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Application of partial derivatives
- 1. Sarvajanik College of Engineering& Technology CALCULAS (2110014) BRANCH : CIVIL ENGINEERING DEPARTMENT (06) APPLICATION OF PARTIAL DERIVATIVES
- 2. Content Tangent Planeand Normal Line Total Differential (Error Approximations)
- 4. Tangent Planes Supposea surface S has equation z=f(x,y) where f has continuous partial derivatives Let P(x0,y0,z0) be a point on S Let T1 and T2 be the tangent lines to curves C1 and C2 at the point P
- 5. Tangent Plane Theequation of the tangent plane to the surface Z=f(x,y) at the point P(x0,y0,z0) is: Δ(x0,y0,z0)⋅(x−x0 ,y−y0 ,z−z0)=0 Fx (x0,y0,z0) (x-x0)+Fy (x0,y0,z0) (y-y0)+Fz (x0,y0,z0) (z-z0)=0
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- 7. Normal Line Thenormal line to a curve at a particular point is the line through that point and perpendicular to the tangent. The slope of any line perpendicular to a line perpendicular to a line with slope m is the negative reciprocal -1/m Thus, changing this aspect of the equation for the tangent line, we can say generally that the equation of the normal line to the graph of f at (x0,f(x0)) is y-f(x0)=(-1/f’(x0))(x-x0)
- 8. Normal Line Letf(x,y,z) define a surface that is differentiable at a point (x0,y0,z0), then the normal line to f(x,y,z) at (x0,y0,z0) is the line with normal vector f(x0,y0,z0) that passes through the point (x0,y0,z0). In particular, the equation of the normal line is x(t) = x0 + fx(x0,y0,z0)t y(t) = y0 + fy(x0,y0,z0)t z(t) = z0 + fz(x0,y0,z0)t
- 9. Error and Approximation If a result R = R(x,y,z) is calculated from a number of data quantities, x, y and z, then the relation: dR = (∂R/ ∂x) dx + (∂R /∂y) dy + (∂R/∂z) dz This is one of the "chain rules" of calculus. This equation has as many terms as there are variables. Then, if the fractional errors are small, the differentials dR, dx, dy and dz may be replaced by the absolute errors ΔR, Δx, Δy, and Δz, and written: ΔR ≈ (∂R/∂x) Δx + (∂R/∂y) Δy + (∂R/∂z) Δz
- 10. Error and Approximations For the quantity x: Δx is known as an Absolute Error in x. Δx/x is known as a Relative Error in x. (Δx/x)*100 is known as a Percentage Error in x.
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