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SOME APLICATIONS OF DIFFERENTIATION AND INTEGRATION Fakhrulrozi Hussain.

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SOME APPLICATIONS OF INTEGRATIONS 1.Area Under a Curve 2.Volume by Slicing 3.Geometric Interpretation

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SOME APPLICATIONS OF INTEGRATIONS 1.Area Under a Curve

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1.Area Under a Curve - example SOME APPLICATIONS OF INTEGRATIONS

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Find the volume of the cylinder using the formula and slicing with respect to the x-axis. A = r 2 A = 2 2 = 4 2.Volume by Slicing-example SOME APPLICATIONS OF INTEGRATIONS

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Area of R 1 – Area of R 2 + Area of R 3 a b R1R1 R2R2 R3R3 3.Geometric Interpretation

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SOME APPLICATIONS OF DIFFERENTATIONS 1.Tangents and Normals 2.Newton's Method for Solving Equations Corollary. 3.Motion 4.Related Rates

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APPLICATIONS OF DIFFERENTATIONS 1.Tangents and Normals we can find the slope of a tangent at any point (x, y) using

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APPLICATIONS OF DIFFERENTATIONS 1.Tangents and Normals - example Find the gradient of (i) the tangent (ii) the normal to the curve y = x 3 - 2x at the point (2,5) Ans : The slope of the tangent is The slope of the normal is found using m 1 × m 2 = -1

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APPLICATIONS OF DIFFERENTATIONS 2.Newton's Method for Solving Equations

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APPLICATIONS OF DIFFERENTATIONS 2.Newton's Method for Solving Equations – example Find the root of 2x 2 x 2 = 0 between 1 and 2. Ans: Try x 1 = 1.5 Then Now f(1.5) = 2(1.5) = 1 f '(x) = 4x 1 and f '(1.5) = 6 1 = 5 So So 1.3 is a better approximation.

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APPLICATIONS OF DIFFERENTATIONS 2.Newton's Method for Solving Equations – example Continuing the process, (better accuracy) Continue for as many steps as necessary to give the required accuracy. Using computer application. The result is: root(2x 2 x 2, x) =

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APPLICATIONS OF DIFFERENTATIONS 3.Motion

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APPLICATIONS OF DIFFERENTATIONS 4.Related Rates If 2 variables both vary with respect to time and have a relation between them, we can express the rate of change of one in terms of the other. We need to differentiate both sides with respect to time ( ).

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APPLICATIONS OF DIFFERENTATIONS 4.Related Rates - example A 20 m ladder leans against a wall. The top slides down at a rate of 4 ms -1. How fast is the bottom of the ladder moving when it is 16 m from the wall?

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APPLICATIONS OF DIFFERENTATIONS 4.Related Rates - example A 20 m ladder leans against a wall. The top slides down at a rate of 4 ms -1. How fast is the bottom of the ladder moving when it is 16 m from the wall? Ans: Now the relation between x and y is: x 2 + y 2 = 20 2 Now, differentiating throughout w.r.t time: That is:

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APPLICATIONS OF DIFFERENTATIONS 4.Related Rates - example A 20 m ladder leans against a wall. The top slides down at a rate of 4 ms -1. How fast is the bottom of the ladder moving when it is 16 m from the wall? Ans: Now, we know and we need to know the horizontal velocity (dx/dt) when x = 16.

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APPLICATIONS OF DIFFERENTATIONS 4.Related Rates - example A 20 m ladder leans against a wall. The top slides down at a rate of 4 ms -1. How fast is the bottom of the ladder moving when it is 16 m from the wall? Ans: The only other unknown is y, which we obtain using Pythagoras' Theorem: So Gives m/s

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MORE APPLICATIONS OF DIFFERENTATIONS AND INTEGRATIONS Area Under a Curve Area in Polar Coordinates Center of Mass Center of Mass of a Curve Center of Mass of an Area Surface of Revolution Volume of Revolution Volume by Slicing The Stirling's Formula for the Factorial and the Gamma Function

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MORE APPLICATIONS OF DIFFERENTATIONS AND INTEGRATIONS Convergence of the Binomial Expansion on [-1, 1] Taylors Expansion with an Integral form of Remainder. Corollary. Theorem (Polygonal Approximation). Theorem (Representationof Polygons). Weierstrass Approximation Theorem Space Curves The Unit Tangent and the Principal Normal Velocity and Acceleration

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