Given f”(x) = – 16 sin(4x) and f'(0) = – 4 and f(0) = – 2. Findf(1) =

Question

Given f”(x) = – 16 sin(4x) and f'(0) = – 4 and f(0) = – 2.

Findf(1) =

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Kim Cúc 5 years 2021-08-10T14:15:36+00:00 1 Answers 17 views 0

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  1. danthu
    0
    2021-08-10T14:16:47+00:00
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    Answer:

    The complete equation is f(x) = \sin 4x -8\cdot x -2. f(1) = -10.756

    Step-by-step explanation:

    Let be f''(x) = -16\cdot \sin 4x, we need to determine the formula of f(x) by integrating twice:

    f'(x) = \int {(-16\cdot \sin 4x)} \, dx

    f'(x) = -16\int {\sin 4x} \, dx

    We apply the following algebraic substitution in expression above:

    u = 4\cdot x and du = 4\,dx

    f'(u) = -4\int {\sin u} \, du

    f'(u) = 4\cdot \cos u + C_{1}

    f'(x) = 4\cdot \cos 4x + C_{1}

    We use the same approach to determine f(x):

    f(x) = \int {(4\cdot \cos 4x)} \, dx + \int {C_{1}} \, dx

    f(u, x) = \int {\cos u} \, du + C_{1}\int \, dx

    f(u,x) = \sin u + C_{1}\cdot x + C_{2}

    f(x) = \sin 4x + C_{1}\cdot x + C_{2}

    If we know that f'(0) = -4 and f(0) = -2, the integration constants are obtained below:

    4 + C_{1} = -4

    C_{1} = -8

    C_{2} = -2

    The complete equation is f(x) = \sin 4x -8\cdot x -2. (Angles are measured in radians) Then:

    f(1) = \sin 4 - 8- 2

    f(1) = -0.756-8-2

    f(1) = -10.756

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