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Mathematics

Mathematics. Session. Applications of Derivatives - 1. Session Objectives. Rate of Change of Quantities Slope and Equation of Tangent Slope and Equation of Normal Angle Between Two Curves. be a function of x. Let. Rate of change of y with respect to x. represents the rate

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Mathematics

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  1. Mathematics

  2. Session Applications of Derivatives - 1

  3. Session Objectives • Rate of Change of Quantities • Slope and Equation of Tangent • Slope and Equation of Normal • Angle Between Two Curves

  4. be a function of x. Let Rate of change of y with respect to x represents the rate of change of y with respect to x at x = x0 Rate of Change of Quantities

  5. Velocity at a time t = t0 can be written as at t = t0 Similarly, rate of change of velocity with respect to time t, represents acceleration. Rate of Change of Quantities Hence, velocity of a point body is defined as the rate of change of displacement with respect to time t.

  6. x rate of change of x with respect to t Rate of Change of Quantities If both x and y are functions of t, then  Rate of change of y with respect to t

  7. Example - 1 An edge of a variable cube is increasing at the rate of 5 cm/s. How fast is the volume of the cube increasing when the edge is 6 cm long.

  8. Solution Let x be the edge of the variable cube and V be the volume at any time t.

  9. Example - 2 The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm.

  10. Solution Let r be the radius of a spherical soap bubble and S be the surface area at any time t. = 1.6 x 22 = 35.2 cm2/sec

  11. Letbe a continuous curve and let (x0 ,y0)be a point on the curve. Tangent The slope of the tangent to curve f(x) at (x0, y0) is The equation of the tangent to the curve at (x0, y0) is

  12. Normal As normal is perpendicular to tangent at the point of contact Equation of normal to the curve at (x0 , y0) is

  13. Example-3 Find the equation of the tangent and normal to the curve y = x4 – 6x3 + 13x2 – 10x + 5 at (0,5). Solution :

  14. Slope of the normal at (0, 5) Solution Cont. Equation of tangent at (0, 5) is y – 5 = -10 (x – 0) Equation of normal at (0, 5) is

  15. If the tangent to the curve at (1, -6) is parallel to the line x – y + 5 = 0, find the values of a and b. Example-4

  16. The tangent is parallel to Therefore, the curve becomes Con. (1, –6) lies on (i)

  17. Angle Between Two Curves

  18. Angle Between Two Curves The other angle is 1800 - q (1) Orthogonal curves: m1m2 = - 1 (2) Curves touch each other: m1 = m2

  19. Example-5 Show that the curves x2 = 4y and 4y + x2 = 8 intersect orthogonally at (2, 1).

  20. Solution We have x2 = 4y and 4y + x2 = 8 m1m2 = 1 x (-1) = -1 Hence, the curves intersect orthogonally at (2, 1).

  21. Thank you

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