Answer:
0.875
Step-by-step explanation:
Point/slope form is:-
y - y1 = m(x - x1) where m = slope and (x1, y1) is a point on the line
In the above diagram the slope is -1 and one point on the line is (0, -2) so we have:-
y - (-2) = -1(x - 0)
y + 2 = -x
x + y + 2 = 0 is the answer
The line perpendicular to x = 2 will be y = something. This line passes through (0,-4) so its equation is y = -4 (answer)
Answer:
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, ∞).
