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Furthermore, a normal vector points towards the center of curvature, and the derivative of tangent vector also points towards the center of curvature.

Differential Calculus | Khan Academy

In summary, normal vector of a curve is the derivative of tangent vector of a curve. We can parameterize the curve by.

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Their cross product is just. In first year calculus, we saw how to approximate a curve with a line, parabola, etc.

Fundamental Theorems of Calculus

Instead we can find the best fitting circle at the point on the curve. The center of the osculating circle will be on the line containing the normal vector to the circle.

In particular the center can be found by adding. How is the normal component of acceleration related to the curvature.

If you remember, the normal component the acceleration tells us how fast the particle is changing direction. If a curve has a sharp bend high curvature then the directional change will be faster. We now show that there is a definite relationship between the normal component of acceleration and curvature.

Parameterization by Arc Length Recall that like parametric equations, vector valued function describe not just the path of the particle, but also how the particle is moving.


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Concepts: Curvature and Normal Vector Consider a car driving along a curvy road. Curvature of a Curve Curvature is a measure of how much the curve deviates from a straight line. Normal Vector of a Curve A unit normal vector of a curve, by its definition, is perpendicular to the curve at given point. The Osculating Circle In first year calculus, we saw how to approximate a curve with a line, parabola, etc. Applications include graph analysis, linear motion, average value, area, volume, and growth and decay models.

Skip to content. Course Objectives: Evaluate limits, including those involving infinity. Define and apply numerical and function derivatives. Understand the relationship between continuity and differentiability. Bibliografia Reading materials. Adams, Calculus, a complete course, Pearson Canada S.

Limits and Continuous Functions - MIT Highlights of Calculus

Lipschutz, M. Materials for further readings will be communicated during class, if needed. Regole d'esame Assessment and grading criteria. Exam: written test; optional oral exam;. At the same time, the exam will ascertain the acquisition of the corresponding skills, such as knowing how to solve single-variable differential calculus problems, integral calculus, linear algebra and geometry. Thus, to know how to reorganize the concepts of matrix, vector, geometrical entity in the plane and in space, surface area, function. The ability to recognize the graph of a function or the skill to represent it will be verified.

The written exam consists of a series of 8 open-ended exercises, of which 2 with a maximum of 7 points available and 6 with a maximum of 3 points available. Maximum points shall be awarded to an exercise if it is complete, correct, logically sound and clearly presented.


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Any unanswered exercise is valued zero. Use of personal notes, textbooks, printed notes and calculator is not permitted. It is possible to use an inventory of formulas specifically prepared by the teacher, which will be rendered available online through the portal.

Course summary

The oral examination must be taken during the same exam period during which the written exam was passed. The oral examination is not administered following a written exam which was not passed. De Gregorio Paolo Mario.