SOME APLICATIONS OF DIFFERENTIATION AND INTEGRATION Fakhrulrozi Hussain.

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

SOME APPLICATIONS OF INTEGRATIONS 1.Area Under a Curve

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

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

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

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

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

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

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

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.

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) =

APPLICATIONS OF DIFFERENTATIONS 3.Motion

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 ( ).

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?

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:

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.

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

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

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