Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 7 views

    10 - HW Multivariable Functions Problems

    Uploaded by

    asdf
    This document provides information about multivariable functions and level curves. It includes definitions of level curves as the projection of the intersection of a surface z=f(x,y) with a horizontal plane z=k. It also notes that a collection of level curves forms a contour map. The document contains examples of common surfaces like spheres, cylinders, cones and paraboloids. It poses multiple calculation and sketching problems involving multivariable functions and finding level curves.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    10 - HW Multivariable Functions Problems

    Uploaded by

    asdf
    7 views3 pages
    This document provides information about multivariable functions and level curves. It includes definitions of level curves as the projection of the intersection of a surface z=f(x,y) with a horizontal plane z=k. It also notes that a collection of level curves forms a contour map. The document contains examples of common surfaces like spheres, cylinders, cones and paraboloids. It poses multiple calculation and sketching problems involving multivariable functions and finding level curves.

    Original Description:

    Original Title

    10_HW Multivariable Functions Problems

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    This document provides information about multivariable functions and level curves. It includes definitions of level curves as the projection of the intersection of a surface z=f(x,y) with a horizontal plane z=k. It also notes that a collection of level curves forms a contour map. The document contains examples of common surfaces like spheres, cylinders, cones and paraboloids. It poses multiple calculation and sketching problems involving multivariable functions and finding level curves.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    7 views3 pages

    10 - HW Multivariable Functions Problems

    Uploaded by

    asdf
    This document provides information about multivariable functions and level curves. It includes definitions of level curves as the projection of the intersection of a surface z=f(x,y) with a horizontal plane z=k. It also notes that a collection of level curves forms a contour map. The document contains examples of common surfaces like spheres, cylinders, cones and paraboloids. It poses multiple calculation and sketching problems involving multivariable functions and finding level curves.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    1

    Calculus WIs ang


    Lecture 10 multivariable functions

    CIRCLE
    (x – a)2 + ( y – b)2 = R2

    x 2 + y 2= 1 y = 1− x2 y = − 1 − x2

    PARABOLA
    𝑦 = 𝑥2, 𝑠ℎ𝑖𝑓𝑡𝑒𝑑: 𝑦 − 𝑏 = (𝑥 − 𝑎)2 𝑥 = 𝑦2 , 𝑠ℎ𝑖𝑓𝑡𝑒𝑑: 𝑥 − 𝑎 = (𝑦 − 𝑏)2

    ( a, b )
    y= x
    ( 0, 0 )
    ( 0, 0 )
    ( a, b )

    HYPERBOLA

    a
    y= or
    x
    xy = a
    a
    or x =
    y

    Level curves

    The projection onto the xy-plane of the intersection of the surface


    z = f (x,y) with the horizontal plane z = k is called the level curve of
    height k.
    A collection of level curves is called a contour map of a function f (x,y)
    2

    1**. Calculate the lengths of the following vectors


    1 1 1
    a) v = (3,-1,2) b) v = (1,2,4) c) v = ( , − , ) d) v = (1, −√3, √5)
    √3 √3 √3

    2**. Find v ⋅ w , |v| , |w| and the cosine of the angle between v and w , which of the pairs
    − − − −

    of vectors are perpendicular (orthogonal)?


    a) v = (3,-1,2) i w = (4,2,-5) b) v = (1,2,-7) i w = (1,3,1)
    c) v = (0,1,2,1) i w = (1,2,5,0)
    3**. Determine all the vectors which are perpendicular (orthogonal) to the vector v.
    a) v = (1,2) b) v = ( 1,2,3) c) v = (1,0,-6).
    4**. Determine the equation of the plane perpendicular to the vector v = [ 2,3,4] and passing
    through point P(5,6,7) .

    5. Sketch and describe the following sets in R2:

    𝑎) 𝑦 ≤ 2𝑥 + 1 𝑏) 𝑥 + 𝑦 < 3 ∧ 𝑥2 + 𝑦2 ≥ 1 𝑐) 1 ≤ 𝑥 2 + 𝑦 2 ≤ 4 ∧ −𝑥 ≤ 𝑦 ≤ 𝑥
    𝑑∗ ) 𝑥 2 + 4𝑦 2 ≤ 4 𝑒) 𝑥 2 − 4𝑥 + 𝑦 2 ≥ 5 𝑓) 𝑥 ⋅ 𝑦 > 2 𝑔) 𝑦 2 < 4𝑥 ℎ) 𝑥 2 < 4𝑦
    𝑖) 𝑦 ≥ (𝑥 − 1)(𝑥 + 2)
    6. Find the natural domain of the following functions, additionally find the range of the
    functions from b), f ), g);

    𝑎) 𝑓(𝑥, 𝑦) = ln(𝑥 ⋅ 𝑦) ⋅ √4 − 𝑥 2 − 𝑦 2 𝑏)∗ 𝑓(𝑥, 𝑦) = arcsin(2𝑦 − 𝑥 2 )


    1
    𝑐) 𝑓(𝑥, 𝑦) = ln(2𝑥 − 𝑥 2 − 𝑦 2 ) 𝑑) 𝑓(𝑥, 𝑦) =
    𝑥2 − 𝑦2
    1
    𝑒) 𝑓(𝑥, 𝑦) = ln(2𝑥 + 4𝑦 − 𝑥 2 − 𝑦 2 + 4) 𝑓) 𝑓(𝑥, 𝑦) =
    √𝑥 ⋅ 𝑦
    ln(2𝑥 2 + 2𝑦 2 − 8)
    𝑔) 𝑓(𝑥, 𝑦) = 𝑒𝑥(𝑥−𝑦) ℎ) 𝑓(𝑥, 𝑦) =
    √𝑥 2 + 2 − 𝑦
    𝑥
    𝑖) 𝑓(𝑥, 𝑦) = √𝑥 ln(𝑥 2 + 𝑦 2 − 3) 𝑗) 𝑓(𝑥, 𝑦) = arccos
    𝑥+𝑦
    𝑘) 𝑓(𝑥, 𝑦) = 𝑥 𝑦 ; 𝑒𝑥𝑝𝑙𝑎𝑖𝑛 𝑤ℎ𝑎𝑡 ℎ𝑎𝑝𝑝𝑒𝑛𝑠 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 (0,0)

    7. Find three level curves and plot the contour map (remember a level curve is the cross-
    section with plane z = k) of the following functions. Can you find a sphere, cylinder, cone
    paraboloid among these surfaces?
    𝑥2 𝑦2
    𝑎) 𝑓(𝑥, 𝑦) = 𝑧 = √16 − 𝑥 2 𝑏)∗ 𝑓(𝑥, 𝑦) = 𝑧 = 4𝑥 2 + 9𝑦 2 𝑐) 𝑓(𝑥, 𝑦) = √1 − −
    4 4
    𝑑) 𝑓(𝑥, 𝑦) = √36 − 9𝑥 2 − 9𝑦 2 𝑒) 𝑓(𝑥, 𝑦) = 2 + 3√𝑥 2 + 𝑦 2

    𝑓) 𝑓(𝑥, 𝑦) = 2 − 2𝑥 2 − 2𝑦 2 𝑔) 𝑓(𝑥, 𝑦) = √2 − 2𝑥 2 − 2𝑦 2
    3

    ℎ∗ ) 𝑓(𝑥, 𝑦) = √16 − 𝑥 2 − 16𝑦 2 𝑖) 𝑓(𝑥, 𝑦) = −√9 − (𝑥 − 1)2 − (𝑦 + 2)2 .


    𝑥
    𝑗) 𝑓(𝑥, 𝑦) = 𝑘) 𝑓(𝑥, 𝑦) = 𝑀𝐼𝑁(𝑥, 𝑦) 𝑙) 𝑓(𝑥, 𝑦) = 𝑀𝐴𝑋(𝑥, 𝑦)
    𝑦
    𝑥
    𝑚) 𝑓(𝑥, 𝑦) = 𝑛) 𝑓(𝑥, 𝑦) = 𝑥𝑦 𝑜) 𝑓(𝑥, 𝑦) = 𝑥 + 𝑦
    𝑦
    𝑝) 𝑓(𝑥, 𝑦) = 3𝑥 2 + 3𝑦 − 12 𝑞) 𝑓(𝑥, 𝑦) = −𝑥 2 + 𝑦 − 4 𝑟) 𝑓(𝑥, 𝑦) = 𝑥 2 + 𝑦 2 − 5
    1
    𝑠) 𝑓(𝑥, 𝑦) = √𝑥 2 − 2𝑥 + 𝑦 2 + 4𝑦 − 4 𝑡) 𝑓(𝑥, 𝑦) = 2 − .
    √𝑥 2 + 𝑦 2 − 4
    1
    𝑡) 𝑓(𝑥, 𝑦) = 2 −
    √4 − 𝑥 2 − 𝑦 2

    8. Find the level curve passing through the given point

    𝑎) 𝑓(𝑥, 𝑦) = 𝑧 = √16 − 𝑥 2 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑃(3,4)


    𝑏) 𝑓(𝑥, 𝑦) = 2 − 2𝑥 2 − 2𝑦 2 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑃(1, 3)
    𝑐) 𝑓(𝑥, 𝑦) = 𝑀𝐼𝑁(𝑥, 𝑦) 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑃( −1,4)
    1
    𝑑) 𝑓(𝑥, 𝑦) = 𝑥𝑦 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑃(4, )
    8
    𝑒) 𝑓(𝑥, 𝑦) = ln(𝑥 2 − 3𝑥 − 𝑦) 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑃(4,1)

    9*. Find three level curves (remember a level curve is the cross-section with z = k) and plot the
    contour map of the following surfaces. Where possible identify the surfaces?

    𝑎) 𝑥 2 + 𝑦 2 + 𝑧 2 = 4 𝑏) (𝑥 − 1)2 + (𝑦 + 2)2 + (𝑧 − 3)2 = 4 𝑐) 𝑧 = −𝑥 2 − 𝑦 2 + 4


    𝑑 ∗ ) 𝑧 = 𝑥𝑦 𝑒 ∗ ) 𝑧 = 2𝑥 2 + 𝑦 2 𝑓) 𝑥 2 + 𝑧 2 = 9 𝑔) 𝑧 = 𝑦 2 ℎ) 𝑧 = 𝑥 2 𝑖) 𝑥 2 + 𝑦 2 = 1

    You might also like