Posted by **sarah** on Friday, February 25, 2011 at 4:29pm.

f(x) passes through (0,2). the slope of f at any point is 3 times the y-coordinate. find the derivative at 1

- elementary calc -
**Reiny**, Friday, February 25, 2011 at 5:11pm
When a derivative of a function is a multiple of the y value of the function, it suggests an exponential function.

let the function be y = a e^(kx), where a and k are constanst

dy/dx = ak e^(kx)

given ak e^(kx) = 3y = 3(a e^(kx))

then ak = 3a

k = 3

so we know y = a e^(3x)

at (0,2)

2 = a e^(0)

a = 2

then y = 2 e^(3x)

then dy/dx = 6 e^(3x)

when x=1

dy/dx = 6 e^3

- elementary calc -
**Damon**, Friday, February 25, 2011 at 5:14pm
dy/dx = 3 y

dy/y = 3 dx

ln y = 3 x + c

y = C e^3x

if x = 0, y = 2

so

y = 2 e^3x

when x = 1

y = 2 e^3 = 40

dy/dx = 3y = 120

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