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SVEN [57.7K]
2 years ago
6

Attached as photo. Please help

Mathematics
1 answer:
Effectus [21]2 years ago
7 0

By Euler's method the <em>numerical approximate</em> solution of the <em>definite</em> integral is 4.189 648.

<h3>How to estimate a definite integral by numerical methods</h3>

In this problem we must make use of Euler's method to estimate the upper bound of a <em>definite</em> integral. Euler's method is a <em>multi-step</em> method, related to Runge-Kutta methods, used to estimate <em>integral</em> values numerically. By integral theorems of calculus we know that definite integrals are defined as follows:

∫ f(x) dx = F(b) - F(a)     (1)

The steps of Euler's method are summarized below:

  1. Define the function seen in the statement by the label f(x₀, y₀).
  2. Determine the different variables by the following formulas:

    xₙ₊₁ = xₙ + (n + 1) · Δx     (2)
    yₙ₊₁ = yₙ + Δx · f(xₙ, yₙ)     (3)
  3. Find the integral.

The table for x, f(xₙ, yₙ) and y is shown in the image attached below. By direct subtraction we find that the <em>numerical</em> approximation of the <em>definite</em> integral is:

y(4) ≈ 4.189 648 - 0

y(4) ≈ 4.189 648

By Euler's method the <em>numerical approximate</em> solution of the <em>definite</em> integral is 4.189 648.

To learn more on Euler's method: brainly.com/question/16807646

#SPJ1

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Answer:

t-shirts: 2790

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x = \frac{-8 \pm \sqrt{8^2 -4(-3)(1)}}{2(-3)}

x = \frac{-8 \pm 8.72}{-6}

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d²p/dx² at x = -0.12: -6(-0.12) + 8 = 8.72 > 0

d²p/dx² at x = 2.79: -6(2.79) + 8 = -8.74 < 0

Then, he should prints 2.79 thousands, that is, 2790 t-shirts to make maximum profits.

Replacing into profit equation:

p(x) = -(2.79)³ + 4(2.79)² + 2.79 =  12.209

that is, $12209

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