Math 20C. Lecture Examples.

Transcription

1 Math 20C. Lecture Eamples. (8/7/0) Section 4.4. Linear approimations and tangent planes A function z = f(,) of two variables is linear if its graph in z-space is a plane. Equations of planes were found in Section 2.5 b using their normal vectors. Here we will need instead equations given in the net theorem for planes in terms of the slopes of their cross sections in the - and -directions. Theorem (a) (The slope-intercept equation of a plane) Suppose that the z-intercept of a plane is b, the slope of its vertical cross sections in the positive -direction is m, and the slope of its vertical cross sections in the positive -direction is m 2 (Figure ). Then the plane has the equation z = m + m 2 + b. () (b) (The point-slope equation of a plane) Suppose that a plane contains the point ( 0, 0, z 0 ), the slope of its vertical cross sections in the positive -direction is m, and the slope of its vertical cross sections in the positive -direction is m 2 (Figure 4). Then the plane has the equation z = z 0 + m ( 0 ) + m 2 ( 0 ). (2) The slope-intercept equation The point-slope equation FIGURE FIGURE 2 Lecture notes to accompan Section 4.4 of Calculus, Earl Transcendentals b Rogawski.

2 Math 20C. Lecture Eamples. (8/7/0) Section 4.4, p. 2 Eample Eample 2 Eample 3 Give an equation of the plane with slope 6 in the positive -direction, slope 7 in the positive -direction, and z-intercept 0. Answer: z = Give an equation of the plane through the point (, 2,3) with slope 4 in the positive -direction and slope 5 in the positive -direction. Answer: z = 3 + 4( ) 5( 2) Find a formula for the linear function z = g(, ) whose values are given in the following table. Values of z = g(, ) = 3 = 0 = 3 = = = Eample 4 Answer: g(, ) = Find a formula for the linear function z = h(, ) whose level curves are given in Figure FIGURE 3 32 Zooming in on level curves of a nonlinear z = f(,) If a function = f() of one variable has a derivative at 0 and the graph = f() is generated b a calculator or computer in a small enough window containing the point ( 0,f( 0 )), the displaed portion of the graph will look like a line. This occurs because the graph is closel approimated b the tangent line near that point. Graphs of functions of two variables with continuous first derivatives are closel approimated b planes when viewed in small windows. Consequentl, their level curves in small windows look like parallel lines. This is illustrated b the level curves of K(,) = in Figures 4 through 6. The level curves look more like equall spaced parallel lines in Figure 5 than Figure 4, and even more like equall spaced parallel lines in Figure 6.

3 Section 4.4, p. 3 Math 20C. Lecture Eamples. (8/7/0) K =, 2, 3,...,7 K = 3,3.5,...,5.5 K = 3.998,...,4.002 FIGURE 4 FIGURE 5 FIGURE 6 Eample 5 (a) The function K(,) = has the value 4 at the point P = (, ) in Figure 7 and the value at the point Q. What is the approimate value of K (, )? (b) K has also has the value at the point R in Figure 8. What is the approimate value of K (, )? (c) Find the eact derivatives in parts (a) and (b) b using the formula K(,) = Q P P R FIGURE 7 FIGURE 8 Answer: (a) K (, ) = 7.4 (b) K (, ) = 9 (c) K (, ) = 7 K = 9

4 Math 20C. Lecture Eamples. (8/7/0) Section 4.4, p. 4 Equations of tangent planes If z = f(,) has continuous first derivatives in an open circle centered at (a,b), then the linear function L(, ) = f(a,b) + f (a,b)( a) + f (a,b)( b) has the same value and the same first derivatives at (a,b) as z = f(,). Consequentl, it has the same directional derivatives in all directions at (a, b) as z = f(,) (see Section 4.5). This implies that all of the tangent lines to vertical cross sections of the graph of f through = a, = b (Figure 9) are in the plane that is the graph of L (Figure 0). That plane is the tangent plane to the graph of f at = a, = b. Tangent lines to vertical cross sections The tangent plane FIGURE 9 FIGURE 0 Definition If z = f(, ) has continuous first derivatives in an open circle centered at ( 0, 0 ), then the tangent plane to its graph at = a, = b is z = f(a,b) + f (a,b)( a) + f (a,b)( b). Eample 6 Give an equation of the tangent plane to the graph of f(,) = 3 4 at =, = 2. Answer: Tangent plane: z = ( ) + 32( 2)

5 Section 4.4, p. 5 Math 20C. Lecture Eamples. (8/7/0) Estimating errors Eample 7 The volume of a cube is determined b measuring its width. The width is measured to be 2 centimeters with an error 0.02 centimeters. Use a tangent-line approimation to estimate the maimum possible error in the calculated volume. Answer: V = w 3 Tangent line at w = 2: V = 8 + 2(w 2) Suppose w is the eact width. [Error] = V (w) 8 2(w 2) 2(0.02) = 0.2 cubic centimeters Eample 8 The radius of a right circular clinder is measured to be r = 0 ± 0.0 centimeters (meaning that it is measured to be 0 centimeters with an error 0.0) and its height is measured to be h = 5 ± centimeters (meaning that it is measured to be 5 centimeters with an error 0.005). Use a tangent-plane approimation to estimate the maimum possible error in the calculated volume of the clinder. Answer: V (r, h) = πr 2 h Tangent plane at r = 0, h = 5: V = 500π + 300π(r 0) + 00π(h 5) Suppose that r and h are the eact radius and height. [Error] = V (r, h) 500π 300π(r 0) +00π(h 5) 300π(0.0) +00π(0.005) = 3.5π cubic centimeters The maimum possible error is approimatel 3.5π. = cubic centimeters. Interactive Eamples Work the following Interactive Eamples on Shenk s web page, http// ashenk/: Section 4.6: Eamples through 7 The chapter and section numbers on Shenk s web site refer to his calculus manuscript and not to the chapters and sections of the tetbook for the course.