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    D DX X X D DX X T D DX T: Antiderivative of A Function

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    Cherry Pye Terrado
    Here are the answers to the matching activity: 1. b 2. a 3. d 4. c 5. e 6. f

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    D DX X X D DX X T D DX T: Antiderivative of A Function

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    Cherry Pye Terrado
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    Here are the answers to the matching activity: 1. b 2. a 3. d 4. c 5. e 6. f

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    Here are the answers to the matching activity: 1. b 2. a 3. d 4. c 5. e 6. f

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    © All Rights Reserved
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    Download as docx, pdf, or txt
    70 views1 page

    D DX X X D DX X T D DX T: Antiderivative of A Function

    Uploaded by

    Cherry Pye Terrado
    Here are the answers to the matching activity: 1. b 2. a 3. d 4. c 5. e 6. f

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    of 1

    Name:_____________________________________Grade & Section: _______________ Score:____________

    School:____________________________________Teacher:_____________________Subject:Basic Calculus
    LAS Writers: _AR JAY S. FRANCO_______ Content Editor ___Johannah Y. Achurra_____________
    Lesson Topic: Illustration of an Antiderivative of a Function (Quarter 4 Wk. 1 LAS 1)
    Learning Targets:Illustrate an antiderivative of a function (STEM_BC11I-IVa-1).
    Reference(s): Larson/Hostetler.1987. Brief Calculus with Applications. Alternate Second Edition. D.C.
    Health and Company,pp.318-319.
    Antiderivative of a Function

    Suppose we are given the following derivatives, then we will find their original functions. If we make some educated guess,
    we might come up with the following functions:

    d
     f’(x) = 2 f(x) = 2x because [2x] = 2
    dx
    d
     g’(x) = 3 x 2 g(x) = x 3 because [ x ¿¿ 3]¿ = 3 x 2
    dx
    d
     s’(x) = 4s s(x) = 2t 2 because ¿2t 2 ¿ = 4t
    dx

    This operation of determining the original function from its derivative is the inverse operation of differentiation, and we call it
    antidifferentiation.

    Definition of an antiderivative. A function F is called an antiderivative of a function f if for every x in the


    domain of f, F’(x) = f(x)
    If F(x) is an antiderivative of f(x), then F(x) + C, where C is any constant, is also an antiderivative of f(x). For instance

    F(x) = x 3 G(x) = x 3−5 H(x) = x 3+ 0.3

    are all antiderivatives of 3 x 2 , because


    d d d 3
    ¿ ¿] = [ x 3−5] = [ x + 0.3] = 3 x 2
    dx dx dx

    Note: As it turns out, all the antiderivatives of 3 x 2 are in the form x 3 + C. The process of antidifferentiation does not determine a
    unique function but rather it determines a family of functions, each differing from the others by a constant.

    Notation for Antiderivatives


     The antidifferentiation process is also called integration and is denoted by the symbol ∫ called an integral sign.
     The symbol ∫ f ( x ) dx is called the indefinite integralof f(x), and it denotes the family of antiderivatives of f(x). That is, if
    F’(x) = f(x) for all x, then ∫ f ( x ) dx = F(x) + C where f(x) is called the integrand and C is the constant of integration.

    Definition of integral notation for antiderivatives. The notation ∫ f ( x ) dx = F(x) + C


    where C is an arbitrary constant, means that F is an antiderivative of f. That is, F’(x) = f(x) for all x in domain of f.

    Activity

    Match the functions in Column A with their corresponding antiderivatives in Column B.

    Column A Column B

    ____1. f(x) = 3x2 + 2x + 1 a. F(x) = 3x3 − x


    ____2. f(x) = 9x2 – 1 b. F(x) = x3 + x2 + x
    1
    ____3. f(x) = x2 – 2 c. F(x) = 2x2 − x3
    3
    1
    ____4. f(x) = (x + 1)(x − 1) d. F(x) = -2x2 + x3
    3
    1
    ____5. f(x) = x(4 − x) e. x3 -2x +1
    3
    1 3
    ____6. f(x) = x(x − 4) f. x -x +1
    3

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