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

Welcome to Scribd!

  • 5 views

    2024cal L2 Note

    Uploaded by

    potato20040927
    This document outlines a lecture on calculus for management majors. It covers topics like calculating the length of a parametric curve using Riemann sums, finding the tangent of a curve, and calculating the area under a curve. It also discusses quadric surfaces like parabolic cylinders, cylinders, ellipsoids, paraboloids, hyperbolic paraboloids, cones, and hyperboloids; and provides examples of each.

    Copyright:

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

    2024cal L2 Note

    Uploaded by

    potato20040927
    5 views24 pages
    This document outlines a lecture on calculus for management majors. It covers topics like calculating the length of a parametric curve using Riemann sums, finding the tangent of a curve, and calculating the area under a curve. It also discusses quadric surfaces like parabolic cylinders, cylinders, ellipsoids, paraboloids, hyperbolic paraboloids, cones, and hyperboloids; and provides examples of each.

    Original Description:

    Original Title

    2024Cal_L2_note

    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 outlines a lecture on calculus for management majors. It covers topics like calculating the length of a parametric curve using Riemann sums, finding the tangent of a curve, and calculating the area under a curve. It also discusses quadric surfaces like parabolic cylinders, cylinders, ellipsoids, paraboloids, hyperbolic paraboloids, cones, and hyperboloids; and provides examples of each.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    5 views24 pages

    2024cal L2 Note

    Uploaded by

    potato20040927
    This document outlines a lecture on calculus for management majors. It covers topics like calculating the length of a parametric curve using Riemann sums, finding the tangent of a curve, and calculating the area under a curve. It also discusses quadric surfaces like parabolic cylinders, cylinders, ellipsoids, paraboloids, hyperbolic paraboloids, cones, and hyperboloids; and provides examples of each.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 24

    Calculus for Management Majors

    mathematical thinking through calculus

    Jungkai Alfred Chen


    Outline

    Lecture 2
    Feb 22, 2024

    I more on calculus of parametric curves


    I length
    I tangents and area
    I quadric surfaces
    I sections
    I level curves
    Length of a Parametric Curve

    ”Approximate” by Riemann sum of length of line segments!


    Length of a Parametric Curve

    ”Approximate” by Riemann sum of length of line segments!


    Z b q
    L= dx 2 + dy 2 .
    a
    Length of a Parametric Curve

    Example (Cycloid)
    x = r (✓ sin ✓), y = r (1 cos ✓), 0  ✓  2⇡.
    Length of a Parametric Curve

    Arc length is independent of parametrization.


    Length of a Parametric Curve

    Since the length (as a function of time) is


    Z tr
    dx dy
    s(t) = ( )2 + ( )2 du.
    ↵ du du

    we have the speed


    r
    dx 2 dy
    0
    v (t) = s (t) = ( ) + ( )2 .
    du du
    Length of a Parametric Curve

    Example
    x = 2t + 5, y = 10 2t 2 , t 0. Find the speed at the point
    (7, 8).
    Tangent of Curves

    Question
    How to determine the tangent of a (parametric) curve?
    Tangent of Curves

    Given x = f (t), y = g(t), then

    dy dy dx
    = · .
    dt dx dt
    Tangent of Curves

    Example
    Find the slope of the curve x = cos t, y = sin 2t at t = ⇡/2 and
    3⇡/2 respectively.
    Tangent of Curves

    Given x = f (t), y = g(t), then

    d 2y
    =?
    dx 2
    Area Associates to a Curve

    Example
    Find the area under the curve x = cos t, y = sin 2t, 0  t  ⇡/2
    and above the x-axis.
    Examples of Quadric Surfaces

    Example
    parabolic cylinder
    z = x2
    Examples of Quadric Surfaces

    Example
    Cylinder
    x 2 + z 2 = 1 (resp. x 2 + y 2 = 1.).
    Examples of Quadric Surfaces

    Example
    Ellipsoid
    x 2 + y 2 /4 + z 2 /9 = 1.
    Examples of Quadric Surfaces

    Example
    (Elliptic) Paraboloid
    z = x 2 + y 2 (resp. z = 4x 2 + y 2 ).
    Examples of Quadric Surfaces

    Example
    Hyperbolic Paraboloid
    z = x 2 + y 2.
    Examples of Quadric Surfaces

    Example
    Cone
    z 2 = x 2 + 4y 2 .
    Examples of Quadric Surfaces

    Example
    Hyperboloid of One Sheet
    1 = x 2 + 4y 2 z 2 .
    Examples of Quadric Surfaces

    Example
    Hyperboloid of Two Sheets
    1 = x 2 4y 2 + z 2 .

    You might also like