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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
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Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
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Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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2. Problem 1
A particle moves along the positive xaxis with velocity 5 units/second. How fast is the
particle moving away from the point (0, 3) (which is on the yaxis) when the particle is
7 units away from (0, 3)?
Problem 2(The antiderivative of
(a)Break the interval [0, a] into a union of n equal subintervals. Using the right
endpoint of each subinterval, compute the Riemann sum approximating the integral,
Leave your answer in the form of a sum. But simplify as much as possible. Show
all work.
Problem 4, continued
(b)For an appropriate choice of a, express the following limit in terms of the Riemann
sum from (a). Use the formula for the antiderivative to compute the limit.
Problem
3. Problem 3
Solve the following separable ordinary differential equation with given initial condition.
�
Problem 4Compute each of the following Riemann integrals. You are not required to
show every step. Please do say what method you use to compute the integral.
(a) �
(b)
Problem 4, continued
(c)
5. Problem 1(A particle moves along the positive xaxis with velocity 5 units/second. How fast is
the particle moving away from the point (0, 3) (which is on the yaxis) when the particle is 7
units away from (0, 3)?
Solution to Problem 1 The independent variable is time, t. Denote by x the position of the
particle on the xaxis. Denote by L the distance of the particle from (0, 3). By the Pythagorean
theorem,
L2
= x2
+ 9.
Differentiating with respect to t gives,
Solving
gives,
When L equals 7, L2 equals 49 and,
2 x = L2 − 9 = 49 − 9 =
40.
Thus x equals 2 √10. By hypothesis, dx/dt equals 5. Solving gives,
Solutions
6. Problem 2
The antiderivative of 1/
(a)Break the interval [0, a] into a union of n equal subintervals. Using the right
endpoint of each subinterval, compute the Riemann sum approximating the
integra
Leave your answer in the form of a sum. But simplify as much as possible. Show
all work. Solution to (a) The partition of [0, a] into n equal subintervals has right
endpoints,
The instructions are to use the right endpoint of the kth subinterval, i.e., xk.
Thus the Riemann sum is,
7. Simplifying gives,
(b)For an appropriate choice of a, express the following limit in terms of the Riemann
sum from (a). Use the formula for the antiderivative to compute the limit.
Solution to (b) The sum in the limit can be rewritten as,
By the Solution to (a),
8. Applying this in the case that a = 4/3 gives,
Therefore,
Since (4/3)2 + 1 equals (5/3)2, and since 4/3 + 5/3 equals 3, the limit
simplifies to,
Problem 3 Solve the following separable ordinary differential equation with
given initial condition.
Solution to Problem 3 Using the exponent rule,
the differential equation is equivalent to,
9. This gives an equation of differentials,
Antidifferentiating gives,
which is,
Substituting the initial condition gives the equation for C,
Solving for C gives,
Substituting this in and simplifying gives,
Taking logarithms and simplifying gives,
10. Problem 4Compute each of the following Riemann integrals. You are not required to
show every step. Please do say what method you use to compute the integral.
(a)
Solution to (a) Using the rule that the antiderivative of xa is xa+1/(a + 1) (for a = −1),
the antiderivative of the integrand is,
Using geometric series, this simplifies to,
Applying the Fundamental Theorem of Calculus,
Evaluating at the limits gives,
In simplest terms, this gives,
11. (b)
Solution to (b) Make the substitution,
This reduces the original integral to the simpler
integral,
Therefore,
(c)
Solution to (c) The function sin(θ) is odd. Therefore the odd power, sin7 (θ) is also an
odd function. The interval [−π/2, π/2] is symmetric with respect to the origin. Therefore
the integral of sin7 (θ) over [−π/2, 0] equals the negative of the integral of sin7 (θ) over
[0, π/2]. So the two integrals cancel to give,
12. Alternatively, rewrite the integral as,
Next use the rule, sin2 (θ) = 1 − cos2(θ) to get
Make the substitution,
to reduce the original integral to,
Since the upper and lower limit are each 0, again this
gives,
(d)
13. Solution to (d) Use the logarithm rule,
to reduce the integral to,
Next make the substitution,
This reduces the original integral to the simpler integral,
Therefore,
(e)
Solution to (e) Use the angle addition formula,
14. to reduce the numerator to,
Thus the integral is,
Applying the Fundamental Theorem of Calculus gives,
Therefore,