• Journal of Educational Research in Mathematics
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• v.14 no.2
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• pp.171-185
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• 2004

Students are exposed to a concept of tangent from a specific context of the relation between a circle and straight lines at the 7th grade. This initial experience might cause epistemological obstacles regarding learning concepts of tangent to additional curves. The paper provides a method of how to introduce a series of concepts of tangent in order to lead students to revise and improve the concept of tangent which they have. As students have chance to reflect and revise a series of concepts of tangent step by step, they realize the facts that the properties such as 'meeting the curve at one point' and 'touching but not cutting the curve' may be regarded as the proper definition of tangent in some limited contexts but are not essential in more general contexts. And finally students can grasp and appreciate that concept of tangent as the limit of secants and the relation between tangent and derivative.

• Communications of Mathematical Education
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• v.24 no.3
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• pp.627-658
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• 2010

The introduce and accurate understanding of the concept of tangent line is very important in Mathematics education and its applications. In this study, we investigated the introduction method in the textbook for tangent line and its concepts. And we studied the effective teaching methods for the accurate understanding using Lakatos learning theory, GSP and related precedence studies. Finally we suggest the teaching plan for the equation of for the tangent line in the high school and apply to the High school students.

• Communications of Mathematical Education
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• v.23 no.3
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• pp.625-642
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• 2009

In the school mathematics the concept of tangent is introduced in several steps in suitable contexts. Students are required to reflect and revise their concepts of tangent in order to apply the improved concept to wider range of contexts. In this paper we consider the tangent as the optimal linear approximation to a curve at a given point and make three discussions on pedagogical aspects of it. First, it provides a method of finding roots of real numbers which can be used as an application of tangent. This may help students improve their affective variables such as interest, attitude, motivation about the learning of tangent. Second, this concept reflects the modern point of view of tangent, the linear approximation of nonlinear problems. Third, it gives precise meaning of two tangent lines appearing two sides of a cusp point of a curve.

• Journal of the Korean Society of Radiology
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• v.15 no.6
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• pp.919-926
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• 2021

In this study, we wanted to find out the optimal angle as a modified tangential projection of the patella. In the experiment, we used Kyoto Kagaku's PBU-50 phantom. In the supine position, the F-T angle was set to 95°, 105°, 115°, 125°, 135°, 145°, and Patella tangential projection images were obtained by varying the X-ray tube angle by 5° so that the angle between the X-ray centerline and tibia at each angle was 5~20°. Image J was used for image analysis and the congruence angle, lateral patellofemoral angle, patellofemoral index and contrast to noise ratio(CNR) were also measured. SPSS 22 was used for statistical analysis, and the mean values of congruence angle, patellofemoral angle, patellofemoral index, and CNR were compared with Merchant method through one-way batch analysis and corresponding sample t-test. As a result of the study, in the case of congruence angle, the angle of incidence of the knee-angle X-ray centerline was 105°-72.5° (20° tangential irradiation), 115°-72.5°, 77.5° (15, 20° tangential irradiation), 125°-82.5° (20° tangential irradiation), lateral patellofemoral angle is 115°-72.5°, 77.5° (15, 20° tangential irradiation), 125°-72.5° (10° tangential irradiation), patellofemoral index is 115°-72.5° (15° tangential irradiation) and 125°-72.5° (10° tangential irradiation) were not significantly different from Merchant method (p> .05). In case of CNR, it is not different from Merchant method at 105°-67.5°, 72.5° (15, 20° tangential irradiation), 115°-67.5°, 72.5°, 77.5° (10, 15, 20° tangential irradiation). (P> .05). Based on the results of this study, high diagnostic value images can be obtained by setting the knee angle and the angle of incidence of the X-ray tube to 115°-72.5° (15° tangential irradiation) during the modified tangential examination of the knee bone. It was confirmed.

• Tunnel and Underground Space
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• v.24 no.5
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• pp.366-372
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• 2014

The generalized Hoek-Brown (H-B) function provides a unique failure condition for a jointed rock mass, in which the strength parameters of rock mass are deduced from the intact values by use of the GSI value. Since it is actually the only failure criterion which accounts for the rock mass conditions in a systematic manner, the generalized H-B criterion finds many applications to the various rock engineering projects. Its nonlinear character, however, limits more active usage of this criterion. Accordingly, many attempts have been made to understand the generalized H-B condition in the framework of the M-C function. This study presents the closed-form expression relating the tangential cohesion to the tangential friction angle, which is derived by the non-dimensional stress transformation of the generalized H-B criterion. By use of the derived equation, it is investigated how the relationship between the tangential cohesion and friction angle of the generalized H-B criterion varies with the quality of rock masses. When only the variation of GSI value is considered, it is found that the tangential friction angle decreases with the increase of GSI, while the tangential cohesion increases with GSI value.

• Tunnel and Underground Space
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• v.28 no.5
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• pp.426-441
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• 2018

Recently, the generalized Hoek-Brown (GHB) failure criterion has been actively employed in various rock engineering calculations, but the analytical form of the corresponding Mohr failure envelope is not available, making it difficult to extend the application of the GHB criterion. In order to overcome this disadvantage, this study proposes a new method to express the tangential friction angle as an explicit function of normal stress by invoking the polynomial best-fitting to the relationship between normal stress and tangent friction angle implied in the GHB failure function. If this normal stress - tangential friction angle relationship is best-fitted with linear or quadratic polynomial function, it is possible to find the analytical root for tangential friction angle. Subsequently, incorporating the root into the relationship between shear stress and tangential friction angle accomplishes the derivation of the approximate Mohr envelope for the GHB criterion. It is demonstrated that the derived approximate Mohr failure envelopes are very accurate in the entire range of GSI value.

• Tunnel and Underground Space
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• v.23 no.3
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• pp.219-227
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• 2013

Although Mohr-Coulomb failure criterion has limitations in that it is a linear criterion and the effect of the intermediate principal stress on failure is ignored, this criterion has been widely accepted in rock mechanics design. In order to overcome these shortcomings, the Hoek-Brown failure criterion was introduced and recently a number of 3-D failure criteria incorporating the effect of the intermediate principal stress on failure have been proposed. However, in many rock mechanics designs, the possible failure of rock mass is still evaluated based on Mohr-Coulomb criterion and most of practitioners are accustomed to understanding the strength of rock mass in terms of the internal friction angle and cohesion. Therefore, if the equivalent Mohr-Coulomb strength parameters of the advanced failure criteria are calculated, it is possible to take advantage of the advanced failure criteria in the framework of the Mohr-Coulomb criterion. In this study, a method expressing the tangential Mohr-Coulomb strength parameters in terms of the stress invariant is proposed and it is applied to the generalized Hoek-Brown criterion and the HB-WW criterion. In addition, a new approach describing the geometric meaning of the ${\sigma}_2$-dependency of failure criteria in 3-D principal stress space is proposed. Implementation examples of the proposed method show that the influence of the intermediate principal stress on the tangential friction angle and cohesion of the HB-WW criterion is considerable, which is not the case for the 2-D failure criterion.

• School Mathematics
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• v.14 no.4
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• pp.409-429
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• 2012

This paper reports the process two 11th grade students went through in reinventing derivatives on their own via a context problem involving the concept of velocity. In the reinvention process, one of the students conceived a tangent line as the limit of a secant line, and then the other student explained to a peer that the slope of a tangent line was the geometric mean of derivative. The students also used technology to concentrate on essential thinking to search for mathematical concepts and help visually understand them. The purpose of this study was to provide meaningful implications to school practices by describing students' process of reinvention of derivatives. This study revealed certain characteristics of the students' reinvention process of derivatives and changes in the students' thinking process.

• Journal of the Korean Geotechnical Society
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• v.17 no.3
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• pp.25-32
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• 2001

보편적인 탄소성 캡 모델은 전통적인 등방 이론에 기초를 두고 있다. 이러한 모델의 응력적분 및 접선 계수의 유도는 여러 가지 논문들에 나타나 있지만 축차 및 체적 거동을 동시에 다루는 내제적인 해석법을 통한 지반해석은 아직까지는 많은 도전이 요구되고 있다. 앞선 동반 논문에서는 비연속적으로 연결된 항복면 사이의 접선 계수는 특이점이 됨을 나타내었고 이에 대하여 새로운 캡 모델의 구성식이 제시되었다. 본 논문에서는 제시된 캡 모델의 비 조건적이고 안정된 내재적 응력적분 및 일관된 탄소성 접선계수를 유도하였다. 또한 간단한 예제를 통하여 모델의 수행능력을 보여주었고 사면안정계산이 수행되었다.

• Journal of Korean Society of Steel Construction
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• v.15 no.5 s.66
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• pp.509-518
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• 2003

In this study, we carried out nonlinear analysis according to various interface nonlinear models by interaction magnitude, and analyzed interface behavior such as distribution of tangential traction and relative slip in steel-concrete composite structure. As a result of this study, tangential traction and relative slip of interface is rapidly increased at the steel plate-concrete interface, especially at the neutral region, rather than tensile, as opposed to the T beam-concrete interface. In transverse direction, it has gradually reduced to go outside from loading position. In longitudinal direction, it was minimum at the central region near the loading point, maximum at 0.6-0.7L from support and gradually reduced as it nears support. Moreover, as the load is increased, the failure of interface gradually expands from the maximum tangential traction position to the entire region. It is expected to provide fundamentality for interface behavior and load-carrying mechanism, and for the design of bending and shear connection of steel-concrete composite structure.