factors are different by , the factors will always have the same parity. That is, if is even then must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.
Answer
[default - edit me]


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Difference Of Squares | Brilliant Math &amp; Science Wiki
ue View solutions What is Note: Try it without using a calculator. Correct! The answer is 1. 74% of people got this right. Continue View solutions Don't use a calculator! Further Extension Edit <span>Since the two factors are different by , the factors will always have the same parity. That is, if is even then must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: Problem Solving Edit The examples and problems in this sections are a bit hard







Flashcard 3199247453452

Question
ou do not understand may seem like an utmost nonsense. Still, an amazing proportion of students commit the offence of learning without comprehension. Very often they have no
Answer
[default - edit me]


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Effective learning: Twenty rules of formulating knowledge | SuperMemo.com
not just learn once but you will repeat the material optimally (as in SuperMemo). The 20 rules of formulating knowledge in learning Do not learn if you do not understand Trying to learn things y<span>ou do not understand may seem like an utmost nonsense. Still, an amazing proportion of students commit the offence of learning without comprehension. Very often they have no other choice! The quality of many textbooks or lecture scripts is deplorable while examination deadlines are unmovable. If you are not a speaker of German, it is still possible to learn







Flashcard 3199249812748

Question
not a speaker of German, it is still possible to learn a history textbook in German. The book can be crammed word for word. However, the time needed for such "blind le
Answer
com here


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Effective learning: Twenty rules of formulating knowledge | SuperMemo.com
nce of learning without comprehension. Very often they have no other choice! The quality of many textbooks or lecture scripts is deplorable while examination deadlines are unmovable. If you are <span>not a speaker of German, it is still possible to learn a history textbook in German. The book can be crammed word for word. However, the time needed for such "blind learning" is astronomical. Even more important: The value of such knowledge is negligible. If you cram a German book on history, you will still know nothing of history. The German history







example is an extreme. However, the materials you learn may often seem well structured and you may tend to blame yourself for lack of comprehension. Soon you may pollute your learning process with a great deal of useless material that treacherously makes you believe "it will be useful some day".

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Effective learning: Twenty rules of formulating knowledge | SuperMemo.com
learning" is astronomical. Even more important: The value of such knowledge is negligible. If you cram a German book on history, you will still know nothing of history. The German history book <span>example is an extreme. However, the materials you learn may often seem well structured and you may tend to blame yourself for lack of comprehension. Soon you may pollute your learning process with a great deal of useless material that treacherously makes you believe "it will be useful some day". Learn before you memorize Before you proceed with memorizing individual facts and rules, you need to build an overall picture of the learned knowledge. Only when individual pieces fit t




Flashcard 3199695457548

Question
[default - edit me]
Answer
your learning process


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example is an extreme. However, the materials you learn may often seem well structured and you may tend to blame yourself for lack of comprehension. Soon you may pollute your learning process with a great deal of useless material that treacherously makes you believe "it will be useful some day".

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Effective learning: Twenty rules of formulating knowledge | SuperMemo.com
learning" is astronomical. Even more important: The value of such knowledge is negligible. If you cram a German book on history, you will still know nothing of history. The German history book <span>example is an extreme. However, the materials you learn may often seem well structured and you may tend to blame yourself for lack of comprehension. Soon you may pollute your learning process with a great deal of useless material that treacherously makes you believe "it will be useful some day". Learn before you memorize Before you proceed with memorizing individual facts and rules, you need to build an overall picture of the learned knowledge. Only when individual pieces fit t







Flashcard 3199899929868

Question
e endings. “And th
Answer
surprise endings. “And then,


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4 Steps to Reading a Textbook Quickly and Effectively
e lame. If you know Bruce Willis is dead, don’t watch the 6th Sense. But textbooks are rarely building to a suspenseful twist at the end. I promise. I’ve read a lot. They don’t come with surpris<span>e endings. “And then, Abraham Lincoln dodged the bullet!” Yep, that’s never going to be in a textbook. Want to try this strategy? Try reading your textbook chapter in this order: Go to the questions at th







Hausdorff's Set theory.

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Flashcard 3203067940108

Tags
#mathematics #polynomials #precalculus
Question
\(5^2-2^2\) = [...] \(\times (5+2) = 3\times 7\)
Answer
\((5−2)\)


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Rewrite 52−22 as a product. We have 52−22=(5−2)×(5+2)=3×7. □

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Difference Of Squares | Brilliant Math &amp; Science Wiki
ction contains examples and problems to boost understanding in the usage of the difference of squares identity: \(a^2-b^2=(a+b)(a-b)\). Here are the examples to learn the usage of the identity. <span>Rewrite \(5^2-2^2\) as a product. We have \[5^2-2^2 = (5-2) \times (5+2) = 3\times 7. \ _\square\] Calculate \(299\times 301\). You can brute force the answer to this problem by using a calculator, but we have a sweeter way. We can apply the difference of two squares identity. At fir







nstead of using a K_TR_D_L for the ’hinge’ group we may use a beam element with a null, or near null, value for the moment of inertia on the right axis.

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As we use a null or near null for one value of moment of inertia 1 it is of course meaningless to try to calculate the related bending stress in these elements, but the forces and moments are meaningful! Last but not least using short beam elements or discrete elements is not strictly equivalent, chapter 3.1 provides a deeper insight into discrete stiffness elements

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Axiomatic set theory by Suppes

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In set theory "naive" and "axiomatic" are contrasting words. The present treatment might best be described as axiomatic set theory from the naive point of view. It is axiomatic in that some axioms for set theory are stated and used as the basis of all subsequent proofs. It is naive in that the language and notation are those of ordinary informal (but for- malizable) mathematics.

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#mathematics #polynomials #precalculus
Since the two factors are different by \2b\, the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.

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Difference Of Squares | Brilliant Math &amp; Science Wiki
lor{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \] Don't use a calculator! Further Extension <span>Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: \[\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)




Flashcard 3209515896076

Tags
#mathematics #polynomials #precalculus
Question
Since the two factors are different by [...], the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.
Answer
2b


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Since the two factors are different by 2b , the factors will always have the same parity. That is, if a−b is even then a+b must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is

Original toplevel document

Difference Of Squares | Brilliant Math &amp; Science Wiki
lor{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \] Don't use a calculator! Further Extension <span>Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: \[\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)







Flashcard 3209517468940

Tags
#mathematics #polynomials #precalculus
Question
Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is [...] then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.
Answer
even


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Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if a−b is even then a+b must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot

Original toplevel document

Difference Of Squares | Brilliant Math &amp; Science Wiki
lor{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \] Don't use a calculator! Further Extension <span>Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: \[\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)







Flashcard 3209519303948

Tags
#mathematics #polynomials #precalculus
Question
Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the [...] is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.
Answer
product


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Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if a−b is even then a+b must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 square

Original toplevel document

Difference Of Squares | Brilliant Math &amp; Science Wiki
lor{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \] Don't use a calculator! Further Extension <span>Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: \[\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)







Flashcard 3209521138956

Tags
#mathematics #polynomials #precalculus
Question
Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be [...], so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.
Answer
even


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Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if a−b is even then a+b must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the differ

Original toplevel document

Difference Of Squares | Brilliant Math &amp; Science Wiki
lor{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \] Don't use a calculator! Further Extension <span>Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: \[\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)







Flashcard 3209522187532

Tags
#mathematics #polynomials #precalculus
Question
Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by [...]. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.
Answer
four


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if a−b is even then a+b must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.

Original toplevel document

Difference Of Squares | Brilliant Math &amp; Science Wiki
lor{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \] Don't use a calculator! Further Extension <span>Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: \[\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)







Flashcard 3209524022540

Tags
#mathematics #polynomials #precalculus
Question

Rewrite \(5^2-2^2\) as a product.


We have

\[5^2-2^2 = (5-2) \times (5+2) = 3\times 7.\]

Answer
5222


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Rewrite 52−22 as a product. We have 52−22=(5−2)×(5+2)=3×7.

Original toplevel document

Difference Of Squares | Brilliant Math &amp; Science Wiki
ction contains examples and problems to boost understanding in the usage of the difference of squares identity: \(a^2-b^2=(a+b)(a-b)\). Here are the examples to learn the usage of the identity. <span>Rewrite \(5^2-2^2\) as a product. We have \[5^2-2^2 = (5-2) \times (5+2) = 3\times 7. \ _\square\] Calculate \(299\times 301\). You can brute force the answer to this problem by using a calculator, but we have a sweeter way. We can apply the difference of two squares identity. At fir







Flashcard 3209525071116

Tags
#mathematics #polynomials #precalculus
Question
Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if [...] is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares.
Answer
ab


statusnot learnedmeasured difficulty37% [default]last interval [days]               
repetition number in this series0memorised on               scheduled repetition               
scheduled repetition interval               last repetition or drill

Parent (intermediate) annotation

Open it
Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if a−b is even then a+b must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not

Original toplevel document

Difference Of Squares | Brilliant Math &amp; Science Wiki
lor{blue}{2014} \times \color{blue}{2014}\color{blue}{2014} - \color{blue}{2014}\color{red}{2013} \times \color{blue}{2014}\color{fuchsia}{2015} = ? \] Don't use a calculator! Further Extension <span>Since the two factors are different by \(2b\), the factors will always have the same parity. That is, if \(a-b\) is even then \(a+b\) must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: \[\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)







如何长时间高效学习[转一优秀答复]



转载,著作权归原作者白诗诗所有

①时间

如果你想要在某个知识领域成为the best of the best,你必须要花至少10000小时实实在在的练习和学习。然而,Josh Kaufman却推翻了这些观念,提出了你只需要20小时就能不错的掌握一个全新的知识和技能。 如果我们能在最初的20小时内熬过初学阶段的不适感和挫败感,我们其实能很轻松地掌握一个全新的领域。哈!20个小时=2个沉睡的夜晚=1周娱乐时间的总合,听起来的确很不错。 嗯,我的意思是用20小时学习一门大学教材。


②案例


高效利用时间的案例:学更多更多的知识,做更多更多的事情。简直是知识、精神双饱满!!斯考特·杨(Scott Young)12个月之内,自学完成了传说中的MIT计算机科学课程表的全部33门课,从线性代数到计算理论。按照他的进度,读完一门课程大概只需要1.5个星期。这是我见过的大学生里在学习方面活得最充实的人。 我曾经对他的学习方法做过超认真的整理:怎样做才能不虚度大学的时光? - 白诗诗的回答

关于他的学习方法,warfalcon 评价如是:


评价a:每天学习10小时,10天左右就要考试一次,并通过,这个学习效率非常高,更不用说重复了33次。在学习一门新的知识时短时间集中注意力不难,但持续一段时间之后,如果做不到很好的放松,只要持续二、三个月左右就会达到一个瓶颈,理解能力和心理状态无法继续,这个瓶颈会重复出现,就象考研时很多人前几个月都能坚持下去,但到6、8月之后就无法坚持。重复三次左右会面临一个临界点,度过这后就很容易继续了。 评价b:刻意练习没有“寓教于乐”这个概念。曾经有个著名小提琴家说过,如果你是练习手指,你可以练一整天;可是如果你是练习脑子,你每天能练两个小时就不错了。高手的练习每次最多1到1.5小时,每天最多4到5小时。没人受得了更多。一般女球迷可能认为贝克汉姆那样的球星很可爱,她们可能不知道的是很少有球员能完成贝克汉姆的训练强度,因为太苦了。

③可实施性


“在学习一门新的知识时,短时间集中注意力不难” + “(刻意练习)高手的练习每次最多1到1.5小时,每天最多4到5小时。没人受得了更多”→→→→结合我们20小时学一门教材的目标,可以得出一个结论:我们进行的不是严格意义刻意练习,我们的强度也没有那个国外小子那么高,而且一轮下来只要20小时,即便是用刻意练习的方式进行,也是可以接受的。学完一科,我们可以进行休整放松,不用严打紧逼。而且就我个人而言,一天高效学习时间安排在八个小时左右,是可以执行的。

但是,要保证利用好八个小时的高效时间。



④如果保证利用好自己的高效时间? 学习仪式感:

人,藉由这种仪式带来的仪式感,来给自己一种强烈的自我暗示---------这种自我暗示能够使自我变革,把自己的专注力、反应能力、运动能力迅速提升。


为了保证高效时段得到利用,我把每一次学习当成月考甚至高考一样的对待。 为了高考你会提前准备什么? 物质准备:吃饱穿暖 精神准备:考前睡眠充足 知识储备:复习再复习 在高考之前,脑海里,我们已经把在考场上的表现重复了多遍。

再来看看,我是怎么为一次高效学习进行仪式感建立的: 物质:水、巧克力、计算器、文具。在正式学习之前,我会在桌上一字排开,以防临时需要某些东西时手忙脚乱去到处翻。


精神:在前一天晚上,我就想好第二天又要进行高效学习了,所以我会安心睡去,而且如果你已经在前一天高效学好的话,会在一种满足感和新的期待中睡去。 时间:比如,八点是我的正式学习时间,我会保证八点之前提前赶到那里,坐在椅子上调整出最舒服的学习姿态。 提取半个小时到达学习现场,我会做以下工作:

仪式第一步:启动 1.把准备好的物品摆在桌上。 2.深呼吸一口气,然后做眼保健操。 3.闭着眼按摩太阳穴一个八拍。 注:眼保健操的第三个八拍和第四个八拍互换,多年的眼保健操经验告诉我,原本第四个八拍(轮刮眼眶)做完之后,眼睛睁开是会一片模糊感,眼液汪汪,十分不舒服。(ps.重新排版此文的时候我才发现,对于眼保健操我也是蛮有见解的嘛!!!哈哈哈。)



仪式第二步:预热

翻看即将在要来到的两个小时之内需要学习的内容,心里有个大概。我会多浏览几遍某些重要的概念以及例题,但是我不会把这个活动当做是我的正式学习的过程,就好像你考试之前翻书,多看几遍重难点考点只是为了考试更好的发挥,而不能把这个过程计入考试本身一样。我现在做的活动,只是为了等一下正式学习更好的强化效果。


仪式最后一步:静心

在八点之前的一分钟,我会盖上书本。静静等着一分钟的流逝,八点钟一到,就带着喜悦感平静地翻开书本。因为有了那么多前戏的酝酿,你都不知道我多么的期待这么一次高效的学习的来临啊。



大学学习的方法千差万别,比如,有人有能耐能泡老师透到题,或者被老师泡也能泡到题,这种做法我认为是最高效的,有了考试题目就只需要锁定知识点夺取高分。这种做法我十分欣赏,但只有一点我不喜欢,那就是逃避了学习过程,没错,我也要考高分,但我只想踏踏实实的通过正常的学习完成这个过程。不过,我的踏踏实实要是和很多人一样一章一节的看下去,那就是侮辱我自己了,因为那样我会看不到最后,就止不住困倒在书上了。


我个人的方法就是对这个踏实过程的一次优化,让自己在持续反馈之中达成纯20小时高效学习一门学科的目的。


第三步:正式高效率学习


步骤如下:

第A步:


第一遍阅读 1.看目录知道这一章重点在哪一节,这一节大概用来解决什么问题 2.看章后习题,圈出术语--------这个术语基本上就是本章的知识点了 3.根据术语去书中划概念和术语解释--------如果有些术语不能理解,请使用网络百度术语名词解释 4.术语理解后带着术语去理解书中的图表和例题以及案例


⑤为什么不首先直接去阅读文字呢?


对此,我还当真有些个人的强烈认知:理工科的书在我看来是不能谈「理解」这个词的,尤其是工科的书,我认为就是一份份的说明书,讲机械那就是机械的说明书,讲制图那就是制图的说明书,这些说明书和商品说明书没有本质区别,只是它们通过系统化的学术语言衔接成书、成册——但本质上一本说明书的集合。洗衣机说明书有什么作用?那就是告诉购买者如何操作、如何保养、注意事项。平时我们会嘲笑某些人看不懂说明书,其实不是对方理解力有问题,而是我们市面上多少说明书简直垃圾,写的不够明确、简洁,导致阅读者操作困难。


同样,我也认为我们的教科书并不都是一本本优秀的说明书合集,因为它们的书写内容是由个人写就,说明的操作细节根据他个人的智识水平书写,他并不曾统计所写的每一句话是不是让读者看起来更加容易形象的理解。我是说,不好的说明书才需要额外的理解,好的说明书只需要我们按着步骤流程一步一步操作就可以达到目的,只有书写不明的说明书才考验读者额外的理解能力。



从某个意义上来说,你学习某一本书,如果看不进去,可能不是你理解有问题,你应该换一本书。我们什么时候讲理解能力?中文阅读理解,英文阅读理解,文学作品,“我爱你,你懂我心吗?”,那是中文,英文。感性表达的时候需要你的理解,文人思维表达的时候需要你的理解,读哈姆雷特的时候需要你的理解,需要一千个读者有一千个哈姆雷特的理解。讲机械机构的时候如果你说不理解,那多半是书本表达的问题,不是你理解力有问题。可以长高宽表达的东西,你说你理解它干嘛?难道某个机械结构脉脉含情的给你传达某种信号,需要你理解她的弦外之音!? 以上是对教科书的微词。我意思是这样的,大学本科教材有可能的话去换几本看看,可能有比较容易“理解”的书呢?中国大学教材烂,这也是为什么国外大学教材受欢迎的原因。 但是,对于多数人而言,外文教材好啊好啊,其实都是喊在嘴巴上,看我这篇文章你都嫌长,你绕个远道去看外文教材?网上的公开课也比中国教授讲的好呀,推荐者万万众,实践者几几人!更何况,多数人外文的书是看不懂的,那么,就踏踏实实的看你们学校的教材,照样可以学好,此时才显得我的这个建议之有含量。


(在重新排版的时候,我请额外圈出一句话:推荐者万万众,实践者几几人!——这句话太好了,且做我的名言2号。) 这个建议是这样:

看书看不进,就牢牢抓住书本的例题、案例、图表。因为例题讲具体情境、图表具有可视化、案例就是讲具体的运用——这些都比理解文字描述容易的多。而且,例题里面包含了对关键知识点的运用,案例和图表其实都是为了辅助你理解正文文字内容的。所以,只要我们配合最少量的文字看懂了案例、图表,就达到了对知识的了解,接着我们再去看例题就知道了知识运用场景,之后,我们再反复地做题目,从而达到了对知识点的掌握。



这里还给你明确指明了什么叫了解、知道、掌握,课本往...

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