We will be sketching 3-dimensional functions. You will be responsible for doing this both by hand and with Mathematica.

Transcription

1 Review polar coordinates before 9.7. Section 9.6 Functions and Surfaces We will be sketching 3-dimensional functions. You will be responsible for doing this both b hand and with Mathematica. Remember: Function notation in -dimensions = f x ( ) is a function of x x, ordered pairs ( ) x independent variable (Domain), dependent variable (Range). Now: Functions in 3-dimensions z = f x, ( ) z is a function of x and x, z, ordered triplets ( ) x and are the independent variables (Domain), z is the dependent variable (Range). In -dimensions, the Domain was written something like this: { x x 3} (this is just an example) In 3-dimensions, the Domain is written something like this: {( x, ) x 1 0, x 1} + + (another example) the domain is an area located in the x-plane In -dimensions In 3-dimensions, f ( x ) Figure 3 near ex. 4 The blue, S part is the function The tan part is the Domain. The function part lies directl above or below its domain. D is an area. x domain is the set of all inputs or x-values range is the set of all outputs or -values or f(x) values

2 Figure 4 shows a ver basic function a linear function in 3-dim., a plane. All variables raised to the first power. (, ) f x = ax + b + c 1443 ( ) f x, = 6 3x 1443 or z = ax+ b+ c z = 6 3x 3x+ + z 6= 0 Recognize the equation of a plane we know? ax + b + cz + d = 0 Notice x,, and z are first degree. No x here. What s the normal vector, n r, to the plane? r n = The domain of a plane is all real numbers. BUT IF we restrict the surface to the first octant onl, D, is the new domain of part of our restricted plane. The region in the x-plane. 3 When x=z=0, =3 and when =z=0, x=. Notice it s an area. Surface S is the plane ling directl above domain D. So choosing all of the points in D would go (up or down) and hit our restricted plane. D x 1 st ex. (, ) = ln( + 1) z = ln ( x+ 1) f x x Do ou remember f ( x) = lnx? a. Evaluate f ( 1,1) b. Evaluate f ( e,1) c. Find and sketch the Domain of f. Since it is a natural log fn., are there an restrictions on x+-1? D : x,? f {( ) } d. Find the Range of f. So once we plug in the appropriate x and -values, what will the natural log of all of that give ou? R : f x,? f { ( ) } { z }

3 Quadratic Surface is the graph of a second degree equations in 3 variables. Right before the HW Exercise is Table (tab this page, ou ll return to it throughout the semester) These are analogous to our conic sections from the past in -dimensions ellipse, hperbola, parabola. These will be centered at the origin. Remember: x + = 1 ellipse a and b stretch the ellipse out along the x and axes respectivel a b Notice the 1 on the right side and the plus sign and the x and. x a 1 = hperbola opens right & left b x 1 If =, it opens up and down b a b Notice the 1 on the right side and the minus sign and the x and. a b a = x tpe of parabola opens up and if it s negative opens down The x = opens right and if negative opens left After Ex.5 To start to sketch graphs of functions of two variables, we find the shapes of cross-sections (slices). To find these shapes, we keep one variable fixed b putting in a constant, k, for that variable and then focus on that equation and figure out what that famil of curves is. There are two vertical slices (one parallel to the xz-plane and the other parallel to the z-plane) and one horizontal slice (parallel to the x-plane; aka the floor ).

4 To find the slices parallel to the z-plane we set x equal to the constant. x = k To find the slices parallel to the xz-plane we set equal to the constant. = k To find the slices parallel to the x-plane we set z equal to the constant. z = k Ellipsoids A general ellipsoid looks like x z + + = 1 a b c notice all + signs, x,, and z are all squared Discuss slices b looking at a picture first. TEC 9.6b Show how changing a, b, and c affect it. All traces are ellipses. Side Note: If x z a= b= c like + + = 1, would this be an ellipsoid? Ellipsoid Example: x z + + = To graph it we look at the TRACES (cross-sections) in the coordinate planes. k = constant Traces in x = k (parallel to the z-plane) Traces in = k (parallel to the xz-plane)

5 Traces in z = k (parallel to the x-plane) Elliptic Paraboloid x = + z c a b Notice, z but x. + sign between squared terms. Discuss slices b looking at a picture first. TEC 9. 6a, the third one, C, is an elliptic paraboloid. Show how changing a, b, and c affect it. Slider a shows slices parallel to the z-plane are all ellipses. Slider b in blue, planes trace out parabolas parallel to xz-plane. Slider c in white, traces out parabolas, parallel to the x-plane. Let s look at the traces: Traces in x = k (z-plane)

6 Traces in = k (xz-plane) Traces in z = k (x-plane) Hperboloid of One Sheet + = 1 x z a b c Notice sign in front of the z, and that the x, and z are all squared. Example: x + z = 1 (parallel to the z-plane) Traces in x = k, x + z = 1 k + z = 1 z = 1 k Notice the minus sign and all variables to the second power. These traces parallel to the z-plane are a famil of (parallel to the xz-plane) Traces in = k, x + z = 1 x + k z = 1 x z = 1 k Notice the minus sign and all variables to the second power. So these traces parallel to the xz-plane are a famil of

7 (parallel to the x-plane) Traces in z = k, x + z = 1 x + k = 1 x + = 1+ k Notice the + sign and all variables to the second power. These traces parallel to the x-plane are a famil of You have this TEC (Tools for Enriching Calculus) CD, so look at some slices on 9.6a. You should be able to see families of hperbolas parallel to the two vertical planes opening up/down and right/left depending on the constant k values. See the famil of circles in the horizontal planes. On the TEC CD, look at some slices on 9.6a. Look at the Quadric Surfaces 9.6b. Homework Hint #1, : You ll need to complete the square to get it into a recognizable form. Just like the did on Example 9 in the book: This is just an extra example for them. Do not go over in class. Hperbolic Paraboloid TEC 9.6a, the second one. or called a Saddle z = x c a b Notice the minus sign. Ver similar to elliptic paraboloid except for this minus sign. Example: z = x x = k z plane z = k z = + k Notice z and. Traces in ( ) The traces parallel to the z-plane are a famil of parabolas that open upside down.

8 Traces in = k ( xz plane) Notice z and x. z = x k z = x k The traces parallel to the xz-plane are a famil of parabolas that open up. Traces in z = k ( x plane) k = x x = k Notice the minus sign and both x and are squared. The traces parallel to the x-plane are a famil of hperbolas.