Curves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent

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1 Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun)

2 Signs of Partial Derivatives on Level Curves Level curves are shown for a function f. Determine whether the following partial derivatives are positive or negative at the point P. (a) f x (b) f y (c) f xx (d) f xy (e) f yy

3 Tangent Planes 1 A surface S given by z = f (x, y), where f has continuous first partial derivativews. 1 we will come back to this topic in 11.6 with gradient vector

4 Tangent Planes 1 A surface S given by z = f (x, y), where f has continuous first partial derivativews. Let P(x 0, y 0, z 0 ) be a point on S. See Figure 1 on page 617, we will come back to this topic in 11.6 with gradient vector

5 Tangent Planes 1 A surface S given by z = f (x, y), where f has continuous first partial derivativews. Let P(x 0, y 0, z 0 ) be a point on S. See Figure 1 on page 617, 11.4 An equation of the tangent plane to the surface z = f (x, y) at the point P(x 0, y 0, z 0 ) is z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) 1 we will come back to this topic in 11.6 with gradient vector

6 Tangent Planes 1 A surface S given by z = f (x, y), where f has continuous first partial derivativews. Let P(x 0, y 0, z 0 ) be a point on S. See Figure 1 on page 617, 11.4 An equation of the tangent plane to the surface z = f (x, y) at the point P(x 0, y 0, z 0 ) is z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) Example: Find the tangent plane to the elliptic paraboloid z = 2x 2 + y 2 at the point (1, 1, 3). [TEC 11.4] 1 we will come back to this topic in 11.6 with gradient vector

7 Tangent Planes 1 A surface S given by z = f (x, y), where f has continuous first partial derivativews. Let P(x 0, y 0, z 0 ) be a point on S. See Figure 1 on page 617, 11.4 An equation of the tangent plane to the surface z = f (x, y) at the point P(x 0, y 0, z 0 ) is z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) Example: Find the tangent plane to the elliptic paraboloid z = 2x 2 + y 2 at the point (1, 1, 3). [TEC 11.4] Practice: Find the tangent plane to the given surface z = e x2 y 2 at the point (1, 1, 1). 1 we will come back to this topic in 11.6 with gradient vector

8 Sensitivity to Change: Total Derivatives The differentials of y = f (x) are dy and dx, which are connected as dy = f (x) dx This measures the sensitivity to the change of a function. In particular, it measures how sensitive the output is to small changes in the input.

9 Sensitivity to Change: Total Derivatives The differentials of y = f (x) are dy and dx, which are connected as dy = f (x) dx This measures the sensitivity to the change of a function. In particular, it measures how sensitive the output is to small changes in the input. Likewise, we define the differentials of z = f (x, y), dx and dy, to be independent variables. Then the differential dz, called total differential, is defined by dz = f x (x, y) dx + f y (x, y) dy (This can be extended to three or more variable functions.)

10 Sensitivity to Change: Total Derivatives The differentials of y = f (x) are dy and dx, which are connected as dy = f (x) dx This measures the sensitivity to the change of a function. In particular, it measures how sensitive the output is to small changes in the input. Likewise, we define the differentials of z = f (x, y), dx and dy, to be independent variables. Then the differential dz, called total differential, is defined by dz = f x (x, y) dx + f y (x, y) dy (This can be extended to three or more variable functions.) Example: The base radius and height of a right circular cone are measured as 10 cm and 25 cm, respectively, with a possible error in measurement of as much as 0.1 cm in each. Estimate the maximum error in the calculated volume of the cone.

11 Sensitivity to Change: Example The volume V = πr 2 h of a right circular cylinder is to be calculated from measured values of r and h. Suppose that r is measured with an error of no more than 2% and h with an error of no more than 0.5%. Estimate the resulting possible percentage error in the calculation of V.