# Math Help - Urgent - Calculus

1. ## Urgent - Calculus

1) A searchlight is kept focus on an aircraft that flies directly above the light at a constant altitude of 2 kilometers and speed of 0.2 kilometer per second. Two seconds after the plane is directly over the light, how fast (in rad/s) is the light rotating?

2)Use a linear approximation (or differentials) to estimate

$15.97^{3/4}$

3)A curve passes through the point (0,4) and has the property that the slope of the curve at every point P is three times the y- coordinate of P. What is the equation of the curve?

2. Originally Posted by treetheta
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3)A curve passes through the point (0,4) and has the property that the slope of the curve at every point P is three times the y- coordinate of P. What is the equation of the curve?
1. You know: $y' = 3y~\implies~\dfrac{y'}{y} = 3$

Integrate wrt x:

$\int\left(\dfrac{y'}{y}\right)dx=\int 3dx~\implies~\ln(y)=3x+k$

Solve for y:

$y = e^{3x+k}$

The graph of this function passes through P(0, 4):

$4=e^{0+k}~\implies~k=\ln(4)$

Thus: $y= e^{3x+\ln(4)} = \boxed{4\cdot e^{3x}}$

3. Omg, thank you so much,

can someone help me with the remaining questions