Calculus 3 Section 14.3 Reading Assignment: Partial Derivatives

I am suck with this question.

Question 1 Answer: Given a 3D surface defined by the multivariate function z=f(x,y), which is a 3D surface, where x and y are horizontal rectangular coordinates and z is the vertical coordinate:

• the partial derivative off(x,y) with respect to x, i.e. xf, at a given point(x0,y0) corresponds to the tangent line that changes along x and z, with y kept constant (y=y0). This tangent line lies within the planey=y0 which is parallel to the x-z plane.
• the partial derivative off(x,y) with respect to y, i.e. yf, at a given point(x0,y0) corresponds to the tangent line that changes along y and z, with x kept constant (x=x0). This tangent line lies within the planex=x0 which is parallel to the y-z plane.

Question 2. Read Example 5 (p. 827). Based on the question, how can you tell which partial derivative should be used for the solution? Explain how this connects with your answer from Question 1 above.

Chapter 14 Partial Derivatives 14.3 Partial Derivatives 827 EXAMPLE 4 Find az/ax assuming that the equation jz - Inz =x+y defines z as a function of the two independent variables x and y and the partial derivative exists. Solution We differentiate both sides of the equation with respect to x, holding y constant and treating z as a differentiable function of x: ax AT (2) - ax In z = ax + Ar az =1+0 With y constant, Y ar ax z ax = 1 ax ax VE -EXAMPLE 5 The plane x = 1 intersects the paraboloid z = x' + y' in a parabola. Plane Find the slope of the tangent to the parabola at (1, 2, 5) (Figure 14.19). Surface Solution The parabola lies in a plane parallel to the yz-plane, and the slope is the value of the partial derivative az /ay at (1, 2): - Tangent 2 ( 8 + 3 2 ) line ay lam (1.2) = zy = 2(2) = 4. (1. 2, 5) As a check, we can treat the parabola as the graph of the single-variable function z = (1) + y' = 1 + y' in the plane x = 1 and ask for the slope at y = 2. The slope, calculated now as an ordinary derivative, is Functions of More Than Two Variables FIGURE 14.19 The tangent to the curve of intersection of the plane * = 1 and The definitions of the partial derivatives of functions of more than two independent vari- surface z = x' + y at the point (1, 2, 5) ables are similar to the definitions for functions of two variables. They are ordinary deriva- (Example 5). tives with respect to one variable, taken while the other independent variables are held constant. EXAMPLE 6 If x, y, and z are independent variables and f(x, ), Z) = xsin (y + 32). then of [x sin () + 32) ] = x 2 Si x * sin () + 32) a held constant = xcos (y + 32) . " ( y + 32) Chain rule = 3x cos () + 3z). y held constant