Instructions

Solve the following problems. Show all steps clearly and provide detailed explanations.

Problem $1$

Compute the following integrals:

a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{2x+1}dx$

b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}sin(x)ln(cos(x))dx$

Problem $2$

Determine the area between the curve $y=x^2-3x+2$ and the line $y=-2x+2.$

Problem $3$

Solve the following initial value problem:

$\frac{dy}{dx}=y*sin(x),y(\frac{}{2})=3$

Problem $4$

A patient is administered a medication, and the amount of the medication in their bloodstream decreases at a rate proportional to the current amount. The rate of decrease is $0\mathrm{.}04$ per minute. Initially, the patient has $50$ milligrams of the medication in their bloodstream.

a$)$ Write a differential equation to model the amount of medication.

b$)$ Solve the differential equation.

c$)$ Determine the amount of medication after $20$ minutes.

MATH$1760-992,$ Calculus for the Life Sciences with Applications II

$2$

Problem $5$

Consider the autonomous differential equation:

$\frac{dy}{dx}=(y-3)(y+2)(y-5)$

where $\frac{dy}{dx}$ is a function of $y.$

a$)$ State the equilibrium solutions.

b$)$ Construct a phase line and determine for each equilibrium solution whether it is locally stable or not.

c$)$ In the $xy-$plane, sketch the equilibrium solutions and the solution curves that pass through the indicated points $(0,-3),(0,-2),(0,0),(0,3),(0,4),(0,5),$ and $(0,6).$

Problem $6$

A local health clinic monitors two conditions: high blood pressure and diabetes. For the patients examined in the clinic, the probability of having high blood pressure is $0\mathrm{.}30,$ the probability of having diabetes is $0\mathrm{.}35,$ and the probability of having both high blood pressure and diabetes is $0\mathrm{.}15.$

a$)$ Find the probability of having either condition or both.

b$)$ Find the probability of not having either condition.