Math 152
Sample Exam 1

1.

Find the general solution of x_n+1 = 2x_n + 2.

2.

Find the general solution of x_n+1 - x_n - 2x_n-1 = 0

3.

Find the general solution of x_n+1 - 8x_n +16x_n-1 = 0

4.

Find a particular solution for 8x_n+1 - 7x_n - 4x_n-1 = 5.

5.

Find a particular solution for 5x_n+1 +2x_n-1 = 21n + 6

6.

Let y_n represent the number of trout in a river at the beginning of year n. Suppose that this trout population can be modeled by the difference equation y_n+1 = 1.25y_n -h where h is a positive constant.

(a)

Explain in words what each term on the right hand side of the difference equation represents.

(b)

Solve the difference equation if initially there are 200 trout in the river.

(c)

Find the value of h, call it h_equil, that causes this trout population to go to equilibrium after a long period of time. That is, for what value of h will $\lim_{n \rightarrow \infty} y_n$ be a constant? What would happen to this trout population over a long period of time if h > h_equil? What would happen to this trout population after a long period of time if h < h_equil?

7.

Given a population of penguins can be modelled by:

x_n+1 = 4x_n - 3x_n-1

(a)

Solve the difference equation.

(b)

Find c₁ and c₂ using x₀ = 500 and x₁ = 475

(c)

Does this population go extinct? If so, when? If not, what is happening to the population, i.e. is it at equilibrium or growing without bound?

8.

Suppose there is a species of fish in the Everglades that has a birth rate of 28% and death rate of 24%. Suppose there is an exact number of fish that fishing liscenses permit fishermen to catch, exactly 2500 fish per year. Find an equation relating x_n+1 to x_n where x_n is the number of fish at time n. Find an equation for x_n if there are 15,000 fish to begin with. When does this species go extinct (if ever)?

9.

If possible, compute the following limits. If a limit does not exist, explain why.

(a)

$\lim_{x \rightarrow \infty} \frac{4-x^2}{x^2+x-2}$

(b)

$\lim_{x \rightarrow -2} \frac{4-x^2}{x^2+x-2}$

(c)

$\lim_{x \rightarrow -1} \frac{x^3-x}{2x^2+3x+1}$

(d)

$\lim_{x \rightarrow \infty} (\frac{1}{2})^n$

(e)

$\lim_{n \rightarrow \infty}(4 + (-\frac{1}{2})^n)$

10.

For each of the following cases below, sketch the graph of the function that satisfies the given condition and state whether or not $\lim_{x \rightarrow 3} f(x)$ exists.

(a)

f is continuous at x=3.

(b)

f has a jump discontinuity at x=3

(c)

f has a vertical asymptote at x=3

(d)

f has a removable discontinuity at x=3