Example: Compute the slope of the tangent line to the parabola \( y=x^{2} \) at the point \( \mathrm{P}(1,1) \).

Determine whether the function is a solution of the differential equation \( y^{(4)}-81 y=0 \) \[ y=8 \sin (3 x) \] Yes No

Consider the following function. \[ y=x^{8} e^{x} \] Find \( y^{\prime} \). \( y^{\prime}=\square \) Determine whether the function is a solution of the differential equation \( x y^{\prime}-8 y=x^{9} e^{x}, x>0 \) Yes No

10) A bucket of uranium ore decays at a rate of \( -0.34 \% \) per year. If the bucket has 3 kgs of uranium at the start, how long will take for the bucket to decay to 1 kg ?

\( \int _{}^{}\frac{1}{x\sqrt {4x^{2}+4x+3}} d x \)

\( x = \theta \sin \theta + \cos \theta , y = \sin \theta - \theta \cos \theta \)