for f ( x, y )
2. Find f xx ( x, y ), f xy ( x, y ), f yx ( x, y ), f yy ( x, y ) for f ( x, y ) x ln( xy )
3. For P ( x, y ) x 2 2 xy 2 y 2 4 x 12 y 5 , find all values of x and y such that
1. Consider the differential equation
(a) Find the equilibrium.
(b) Graph the rate of change as a function of x for 2 x 2 .
(c) Draw the phase-line diagram.
(d) Use the stability theorem to evaluate the sta
1. The sample space is S 0,1, 2,3, 4 . Suppose that
Pr 0 0.1 , Pr 1 0.3 , Pr 2 0.4 , Pr 3 0.1 , Pr 4 0.1
(a) Find Pr A and Pr Ac if A= 0, 2
(b) Find Pr B and Pr B c if B = 3, 4
(c) Is Pr A B Pr A Pr B ? Why or why not?
1. Zoe is taking two books along on her holiday vacation. With probability 0.5, she
will like the first book; with probability 0.4, she will like the second book; and
with probability 0.3, she will like both books. What
1. Show that
dx is divergent.
dx is convergent.
2. Show that
3. Identify the following as pure-time, autonomous, or nonautonomous differential
equations. In each case, identify the state variable.
1. Let X be the binomial random variable with the parameters (n, p) . Prove that
E ( X 2 ) k 2Ckn p k (1 p) n k n 2 p 2 np 2 np
2. Only 60% of certain kinds of seeds germinate when planted under normal
1. Four balls are to be randomly selected, without replacement, from an urn that
contain 20 balls numbered 1 through 20. Let X is the largest number to be
(a) What are the possible values of X and the probabi
1. Find the area between f ( x) (1 3x)3 and the x-axis from x 0 to x 2.
2. Find the area between f ( x) x 2 and g ( x) x3 for 0 x 2 .
3. Find the area of the region bounded by y x 2 , y ( x 2)2 , y 0 from x 0
to x 2.
1. Suppose the following functions are solutions of some differential equations.
Graph these as functions of time and as phase-plane trajectory for 0 t 2 . Mark
the position at t 0 , t 1 , and t 2 .
x(t ) 1 t (t 2)
1. The spring equation or simple harmonic oscillator
describes how acceleration (the second derivative of the position x) is equal to the
negative of the position. The spring constant k has been set to 1 f
1. Are inverse trigonometric functions such as arcsin, arccos, and arctan functions
are continuous? Explain why.
f ( x) e x
if x 0
if 0 x 1
if x 1
(a) Sketch the graph of f.
(b) Find the numbers at which f i
1. Explain in your own words what it means to say that
lim f ( x ) 3 and lim f ( x ) 7
In this situation, is it possible that lim f ( x ) exists? Explain.
2. Sketch the graph of the following function and u
1. Using the graph of the function g ( x ) , sketch a graph of the shifted or scaled
function, say which kind of shift or scale it is, and compare with the original
(a) g ( x / 3)
(b) g ( x 1)
2. Determine whe
1. Use the tangent line approximation to evaluate ln( 0.98) in two ways. First,
find the tangent line to the whole function using the chain rule. Second, break the
calculation into two pieces by writing the function a
1. For the following figure, label
(a) One critical point
(b) One point with a positive derivative
(c) One point with a negative derivative
(d) One point with a positive second derivative
(e) One point with a negative s
by implicit differentiation.
(a) x 2 xy y 2 4
(b) sin( x y ) y 2 cos x
(c) e y x y
(d) tan 1 ( x 2 y ) x xy 2
2. Show that
x x2 1
3. f ( x) 2 x e x , x R , find ( f 1 ) '(1) .
4. f ( x
1. Suppose a population of bacterial doubles every hour, but that 1106 individuals
are removed before reproduction to be converted into valuable biological byproducts. Suppose the population begins with b0 3 106 bacter
1. Find the derivative of the following polynomial function. State where you used
the sum, constant product, and power rules.
x2 x3 x4
2. Compute the derivative of the following function in the two ways given.
1. Sketch the graph of the function y
e 1 and determine its domain and
2. Find the domain of each function.
(a) f ( x)
1 e1 x
(b) g (t ) sin(e t )
3. Sketch the graphs of f ( x) 1 x and its inver
1. Use the Intermediate Value Theorem to show that the following equation has
solution for 0 x 1.
e x x 2 2 cos(2 x ) 1
2. Suppose that f is a differentiable function and that f (0) 3 , and f ( x ) 5 for
all value of x.