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natalie2sheffield
12.10.2020 •
Mathematics
How do you these two questions?
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Ответ:
dy/dx = e⁻²ᵗ (1 − t)
d²y/dx² = -e⁻³ᵗ (3 − 2t)
(1.5, ∞)
Step-by-step explanation:
x = eᵗ
y = te⁻ᵗ
dx/dt = eᵗ
dy/dt = -te⁻ᵗ + e⁻ᵗ
dy/dt = e⁻ᵗ (1 − t)
dy/dx = (dy/dt) / (dx/dt)
dy/dx = e⁻ᵗ (1 − t) / eᵗ
dy/dx = e⁻²ᵗ (1 − t)
d²y/dx² = d(dy/dx)/dt / (dx/dt)
d²y/dx² = (-e⁻²ᵗ − 2(1 − t) e⁻²ᵗ) / eᵗ
d²y/dx² = -e⁻²ᵗ (1 + 2(1 − t)) / eᵗ
d²y/dx² = -e⁻³ᵗ (3 − 2t)
The function is concave up when the second derivative is positive. -e⁻³ᵗ is always negative, so the second derivative is positive when 3 − 2t is negative.
3 − 2t < 0
3 < 2t
t > 3/2
Or, in interval notation, (1.5, ∞).
Ответ:
Because there's a negative sign, it suggests that you should inverse the expression. After inverting it, the power will be positive, and you can square like normal.