Calculus II
Homework 6
1. Find
Due 2016/4/12
f
f
x
and
for f ( x, y )
y
y
x
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
Calculus II
Homework 3
1. Consider the differential equation
Due 2016/3/15
dx
1 ex
dt
(a) Find the equilibrium.
(b) Graph the rate of change as a function of x for 2 x 2 .
(c) Draw the phase-line dia
Calculus II
Homework 7
Due 2016/4/26
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
Calculus II
Homework 8
Due 2016/5/3
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; a
Calculus II
Homework 2
1. Show that
Due 2016/3/8
1
dx is divergent.
x x
1
dx is convergent.
2. Show that
1
1 x4
3. Identify the following as pure-time, autonomous, or nonautonomous differential
1
equ
Calculus II
Homework 10
Due 2016/5/17
1. Let X be the binomial random variable with the parameters (n, p) . Prove that
n
E ( X 2 ) k 2Ckn p k (1 p) n k n 2 p 2 np 2 np
k 0
2. Only 60% of certain kinds
Calculus II
Homework 9
Due 2016/5/10
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
selected
Calculus II
Homework 1
Due 2016/3/1
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
Calculus II
Homework 5
Due 2016/3/29
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
th
Calculus II
Homework 4
Due 2016/3/22
1. The spring equation or simple harmonic oscillator
d 2x
x
dt 2
describes how acceleration (the second derivative of the position x) is equal to the
negative of
Calculus I
Homework 5
Due 2015/10/28
1. Are inverse trigonometric functions such as arcsin, arccos, and arctan functions
are continuous? Explain why.
2.
x 2
f ( x) e x
2 x
if x 0
if 0 x 1
if x 1
(a) S
Calculus I
Homework 4
Due 2015/10/21
1. Explain in your own words what it means to say that
lim f ( x ) 3 and lim f ( x ) 7
x 1
x 1
In this situation, is it possible that lim f ( x ) exists? Explain.
Calculus I
Homework 1
Due 2015/9/30
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
Calculus I
Homework 10
Due 2015/12/16
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 th
Calculus I
Homework 8
Due 2015/12/2
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 pos
Calculus I
Homework 7
Due 2015/11/18
dy
by implicit differentiation.
dx
1. Find
(a) x 2 xy y 2 4
(b) sin( x y ) y 2 cos x
x
(c) e y x y
(d) tan 1 ( x 2 y ) x xy 2
d
1
(sec1 x)
2. Show that
dx
x x2 1
Calculus I
Homework 3
Due 2015/10/14
1. Suppose a population of bacterial doubles every hour, but that 1106 individuals
are removed before reproduction to be converted into valuable biological byprodu
Calculus I
Homework 6
Due 2015/11/4
1. Find the derivative of the following polynomial function. State where you used
the sum, constant product, and power rules.
x2 x3 x4
2
6 24
2. Compute the deriv
Calculus I
Homework 2
1. Sketch the graph of the function y
Due 2015/10/7
1 x
e 1 and determine its domain and
2
range.
2. Find the domain of each function.
(a) f ( x)
1 ex
2
2
1 e1 x
(b) g (t ) sin
Calculus I
Homework 9
Due 2015/12/9
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 functi