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    Define Bounded Area Formula

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    Define A(x) to be the area bounded by the t-axis, the line y = 2t and a vertical line at t = x.
    (a) Find a formula for A(x).
    (b) Determine A'(x)

    The figure below shows the graph of the derivative of a continuous function f .
    (a) List the critical numbers of f .
    (b) What values of x result in a local maximum?
    (c) What values of x result in a local minimum?

    Use information from the derivative of each function to help you graph the function. Find all local maximums and minimums of each function.

    g(x)=2x^3-〖15x〗^2+6

    In 4 and 5, a function and values of x so that f '(x) = 0 are given. Use the Second Derivative Test to determine whether each point (x, f (x)) is a local maximum, a local minimum or neither.
    h(x)=x^4-8x^2-2; x=-2,0,2

    f(x)=x∙In(x); x=1/e

    Lest you have forgotten, the formulas you will need are as follows:
    For a right circular cylinder of radius "r" and height "h":
    The volume V = πr2h.
    The surface area S = circular ends plus the cylindrical wall = 2πr2 + 2πrh.

    You have been asked to design a one-liter (i.e., 1000 cm3) can shaped like a right circular cylinder (figure below). What dimensions will use the least material?

    So the question is: What should "r" be and what should "h" be such that the volume is 1 liter but the surface area is least?

    © BrainMass Inc. brainmass.com December 24, 2021, 11:53 pm ad1c9bdddf
    https://brainmass.com/math/functional-analysis/define-bounded-area-formula-614461

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    SOLUTION This solution is FREE courtesy of BrainMass!

    1. Define A(x) to be the area bounded by the t-axis, the line y = 2t and a vertical line at t = x.
    (a) Find a formula for A(x).
    (b) Determine (x)

    Solution:

    (a)

    (b)

    2. The figure below shows the graph of the derivative of a continuous function f .
    (a) List the critical numbers of f .
    (b) What values of x result in a local maximum?
    (c) What values of x result in a local minimum?

    Solution:

    (a)
    Critical numbers are x = 2, 7

    (b)
    Function has local maximum at x = 7.

    (c)
    Function has local minimum at x = 2.

    3. Use information from the derivative of each function to help you graph the function. Find all local maximums and minimums of each function.

    Solution:

    To find critical points, put g'(x) = 0

    Critical points are x = 0, 5

    Since g"(0) < 0 so g(x) will have local maximum at x = 0.
    Since g"(5) > 0 so g(x) will have local minimum at x = 5.

    Interval Test value Value of f'(x) at test value Sign of f'(x)
    (-∞,0) -1
    positive
    (0,5) 1
    negative
    (5,∞) 6
    Positive

    Since value of g'(x) is positive in the intervals (-∞,0) and (5,∞) and value of g'(x) is negative in the interval (0,5). Thus, function is increasing in the intervals (-∞,0) and (5,∞)and decreasing in the interval (0,5).

    Find g''(x)

    To find inflection points, put g''(x) = 0

    12x-30 = 0
    12x = 30
    x = 30/12 = 5/2

    Test value Value of f''(x) at test value Sign of f''(x)
    (-∞,5/2) 0
    negative
    (5/2,∞) 3
    Positive

    Since g"(x) is negative in the interval (-∞,5/2) so g(x) will be concave down in the interval (-∞,5/2).
    Since g"(x) is positive in the interval (5/2,∞) so g(x) will be concave up in the interval (5/2,∞).

    In 4 and 5, a function and values of x so that f '(x) = 0 are given. Use the Second Derivative Test to determine whether each point (x, f (x)) is a local maximum, a local minimum or neither.
    4.

    Solution:

    Find the value of h''(x) on x = -2,0,2

    Since h"(-2) and h"(2) greater than 0 so h(x) has local minimum at x = -2 and x = 2.

    Since h"(0) < 0 so h(x) has local maximum at x = 0.

    Thus function has local minimum at (-2, -18) and (2,-18).
    Function has local maximum at (0, -2).

    5.

    Solution:

    Find the value of f''(x) on x = 1/e

    Since f"(1/e) >0 so f(x) has local minimum at x = 1/e.

    Thus function has local minimum at (1/e, -1/e).

    6. Lest you have forgotten, the formulas you will need are as follows:
    For a right circular cylinder of radius "r" and height "h":
    The volume V = πr2h.
    The surface area S = circular ends plus the cylindrical wall = 2πr2 + 2πrh.

    You have been asked to design a one-liter (i.e., 1000 cm3) can shaped like a right circular cylinder (figure below). What dimensions will use the least material?

    So the question is: What should "r" be and what should "h" be such that the volume is 1 liter but the surface area is least?

    Solution:

    V = πr2h

    Given V = 1000 cm3
    πr2h = 1000
    h = 1000/πr2

    We know that

    S = 2πr2 + 2πrh
    S = 2πr2 + 2πr*1000/πr2

    Find S'(r)

    Put S'(r) = 0

    cm

    Thus, S"(r) is positive throughout the domain. Thus, S(r) will have least value at r = 5.42 cm

    We know h = 1000/πr2

    cm

    Answer: r = 5.42 cm and h = 10.84 cm

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:53 pm ad1c9bdddf>
    https://brainmass.com/math/functional-analysis/define-bounded-area-formula-614461

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