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Find the equation of the tangent line to the curve \[y = 3 \arctan(2x)\] at the point \(\displaystyle x = \frac{1}{2}\).

Find the total mass of the \(1\) m rod whose linear density function is given by \[\rho(x) = 4x(1+x^2)^3 \;\;\text{    kg/m}\] for \(0 \leq x \leq 1\).

Order: Omeprazole 11mg, gastrostomy tube, every 12 hours, for reflux  Child’s weight: 22 kg Safe dose range: 1-1.5mg/kg/day in two divided doses   a) What is the safe low dose (mg/dose) for this child? b) What is the safe high dose (mg/dose) for this child?  c) Is the ordered dose safe for this child?

Which of the following is the partial fractions decomposition of the rational function \[f(x)=\frac{5x^2+1}{(x^2-1)^2(x^2+9)}\;\;?\]

Order: Vancomycin 600mg, intravenously, every 6 hours, for moderate infection  Child’s weight: 50 kg  Safe dose range: 10-15mg/kg/dose (max dose 750mg) every 6 hours    a) What is the safe low dose (mg/dose) for this child?  b) What is the safe high dose (mg/dose) for this child? c) Is the ordered dose safe for this child?  

Evaluate the indefinite integral \[\int_1^e  \ln\left(x^2\right)\;dx.\] (Hint: Integration by Parts.)

Evaluate the integral \[\int_0^{\pi/2} \cos^3{x}\;dx.\](Hint: \(\sin^2{x}+\cos^2{x}=1\))

Order: Acetaminophen 375mg, oral, PRN every 4 hours, for fever  Child’s weight: 35 kg  Safe dose range: 10-15mg/kg/dose every 4 hours    a) What is the safe low dose (mg/dose) for this child?  b) What is the safe high dose (mg/dose) for this child?   c) Is the ordered dose safe for this child?

Which of the following definite integrals represents the volume of the solid obtained by rotating the region bounded by

Order: Morphine 0.5mg, intravenously, every 4 hours, for pain  Patient’s weight: 6.31 kg  Safe dose range: 0.05-0.1mg/kg/dose every 4 hours    a) What is the safe low dose (mg/dose) for this child? Round to the nearest hundredth.  b) What is the safe high dose (mg/dose) for this child? Round to the nearest hundredth.   c) Is the ordered dose safe for this child?