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11 April, 05:03

Suppose f left-parenthesis x right-parenthesis equals StartFraction 1 Over 4 EndFraction x Superscript 4. Estimate f Superscript prime Baseline left-parenthesis 2 right-parenthesis, f Superscript prime Baseline left-parenthesis 3 right-parenthesis, and f Superscript prime Baseline left-parenthesis 4 right-parenthesis. What do you notice? Guess a formula for f Superscript prime Baseline left-parenthesis x right-parenthesis

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  1. 11 April, 08:03
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    f' (x) = x/2

    Step-by-step explanation:

    Given the function

    f (x) = (1/4) x²

    f' (x) will give the first derivative of the function

    Using the differentiation formula

    Generally, if f (y) = ayⁿ

    f' (y) = nay^n-1

    Applying this to differentiate the given function,

    f' (x) = 2 (1/4) x^2-1

    f' (x) = (1/2) x

    When x = 2

    f' (2) = (1/2) * 2

    f' (2) = 1

    When x = 3

    f' (3) = (1/2) * 3

    f' (3) = 3/2

    When x = 4

    f' (4) = (1/2) * 4

    f' (4) = 2

    Based on the answers, it can be seen that the values keeps increasing arithmetically. The values 1, 1 1/2, and 2 are in arithmetic progression.

    The formula for calculating the nth term Tn of an arithmetic progression is expressed as:

    Tn = a + (n-1) d

    a is the first term of the sequence

    n is the number of the terms,

    d is the common difference

    According to the sequence

    a = 1/2, d = 3/2 - 1 = 2 - 3/2 = 1/2

    Note that we started with when x=2, the first term will be at when x = 1 which will give 1/2 hence the reason for a = 1/2 instead of 1

    Substituting the values in the formula

    Tn = 1/2 + (n-1) 1/2

    Tn = 1/2+n/2-1/2

    Tn = n/2

    Therefore the formula for f' (x) can be expressed as x/2

    f' (x) = x/2
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