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SOME APLICATIONS OF DIFFERENTIATION AND INTEGRATION Fakhrulrozi Hussain.

SOME APLICATIONS OF DIFFERENTIATION AND INTEGRATION Fakhrulrozi Hussain. http://fakhrulrozi.com/. Area Under a Curve Volume by Slicing Geometric Interpretation. SOME APPLICATIONS OF INTEGRATIONS. SOME APPLICATIONS OF INTEGRATIONS. Area Under a Curve.

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SOME APLICATIONS OF DIFFERENTIATION AND INTEGRATION Fakhrulrozi Hussain.

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  1. SOME APLICATIONS OF DIFFERENTIATION AND INTEGRATION Fakhrulrozi Hussain. http://fakhrulrozi.com/

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

  3. SOME APPLICATIONS OF INTEGRATIONS Area Under a Curve

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

  5. SOME APPLICATIONS OF INTEGRATIONS Volume by Slicing-example Find the volume of the cylinder using the formula and slicing with respect to the x-axis. A = r2 A = 22 = 4

  6. Geometric Interpretation R1 R3 a b R2 Area of R1 – Area of R2 + Area of R3

  7. Tangents and Normals • Newton's Method for Solving Equations Corollary. • Motion • Related Rates SOME APPLICATIONS OF DIFFERENTATIONS

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

  9. Tangents and Normals - example • Find the gradient of • (i) the tangent (ii) the normal • to the curve y = x3 - 2x2 + 5 at the point (2,5) • Ans : • The slope of the tangent is • The slope of the normal is found using m1 × m2 = -1 APPLICATIONS OF DIFFERENTATIONS

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

  11. Newton's Method for Solving Equations – example Find the root of 2x2 − x − 2 = 0 between 1 and 2. Ans: Try x1 = 1.5 Then Now f(1.5) = 2(1.5)2 − 1.5 − 2 = 1 f '(x) = 4x − 1 and f '(1.5) = 6 − 1 = 5 So So 1.3 is a better approximation. APPLICATIONS OF DIFFERENTATIONS

  12. 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(2x2 − x − 2, x) = 1.2807764064044 APPLICATIONS OF DIFFERENTATIONS

  13. Motion APPLICATIONS OF DIFFERENTATIONS

  14. 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 ( ). APPLICATIONS OF DIFFERENTATIONS

  15. 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? APPLICATIONS OF DIFFERENTATIONS

  16. 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: x2 + y2 = 202 Now, differentiating throughout w.r.t time: That is: APPLICATIONS OF DIFFERENTATIONS

  17. 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. APPLICATIONS OF DIFFERENTATIONS

  18. 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 APPLICATIONS OF DIFFERENTATIONS

  19. 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 MORE APPLICATIONS OF DIFFERENTATIONS AND INTEGRATIONS

  20. Convergence of the Binomial Expansion on [-1, 1] • Taylor’s 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 MORE APPLICATIONS OF DIFFERENTATIONS AND INTEGRATIONS

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