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MATH23 MULTIVARIABLE CALCULUS

PARTIAL DERIVATIVES. MATH23 MULTIVARIABLE CALCULUS. GENERAL OBJECTIVE. Determine the geometric interpretation of partial derivatives and its derivation. At the end of the lesson the students are expected to:. PARTIAL DERIVATIVES. Definition.

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MATH23 MULTIVARIABLE CALCULUS

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  1. PARTIAL DERIVATIVES MATH23 MULTIVARIABLE CALCULUS

  2. GENERAL OBJECTIVE • Determine the geometric interpretation of partial derivatives and its derivation At the end of the lesson the students are expected to:

  3. PARTIAL DERIVATIVES

  4. Definition Let z = f(x,y). The partial derivatives of f are defined as follow: Partial Derivative with respect to x: Partial Derivative with respect to y:

  5. Examples Let f(x,y) = 3x3y2. Find a. fx(x,y) b. fy(x,y) c. fx(x,1) d. fy(1,y) e. fx(1,2) f. fy(1,2) Note: Partial differentiation entails getting the ordinary derivative with respect to the indicated variable while holding the other variable constant.

  6. Geometric Interpretation of Partial Derivatives with respect to x fx(x0,y0) – slope of the surface in the x-direction at (x0,y0)

  7. Geometric Interpretation of Partial Derivatives with respect to x fy(x0,y0) – slope of the surface in the y-direction at (x0,y0).

  8. Example Let f(x,y) = (3x + 2y)1/2. a) Find the slope of the surface z = f(x,y) in the x-direction at the point (4,2). b) Find the slope of the surface z = f(x,y) in the y-direction at the point (4,2). Find the slope of the sphere x2 + y2 + z2 = 1 in the y-direction at the points (2/3,1/3,2/3) at (2/3,1/3,-2/3).

  9. Tangent Plane and Normal Line Example Find an equation for the tangent plane and parametric equations for the normal line to the surface z = x2y at the point (2,1,4).

  10. Tangent Plane and Normal Line

  11. Example Consider the ellipsoid x2 + 4y2 + z2 = 18. a) Find the equation of the tangent plane at the point (1,2,1). b) Find the parametric equations of the line normal at the point (1,2,1).

  12. Functions with more than 2 variables. Let f(x,y,z) = zln(x2ycosz). Find a) fx b) fy c) fz

  13. Higher Order Partial Derivatives Let z = f(x,y). Second order Partial Derivatives: fxx fyy fxy fyx Third order Partial Derivatives: fxxx fyyy fxxy fxyx fyxx fyyx ….

  14. Example: Prove fxy = fyx Note: fxy = fyx

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