Partial derivatives and their application.

Transcription

1 Math 2080 Week 10 Page 1 Gentry Publishing Chapter 10 Partial derivatives and their application Partial Derivatives 10.2 Tangent Planes and slopes of surfaces Linear approximations and the differential of F(x, y) Linear Stability analysis of multivariate dynamical systems.

2 Math 2080 Week 10 Page 2 Gentry Publishing The first partial derivative of z = F(x, y) with respect to x is M Mx F(x, y) = lim h60 F(x%h, y)&f(x, y) h provided the limit exists. This partial derivative is also denoted by and M z Mx z x F x (x, y) or, simply F x.

3 Math 2080 Week 10 Page 3 Gentry Publishing The first partial derivative of F(x, y) with respect to y is M My F(x, y) = lim h60 F(x, y%h)&f(x, y) h provided the limit exists. This partial derivative is also denoted by and M z My, z y, F y (x, y) or simply F y.

4 Math 2080 Week 10 Page 4 Gentry Publishing The four second partial derivatives of z = F(x, y) are : The second partial derivative of F with respect to x: M 2 M 2 /Mx 2 F(x, y) = F(x, y) = F xx (x, y) Mx 2 M Mx M Mx M Mx / F x (x, y) = F(x, y)

5 Math 2080 Week 10 Page 5 Gentry Publishing The second partial derivative of F with respect to y: M 2 M 2 /My 2 F(x, y) = F(x, y) = F yy (x, y) My 2 M My M My M My / F y (x, y) = F(x, y)

6 Math 2080 Week 10 Page 6 Gentry Publishing The (mixed) second partial derivative of F with respect to x and then y: M 2 M 2 /MyMx F(x, y) = F(x, y) = F xy (x, y) M My MyMx M My M Mx / F x (x, y) = F(x, y) The (mixed) second partial derivative of F with respect to y and then x: M 2 M 2 /MxMy F(x, y) = F(x, y) = F yx (x, y) MxMy M Mx M Mx M My / F y (x, y) = F(x, y)

7 Math 2080 Week 10 Page 7 Gentry Publishing Problem Determine all first and second order partial derivatives of the given function. (a) F(x, y) = x 3 + x 2 y - y 4 (b) F(x, y) = x sin(y)

8 Math 2080 Week 10 Page 8 Gentry Publishing (c) F(x, y) = x ln(y 2 ) - e yx

9 Math 2080 Week 10 Page 9 Gentry Publishing EQUALITY OF MIXED PARTIAL DERIVATIVES. If the partial derivatives F xy and F yx are continuous in a circular region about a point (x, y) then F xy (x, y) = F yx (x, y).

10 Math 2080 Week 10 Page 10 Gentry Publishing These concepts extends to functions of more than two variables, e.g.. If G(x,y,z) = xy -yz 2 then G x = G z = G zy = G zx =

11 Math 2080 Week 10 Page 11 Gentry Publishing For a function G of x, y, and z, there are 24 mixed partial derivatives that differentiate twice with respect to x, twice with respect to y and once with respect to z: G xxyyz G xxyzy G xxzyy G xzxyy G zxxyy G xyxyz G xyxzy G xyzxy G xzyxy G zxyxy G yyxxz G yyxzx G yyzxx G yzyxx G zyyxx G yxyxz G yxyzx G yxzyx G yzxyx G zyxyx G yxxyz G yxxzy G yxzxy G yzxxy G zyxxy G xyyxz G xyyzx G xyzyx G xzyyx G zxyyx If G is smooth enough, all of these will have the same value. What does this mean? Eg. if G(x,y,z) = x sin(y) + xe -zy - x%ln(x%z 2 ) x z &2x 2 %cos(x/z) y which way would you evaluate G zxyyx?

12 Math 2080 Week 10 Page 12 Gentry Publishing Planes in 3-dimensional space. Linear FUNCTION EQUATIONS 2-Dimensional Line: y = mx + b 3-Dimensional Plane: z = Ax + By +C more generally: 4-dimensions: Hyper-space In N-dimensions, a linear surface is an N-1 dimensional Hyper space. N-dimensional vector:

13 Math 2080 Week 10 Page 13 Gentry Publishing POINT-SLOPE EQUATIONS 2-D Line: y = y 0 + m(x - x 0 ) y x 3-D Plane : z = z 0 + A(x - x 0 ) + B(y - y 0 )

14 Math 2080 Week 10 Page 14 Gentry Publishing 2-D Line: INTERCEPT EQUATIONS y x a % y b ' x

15 Math 2080 Week 10 Page 15 Gentry Publishing 3-D Plane: x a % y b % z c ' 1 z x y X/4 + y/2 + z/3 = 1

16 Math 2080 Week 10 Page 16 Gentry Publishing Planes parallel to two of the coordinate axes. Each pair of coordinate axes determine a plane, e.g., the x-y plane, the y-z plane. These planes are characterized by the fact that the third coordinated is zero at each point in the plane. The x-y plane is The x-z plane is The y-z plane is

17 Math 2080 Week 10 Page 17 Gentry Publishing More generally, planes that are parallel to one of these two-axis planes are characterized by the property that one of their coordinates is constant. { (x, y, z) } This plane is then parallel to the plane formed from the two non-constant coordinate axes.

18 Math 2080 Week 10 Page 18 Gentry Publishing Parallel Lines L 1 : y = y 0 + m(x - x 0 ) Slope: Point: Two lines are parallel if. A line parallel to L 1 is L 2 : y = What if the slope is m = P/Q? Then we could write the line L 1 as (y - y 0 ) - P/Q (x - x 0 ) = 0 or (x - x 0 ) (y - y 0 ) = 0

19 Math 2080 Week 10 Page 19 Gentry Publishing y x grid axis -6 to 6

20 Math 2080 Week 10 Page 20 Gentry Publishing What is a vector? What is the (position) vector v = (Q, P)? What is the relationship of the vector v = (Q, P) to the line L 1? y x

21 Math 2080 Week 10 Page 21 Gentry Publishing (Parametric) Vector Equation of a Line L 1 = { (x, y) (x, y) = (x 0, y 0 ) + t(q, P) for t 0 (-4, 4) }

22 Math 2080 Week 10 Page 22 Gentry Publishing Perpendicular Lines Two lines are perpendicular if What is the slope of the line perpendicular to the line L 1? Slope of :L 1 : m = Slope of perpendicular line L 1-N : 8m = Equation of perpendicular line: Do you know another word that in mathematics means perpendicular? The line L 1-N is to the line L 1 at the point (x 0, y 0 ).

23 Math 2080 Week 10 Page 23 Gentry Publishing Dot Products The dot product or inner product of two vectors is the sum of the product of their respective components: Let V = (v 1, v 2 ) U = (u 1, u 2 ) the dot product of V and U is VCU / v 1 u 1 + v 2 u 2 If V = (2, -3) U = (4, 5) then VCU = In 3-dimensions, if N = (2, 3, 5) and T = (-1, 7, 4) then NCT =

24 Math 2080 Week 10 Page 24 Gentry Publishing The dot-product of two vectors can also be expressed as ucv = u v cos(θ) where θ is the angle between the two vectors, and u is the length of the vector u: (u u 2 2 ) ½ O- v_ U_ Notice that since -1 # cos(θ) # 1 we have - u v # ucv # u v

25 Math 2080 Week 10 Page 25 Gentry Publishing Geometrically, the dot product is the product of the length of the vector u and the length of the projection of the vector v onto u this is how much v goes in the direction of u In the diagram at the right, the length A is v, and C is the length of the projection of v onto u, A O- C Pv v_ B U_ This is the distance from the common vertex to the point Pv which is given as C = v cos(θ) since cos(θ) = C/A.

26 Math 2080 Week 10 Page 26 Gentry Publishing What is a vector perpendicular to a given vector v = (v 1, v 2 )? N = (, ) If two vectors are perpendicular what is their dot product? If N is perpendicular to v then N C v =

27 Math 2080 Week 10 Page 27 Gentry Publishing Returning to the line L 1 = { (x, y) (x, y) = (x 0, y 0 ) + t(q, P) for t 0 (-4, 4) } What is the vector N = (P, -Q)? What is the relationship of the vector N = (P, -Q) to the line L 1? y x

28 Math 2080 Week 10 Page 28 Gentry Publishing The Normal Equation of the line through the point (x 0, y 0 ) perpendicular to the vector N = (P, -Q): L 1 = {(x, y) [(x, y) - (x 0, y 0 )] C (P, -Q) = 0} or, without using the set notation L 1 : (x - x 0, y - y 0 ) C(P, -Q) = 0 or L 1 : P(x - x 0 ) - Q(y - y 0 ) = 0

29 Math 2080 Week 10 Page 29 Gentry Publishing Exercises. What is a point on the given line and what is a normal vector to the line? (a) 2(x - 3) + 3(y + 2)= 0 Point: (x 0,y 0 ) = Normal vector: N = (b) y = 4x+ 5 Point: (x 0,y 0 ) = Normal vector: N = (c) x/2 + y/3 = 1 Point: (x 0,y 0 ) = Normal vector: N =

30 Math 2080 Week 10 Page 30 Gentry Publishing The Normal Equation of a Plane through the point (x 0, y 0, z 0 ) perpendicular to the vector (n x, n y, n z ): P 1 = {(x, y, z) [(x, y, z) - (x 0, y 0, z 0 )] C (n x, n y, n z ) = 0} or, without the set notation P 1 : [(x, y, z) - (x 0, y 0, z 0 )] C (n x, n y, n z ) = 0 or P 1 : n x (x - x 0 ) + n y (y - y 0 ) + n z (z - z 0 ) = 0 If we expand the last equation we get the more common equation for a plane x + y + z = or, if n z 0, solving for z z = x + y +

31 Math 2080 Week 10 Page 31 Gentry Publishing What is a normal vector to the plane (d) 2(x - 3) + 3(y + 2) - (z -5) = 0 N = (e) z = 4x - 2y + 5 N = (f) x/2 + y/3 + z/5 = 1 N =

32 Math 2080 Week 10 Page 32 Gentry Publishing TANGENT Lines and SURFACES as lines or planes perpendicular to a normal vector: 2-D: Tangent line to y = f(x) at (x 0, y 0 ) is the line perpendicular to the Normal vector: (fn(x 0 ), -1) fn(x 0 )(x - x 0 ) + -1(y - y 0 ) = 0 y x

33 Math 2080 Week 10 Page 33 Gentry Publishing 3-D: Tangent Plane to z = F(x, y) at (x 0, y 0, z 0 ) is the plane perpendicular to the normal vector N = ( F x (x 0, y 0 ), F y (x 0, y 0 ), -1) F x (x 0, y 0 )(x - x 0 ) + F y (x 0, y 0 )(y - y 0 ) - 1(z - z 0 ) = 0 or z = z 0 + F x (x 0, y 0 )(x - x 0 ) + F y (x 0, y 0 )(y - y 0 )

34 Math 2080 Week 10 Page 34 Gentry Publishing TANGENT Lines and SURFACES 2-D: Tangent line to y = f(x) at (x 0, y 0 ) y y = L(x) = x 3-D: Tangent Plane to z = F(x, y) at (x 0, y 0, z 0 ) z = P(x, y) =

35 Math 2080 Week 10 Page 35 Gentry Publishing In 2-D The intersection of 2 lines is In 3-D. The intersection of two planes is Line of intersection of two planes. z y x

36 Math 2080 Week 10 Page 36 Gentry Publishing Line in three-space as the intersection of two planes. L = {(x, y, z) * z = z 0 + A 0 x + B 0 y and z = z 1 + A 1 x + B 1 y }

37 Math 2080 Week 10 Page 37 Gentry Publishing Line in three-space as the intersection of two planes. If (x 0, y 0, z 0 ) is a point on the line L which is the intersection of the planes z = z 0 + A 0 x + B 0 y and z = z 1 + A 1 x + B 1 y Then the line L is the set of points that simultaneously satisfies the point-slope equations A 0 (x - x 0 ) + B 0 (y - y 0 ) + C 0 ( z - z 0 ) = 0 and A 1 (x - x 0 ) + B 1 (y - y 0 ) + C 1 ( z - z 0 ) = 0 with C 1 = C 0 = Example: What is the line given by x= y and y = z? What is the line given by y = 2 and z = 3