Workshop 4
October 3, 2016
Problem 1. Use appropriate trig substitutions to evaluate
Z
Z
Z
2
2
4 x dx,
4 + x dx, and
5 + 2x + x2 dx.
Problem 2. Find
and then find
Z
Z
e2x
dx,
e2x + 1
ex
dx.
e2x + 1
Comment. These integrands may appear similar, but diff
Workshop 2
September 19, 2016
Problem 1. A sort of raindrop is obtained by revolving the profile curve
y = x(x C)2 for 0 x C
about the x-axis. Here C is a positive constant.
(a) Sketch the profile curve and the solid of revolution.
(b) For which value of
Workshop 3
September 26, 2016
Problem 1.
(a) Suppose that m and n are integers. Compute
Z 2
cos(mx) cos(nx) dx.
0
(Be careful: there will be three different results, one when m = n = 0,
another when m = n 6= 0, and the third when m 6= n.)
(b) Suppose f (x
Workshop 1
September 12, 2016
Problem 1. Suppose that f is a continuous function (defined for all x) and
that the values of the following integrals are known:
Z 1
Z 1
Z 2
Z 4
f (x)dx = 5,
f (x)dx = 3,
f (x)dx = 8,
f (x)dx = 11.
0
1
0
0
Determine the value
Workshop 5
October 24, 2016
Problem 1. Consider the four integrals:
(a)
Z
x
dx,
x4 + 1
x2
dx,
x4 + 1
x3
dx,
x4 + 1
x4
dx.
x4 + 1
0
(b)
Z
0
(c)
Z
0
(d) and
Z
0
Which of these integrals converge? Compute the exact value of the convergent integrals.
ProblemR
Workshop 6
October 31, 2016
Problem 1.
(a) Show that the surface area of a spherical cap of height h and radius R
is 2Rh.
(b) Suppose you are at a height d above a sphere of radius R. Show that
dR2
the portion of the spheres surface that you can see has a
Sample Workshop
September 12, 2016
Problem 1
2
Suppose n is a positive integer, and f is the function f (x) = nx(n ) . For
example, if n = 5, f (x) = 5x25 .
(a) What is the largest value of f on the unit interval, [0, 1]? Your answer
will depend on n. Wha
Workshop 7
November 7, 2016
Problem 1. Under the hypotheses of the integral test, if an = f (n) then for
any positive integer N ,
Z
X
an f (x) dx.
N +1
N
(a) How large does N have to be to insure that
N
X
1
n5
n=1
is within 106 of
X
1
?
n5
n=1
(b) And how
4.Question:
(a)
(b)
(c)
(d)
Calculate the derivative of: F(x)=ln(cos(3x)
Calculate the derivative of: G(x)=(e*sin(ex)*cos(x+x2)
Calculate the derivative of: H(x)= (ecos4xn
+ln *cos(x)2
More generally, show that if f,g,and h are differentiable, then :
[f(g
3. Problem Statement: Suppose f(x) is a piecewise function defined as follows:
f(x)=
2x2+2
,if x<1
2
ax +bx ,if 1x2
2- (6/x)
. if x>2
a) Suppose that a=2 and b=-3. Graph f(x) for 0 x 3. Find the left and
right hand limits of f(x) as x approaches 1 and as
3. Problem Statement: Suppose f(x) is a piecewise function defined as follows:
f(x)=
2x2+2
,if x<1
2
ax +bx ,if 1x2
2- (6/x)
. if x>2
a) Suppose that a=2 and b=-3. Graph f(x) for 0 x 3. Find the left and
right hand limits of f(x) as x approaches 1 and as
1.Problem Statement: Two circles have the same center. The inner circle has radius r which is
increasing at the rate of 3 inches per second. The outer circle has radius R which is increasing at
the rate of 2 inches per second. Suppose that A is the area o
Raymond Wan
Calculus 151
Workshop #7
David Sheiman
11/3/16
1. Problem Statement: A square and a circle are placed so that the circle is outside the
square and tangent to the one side of the square. The sum of the length of side of the
square and the circl
4.Question:
a) If you substitute h by -h in the definition of derivative, what conclusion will you get?
b) Now take the average of the FDQ and BDQ, what conclusion can you draw from this?
Can we define the derivative by taking the limit of SDQ?
c) Conside
4. Problem Statement Suppose that f is the function defined by the formula
f(x)= (arctan(ln(x-1)3
A) What are the domain and range of f? Answers should not be numerical approximations,
but should be written if needed in terms of traditional constants such
3. Problem statement: An object is moving along the parabola y=3x2.
a) When it passes through the point (2,12), its horizontal velocity is dx/dt= 3. What is its
vertical velocity at that instant?
b) If it travels in such a way that dx/dy=3 for all t, then
Math
135
Quiz
f1b
Summer 2OL4
Solulru^s
SHOW ALL YOUR WORK TO RECEIVE FULL CREDIT.
NO CALCULATOR ALLOWED.
1.(aPts.)Evaluateandsimp1ifythedifferencequotientryforthefirtrction
(') : 3a2 - 2r'
"f
cfw_ (*+h)
- 16)
h
Lt-)]'
(3
xz
(x.nt^)
r
* t 3(x+Ll*t3(^'+
+i
t4
Math
$lutrnt*
ff\a
Qt:jz
135
Fall
201-4
Name:
SHowALLYoURwoRI(ToRECEIVEFULLCREDIT.
CALCULATOR ALLOWED.
1. (5 Pts.)
(a) Find aatd, for E denned impricitr,:;j:lr)ri,.,*" o,
(H@*d, (z-#r) = X+^#
2tK px-%) cn(Yr-DH= W+"h
M
dx
t+ w0-*-2)
point
(b) Write an
Math 135, Fall 2014, Midterm II Study Guide
Midterm II will take place in lecture on Thurs, Nov 13, and will cover Sec 3.6-3.8 and 4.1-4.6 of the textbook.
Below is a list of types of problems you are expected to be familiar with.
Sec 3.6
Given an equatio
Math 135, Fall 2014, Final Exam Study Guide
The final exam will take place on Mon, Dec 15, 4-7pm, in Loree 022 on the Cook Douglass campus.
The final will be cumulative and will cover all sections discussed in lecture from Ch 1-5 of the
textbook. Below is