2_1_2_2 - Sections 2.1 and 2.2 Limit Instructor Ms Hoa...

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Sections 2.1 and 2.2: Limit Instructor: Ms. Hoa Nguyen ([email protected]) What is Calculus? Calculus is the mathematics of motion and change. Why is Limit Theory the basis of Calculus? The major concepts of Calculus such as continuity, derivative and definite integral are all defined by limit. How do limits arise in the tangent problem? The tangent problem: Given a point P on a curve C , find an equation of the tangent line to the curve C at the point P . Steps : 1) Draw a secant line PQ where Q is a nearby point. 2) The slope m of the tangent line at P is the limit of the slopes of the secant line PQ as Q approaches P ( slope = rise over run ). Remark : The slope m of the tangent line to the curve y = f ( x ) at the point P ( a, f ( a )) is: m = lim x a f ( x ) - f ( a ) x - a Let h = x - a , then m = lim h 0 f ( a + h ) - f ( a ) h 3) The equation of the tangent line at the point P is an equation of the line passing through the point P and having slope m : y - y P = m ( x - x P ) ( point-slope form of the equation of a line ).
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