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    Sketch The Domain of Functions

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    Pedro
    The document provides examples of functions and their domains, evaluating functions at given values, sketching graphs of functions, and solving probability problems involving probability density functions. It contains 5 sections that demonstrate: 1) sketching domains of functions, 2) evaluating functions, 3) sketching graphs, 4) determining properties of a probability density function such as its mean and median, and 5) calculating a probability using a given density function.

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    © All Rights Reserved
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    Sketch The Domain of Functions

    Uploaded by

    Pedro
    11 views3 pages
    The document provides examples of functions and their domains, evaluating functions at given values, sketching graphs of functions, and solving probability problems involving probability density functions. It contains 5 sections that demonstrate: 1) sketching domains of functions, 2) evaluating functions, 3) sketching graphs, 4) determining properties of a probability density function such as its mean and median, and 5) calculating a probability using a given density function.

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    Sketch the Domain of functions

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    The document provides examples of functions and their domains, evaluating functions at given values, sketching graphs of functions, and solving probability problems involving probability density functions. It contains 5 sections that demonstrate: 1) sketching domains of functions, 2) evaluating functions, 3) sketching graphs, 4) determining properties of a probability density function such as its mean and median, and 5) calculating a probability using a given density function.

    Copyright:

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

    Sketch The Domain of Functions

    Uploaded by

    Pedro
    The document provides examples of functions and their domains, evaluating functions at given values, sketching graphs of functions, and solving probability problems involving probability density functions. It contains 5 sections that demonstrate: 1) sketching domains of functions, 2) evaluating functions, 3) sketching graphs, 4) determining properties of a probability density function such as its mean and median, and 5) calculating a probability using a given density function.

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    © All Rights Reserved
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    You are on page 1of 3

    1. Find and sketch the domain of the following functions.

    √
    (a) f (x, y) = x + y + 2

    Solution: We need x+y +2 ≥ 0, which is drawn below. The boundary is the line x+y +2 = 0.
    -1 -0.5 0 0.5 1
    -1

    -1.5

    -2

    -2.5

    -3

    √ √
    (b) f (x, y) = x+ y−3

    Solution: We need x ≥ 0 and y − 3 ≥ 0.


    5

    4.5

    3.5

    2.5

    2
    -1 -0.5 0 0.5 1 1.5 2

    xy
    (c) f (x, y) = x+y−1

    Solution: We need x + y − 1 6= 0. The line x + y − 1 = 0 is drawn below; the domain of f is


    everything else in the plane.
    1

    0.5

    -0.5

    -1
    -1 -0.5 0 0.5 1
    2
    (d) f (x, y) = xey +y+1

    Solution: The domain is the whole plane.

    (e) f (x, y) = ln(1 − x2 − y 2 )

    Solution: We need 1 − x2 − y 2 > 0, or equivalently, 1 > x2 + y 2 . This is the interior of the


    unit circle.
    2

    1.5

    0.5

    -2 -1.5 -1 -0.5 0.5 1 1.5 2


    -0.5

    -1

    -1.5

    -2

    2. Find the values of the function


    (a) f (x, y) = x2 − 4xy − y 2
    i. f (3, 2)

    Solution: f (3, 2) = 19

    ii. f (x, x1 ) f (x, x1 ) = x2 − 4 − 1


    x2
    p
    (b) f (x, y, z) = x2 + y 2 + x2
    i. f (1, 2, 3)
    √
    Solution: f (1, 2, 3) = 14

    ii. f (2, 4, 6)
    √
    Solution: f (2, 4, 6) = 2 14

    3. Sketch the graph of the following functions.


    (a) f (x, y) = 8 + x
    (b) f (x, y) = x + y
    (c) f (x, y) = x2 + y = 2
    p
    (d) f (x, y) = 5 − x2 − y 2 = 1
    2
    +2x+1+y 2
    (e) f (x, y) = ex

    Solution: Searching “3D plotter” will yield many options for graphing these.

    x3
    4. Let f (x) = C (10 − x) for 0 ≤ x ≤ 10 where C is a constant, and f (x) = 0 for all other values of x.

    Page 2
    (a) if f (x) is a probability density function, determine the value of C.

    Solution:
    ∞ 10 10
    x3
    Z Z Z
    1= f (x) dx = (10 − x) dx =⇒ C = x3 (10 − x) dx = 5000
    −∞ 0 C 0

    The function you are integrating is a polynomial.

    (b) For that value of C, find P (1 < X < 4) and P (X > 6) where X is the random variable associated
    to f (x).

    Solution:
    4
    x3
    Z
    4329
    P (1 < X < 4) = (10 − x) dx =
    1 5000 50000
    10
    x3
    Z
    2072
    P (X > 6) = (10 − x) dx =
    6 5000 3125

    (c) Find the mean and median of X given above.

    Solution: The mean is


    10 10
    x4
    Z Z
    20
    x · f (x) dx = (10 − x) dx = .
    0 0 5000 3
    Rm
    The median is the number m such that 0
    f (x) dx = 1/2. To find it:
    m
    x3
    Z    
    1 1 5 4 1 5 m 1 5 4 1 5
    = (10 − x) dx = x − x = m − m
    2 0 5000 5000 2 5 0 5000 2 5

    It’s not obvious to me how you are expected to solve this equation, but the unique solution in
    the inteval [0, 10] is approximately m ≈ 6.86.

    5. Given the probability density function f (x) = xe−x for x ≥ 0 and f (x) = 0 for all other values of x.
    Find the probability P (X ≥ 1).

    Solution: Z ∞
    2
    P (X ≥ 1) = f (x) dx =
    1 e
    You can do this integral using integration by parts with u = x and dv = e−x dx.

    Page 3

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