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    Derivatives

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    Application Word Problem: Continuity and Derivatives

    It costs Sugarco 25 cents/lb to purchase the first 100 lb of sugar 20 cents/lb to purchase the next 100 lb and 15 cents to buy each additional pound. Let f(x) be the cost of purchasing x pounds of sugar. Is f(x) continuous at all points? Are there any points where f(x) has no derivative?

    Differentiation

    Please solve and explain. Two factories are located at the coordinates (-x,0) and (x,0), with their power supply located at (o,h). Find y such that the total amount of power line from power supply to the factories is a minimum.

    Differentiation

    A rectangular package can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume. Assume cross section is square.

    Maximum area with given perimeter

    A Norman window is constructed by adjoining a semicircle to the top of a rectangular window. Find the dimensions of the Norman window of maximum area if the total perimeter is 16 feet.

    Differentiation

    Use a graphing utility to graph f and g in the same window and determine which is increasing at the faster rate for "large" values of x. What can you conclude about the rate of growth of the natural logarithmic function? f(x) = ln x, g(x) = the square root of x

    Differentiation over Interval Values

    1. Suppose f'(t) <0 for all t in the interval (2,8). Explain why f(3) > f(5) 2. Suppose f(0) = 3 and 2 is less than or equal to f'(x) which is less than or equal to 4 for all x in the interval [-5,5]. Determine the greatest and least possible values of f(2).

    Differentiation

    Find the length and width of a rectangle that has an area of 64 square feet and a minimum perimeter.

    Differentiation - Point on a graph

    Please solve and explain how to do so. Find the point on the graph of the function that is closest to the given point. f(x) = the square root of x Point: (4,0)

    Differentiation and chemical reaction

    Please solve to the specified answer and explain how to do so. In an autocatalytic chemical reaction, the product formed is a catalyst for the reaction. If Q sub zero is the amount of the original substance, and x is the amount of catalyst formed, the rate of the chemical reaction is dQ/dx = kx(qsubzero - x) For what

    Inflection and concavity

    Please find the points of inflection and discuss the concavity of the graph of the function. Please explain as much as possible. Please show how to obtain these answers. f(x) = x/x^2 + 1

    Determining Concavity in Intervals

    For each of the following equations, would you please state the intervals for which it is concave up and for which it is concave down? y = 10xe^-x f(x) = x^2 - 4/x + 1

    Differentiation Derivative Proofs

    Please state whether each statement is true or false, and if false please explain why The maximum slope of the graph of y = sin(bx) is b If f''(2) = 0, then the graph of f must have a point of inflection at x = 2.

    Differentiation for Particular Beams

    Please solve and explain to the specified answer. Please explain solution. The deflection D of a particular beam of length L is D = 2x^4 - 5Lx^3 + 3L^2x^2 where x is the distance from one end of the beam. Find the value of x that yields the maximum deflection.

    Graphing from the derivative

    Please sketch a graph of a function f have the indicated characteristics. Please explain. (a) f(0) = f(2) = 0 f'(x) > 0 if x <1 f'(1) = 0 f'(x) < 0 if x > 1 f''(x) < 0 (b) f(0) = f(2) = 0 f'(x) < 0 if x < 1 f'(1) = 0 f'(x) > 0 if x > 1 f''(x)

    S represents weekly sales of a product.

    S represents weekly sales of a product. What can be said of S' and S'' for each of the following? (a) the rate of change of sales is increasing (b) sales are increasing at a slower rate (c) the rate of change of sales is constant (d) sales are steady (e) sales are declining, but at a slower rate (f) sal

    Differentiation and graph a function

    Please solve the following. All explanation is welcome. Consider a function f such that f' is decreasing. Sketch graphs of f for (a) f' is less than zero and (b) f' is greater than zero.

    Differentiation Critical Number

    Please solve and offer explanation for the following: A differentiable function f has one critical number at x = 5. Identify the relative extrema of f at the critical number if f'(4) = -2.5 and f'(6) = 3.

    Differentiation and air velocity

    Please show how to solve the problem to the specified answer. Please offer as much explanation as possible. Coughing forces the trachea to contract, which affects the velocity v of the air passing through the trachea. Suppose the velocity of the air during coughing is v = k(R - r)r^2, 0 is less than or equal to r w

    Sketching of a function knowing its derivative

    The function f is differentiable on the interval [-1,1]. The table shows the values of f' for selected values of x. Sketch the graph of f, approximate the critical numbers, and identify the relative extrema. x -1 -0.75 -0.50 -0.25 f'(x) -10 -3.2 -0.5 0.8 x 0 0.25 0.50

    Mean value theorem proof

    Please explain how to prove the following. As much explanation as possible would be great Let p(x) = Ax^2 + Bx + C. Prove that for any interval [a,b], the value c guaranteed by the Mean Value Theorem is the midpoint of the interval.

    Question about Differentiation proof

    Please indicate if each statement is true or false and if false please explain why If the graph of a function has three x intercepts, then it must have at least two points at which its tangent line is horizontal. If f'(x) = 0 for all of x in the domain of f, then f is a constant function.

    Finding Derivatives Descriptions

    (See attached file for full problem description with proper equations) --- First: solve these problems. Second: check my answers (they're not simplified). Third: if my answers are wrong explain why. Find . My Answers: 1. y = (x3 + 1)20 2. y = (x3 + y3) 20 ---