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[nrumble42]

Any help pls guys

 

[pikachu975]

1) x = e^5t + te^5t
dx/dt = 5e^5t + e^5t + 5te^5t
= 6e^5t + 5te^5t

d^2 x / dt^2 = 30e^5t + 5e^5t + 25te^5t
= 35e^5t + 25te^5t

LHS = d^2 x / dt^2 - 10dx/dt + 25x
= 35e^5t + 25te^5t - 60e^5t - 50te^5t + 25e^5t + 25te^5t
= 0 Q.E.D.

2) y = e^x crosses the y axis when x = 0
y = e^0
y = 1
y' = e^x, sub in x = 0
y' = e^0
y' = 1

Now using gradient one point formula
y - 1 = 1 (x-0)
y = x+1

3) y = x^(2x)
y = e^ln(x^2x)
y = e^(2x*lnx)
y' = d(2x*lnx)/dx * e^(2x*lnx)
y' = [2lnx + (2x*1/x)] * e^(2x*lnx)
y' = 2(lnx + 1) * (x^2x)
when x = 1, y' = 2(ln1+1) * 1^2)
y' = 2 * 1
y' = 2

Tangent is given by:
y - 1 = 2 (x-1)
y - 1 = 2x - 2
y = 2x - 1

a) intersects x axis when y = 0
0 = 2x - 1
x = 1/2

b) intersects y axis when x = 0
y = 0 - 1
y = -1