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Practice Problems

Exam I Problems

1.	a) Suppose a population grows logistically with carrying capacity $L=200$. If the growth is near exponential for small $t$ with $k=.13$, write down a differential equation describing the growth of the population.
	b) Solve the differential equation in a) if $P(0)=25$.
2.	Compute the following integrals:
	a) $\int \frac{dx}{1+x^2}$
	b) $\int (1+x^2)e^x dx$
	c) $\int \frac{dx}{x^2+4x+5}$
	d) $\int \sin^3(2x)dx$
3.	Sketch the slopefield for the differential equation $\frac{dy}{dt}=-\frac{t}{y}$.
4.	Let $f(x)=\frac{1}{x}$ throughout this problem and the definite integral referred to be $$\int_{1}^{2}\frac{1}{x} dx$$ (1)
	a) What is the minimum number of subdivisions that you would need to use if you wanted to estimate the definite integral to an accuracy of 0.1?
	b) Find the left and right Riemann sums for the definite integral when $N=5$ subdivisions are used. (This does not mean that the answer to part a) is 5)
	c) Using a sketch of $y=\frac{1}{x}$, show how the left and right Riemann sums are related to the definite integral (i.e which is larger, which is smaller).
	d) Verify that the function $F(x)=\ln(x)$ is an antiderivative of $f(x)$, and use the Fundamental Theorem of Calculus to find the exact value of the definite integral.
	e) Using your knowledge of Riemann sums and noting answer to d), can you propose a method for computing natural logs for the first ten integers without using the $\ln$ key of your calculator?
5.	In this problem, the differential equation is always $$\frac{dy}{dx}=0.5(y-3)(y+2)$$ (2)
	a) Find the equilibrium solutions of the differential equation.
	b) Using a sheet of graph paper and a set of coordinate axes, carefully sketch the slope field of the differential equation. Sketch and label the equilibrium solutions on the slope field.
	c) Are the equiblibrium solutions stable or unstable? Explain your answer.
	d) Use the technique of seperation of variables to find a formula for the solution to the differential equation with the initial condition $y(0)=1$.
6.	Solve the following initial value problems:
	a) $\frac{dx}{dt}=\sec^2(t)$, $x(0)=1$
	b) $\frac{dy}{dx}=xy$, $y(0)=.5$
7.	Match the slope fields below with the given differential equations:
	$\frac{dy}{dt}=\frac{1}{2}y$
	$\frac{dy}{dt}=\frac{3}{10}y(3-y)$
	$\frac{dy}{dt}=\frac{3}{10}t(3-t)$
	a) b)
	c) d)
8.	State the Fundamental Theorem of Calculus, including all hypotheses necessary.
9.	A water balloon thrown upwards from the top of a 200 foot high building at 50 $\frac{feet}{sec}$ falls back to the street below.
	a) How long does the balloon take to reach the street?
	b) What is its maximum height?
	c) How long does the balloon take to reach this height?
	d) What is the balloons velocity when it strikes the street?
10.	Evaluate
	a) $\int (e^{2r})dr$
	b) $\int (\cos(t) + \frac{1}{\cos^2(t)})dt$

Exam II Problems

1.	Do the following series converge or diverge?
	a) $\sum_{k=0}^{\infty} \frac{1}{(k+\sin(k))(k+\cos(k))}$
	b) $\sum_{j=0}^{\infty} \frac{j}{j!}$
2.	A certain test for a disease is $94\%$ accurate when a patient has the disease and $98\%$ accurate if the do not. If the disease occurs in $.2\%$ of the population, what are the odds that a person has the disease if they test positive?
3.	You perform an experiment with two outcomes H and T where H has probability $p$ and T has probability $1-p$. Suppose you perform this experiment five times in a row and record the results. Assign a random variable X to this event given by the number of H's minus the number of T's.
	a) Sketch a graph of the mass density function for X.
	b) Compute the expected value of X.
4.	Find the volume of the solid gotten by revolving the area between the curves $y=x$, $y=\sqrt{x}$, $x=0$ and $x=1$ about the $x$-axis.
5.	Answer the questions from the Probability and Geometric Series lab.
6.	Decide whether the improper integral
	$$\int_{0}^{\infty} \frac{dx}{x+e^x}$$ (3)
	converges or diverges. Explain your reasoning (calculator work is not sufficient).
7.	Consider the definite integral $\int_{1}^{2} \ln(x) dx$.
	a) State whether the left-hand sum, right-hand sum, trapezoidal rule and midpoint rules over or underestimate the definite integral.
	b) Put the four approximations of the definite integral and the definite integral in order by size.
	c) Compute Simpson's rule for this definite integral with $N=2$.
	d) Use the Fundamental Theorem of calculus to evaluate the definite integral exactly.
8.	For what values of $k$ does the integral
	$$\int_{0}^{1} \frac{1}{x^{k+1}} dx$$ (4)
	converge?
	a) $k>0$
	b) $k<0$
	c) $k<1$
9.	A fighter base stores jet fuel in an underground spherical tank of radius $10$ feet with its top just at ground level. If the tank is full and if jet fuel weighs $40\frac{lbs}{ft^3}$, how much work must be done to pump all the fuel out of the tank (to ground level)?

Exam III Problems

1.	Suppose $p(x)=ax$ for $0\leq x\leq 1$, $\frac{-a}{3}(x-4)$ for $1\leq x \leq 4$ and 0 elsewhere is the density function for a population.
	a) Find the value of $a$.
	b) Find the mean and median of the variable $x$.
	c) Find the function $P(x)$ for the cumulative distribution function of $x$.
2.	Let $f(x)$ be the function defined by $f(x) = 1$ if $\frac{\pi}{2}\leq x<\frac{3\pi}{4}$ and zero otherwise on the interval $[0,\pi]$
	a) Find the first three Fourier approximations of $f(x)$.
	b) Plot the harmonics for $k=0,1,2,3$.

3.	Let $f(x)=\frac{1}{x}$.
	a) Find the first three Taylor polynomials for $f(x)$ at $x=1$.
	b) Based on part a), write down the Taylor series representation of $f(x)$ near $x=1$.
	c) What is the interval of convergence for the Taylor series you found in $b)$?

4.	Let $S(x)=\sum_{k=1}^{\infty}\frac{(x-7)^k}{4^k k^2}$.
	a) About what point in the real number line does this Taylor series converge?
	b) Find the interval of convergence for this function and determine whether or not the series converges at the endpoints of the interval.

5.	Find the general solution to the following second order differential equations
	a) $\frac{d^2s}{dt^2}+49y=0$
	b) $\frac{d^2s}{dt^2}+9y=0$
	c) $\frac{d^2s}{dt^2}+y=0$

6.	Given the system of differential equations $$\frac{dx}{dt}=-x+xy\;\;\;\frac{dy}{dt}=3y(1-\frac{y}{2})-xy$$ (5) describing the populations $x(t)$ and $y(t)$ of two competing populations at time $t$
	a) describe the relationship between the two species in words and
	b) analyze the phase plane of $\frac{dy}{dx}$ completely.
7.	Given the system of differential equations $$\frac{dx}{dt}=xy\;\;\;\frac{dy}{dt}=-x+3xy$$ (6)
	a) Find the nullcline(s) and equilibrium point(s) for this system.
	b) Using your nullclines and any other information that might help, sketch a phase plane for x versus y, indicating the direction that trajectories move in each region defined by the nullclines (one arrow is enough in each region).
	c) Sketch a possible trajectory starting at $x(0)=1$, $y(0)=-1$.
	d) Write down and solve a differential equation for $\frac{dy}{dx}$ with initial condition $y(0)=1$.
8.	Compute the Taylor series for the following functions
	a) $f(x)=\frac{1}{1-x}$
	b) $f(x)=\frac{1}{1+x}$
	c) $f(x)=\frac{1}{1-x^2}$
	d) $f(x)=\arctan{x}$
9.	In the SIR lab, what did the term $\mu R$ represent? How did changing the position of the vertical nullcline change the behaviour of trajectories in the phase plane? Which parameters would you change in order to move this nullcline? What effect does this movement have on the spread of the disease being studied?

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