Honors Calculus I
Quiz 6
1.
Determine derivative for the following functions. Use the limit definition of the
derivative.
(a) f ( x ) = x 3
(b) f ( x ) = x
(c) f ( x ) = sin x
2.
Use the limit definition of the derivative to show the following that the fo
Honors Calculus I
Quiz 1
!
1.
Show that
"x
n=0
n
converges to
1
for all !1 < x < 1 . We denote this by
1! x
!
1
" xn = 1 # x .
n=0
2.
Determine the rational number given by the decimal expansion: 3.233233 .
3.
Suppose that .a1a2 a3 ! = .b1b2b3 ! . That is
Solution of Ordinary Dierential Equations
James Keesling
1
General Theory
Here we give a proof of the existence and uniqueness of a solution of ordinary dierential
equations satisfying certain conditions. The conditions are fairly minimal and usually
sati
Closed Newton-Cotes Integration
James Keesling
This document will discuss Newton-Cotes Integration. Other methods of numerical
integration will be discussed in other posts. The other methods will include the Trapezoidal
Rule, Romberg Integration, and Gaus
The Newton Method
James Keesling
1
The Newton Formula
Suppose that we are trying to nd a solution to the equation f (x) = 0 and we have an
approximation, x0 , near a solution, z . The Newton Method nds a new approximation by
following the tangent line fro
The Intermediate Value Theorem and the Mean Value Theorem
We give the correct statements of the Intermediate Value Theorem and the Mean
Value Theorem.
Theorem (Intermediate Value). Suppose that f : [ a, b ] ! R is a continuous function.
Suppose that d is
Study Sheet for Test 4
Honors Calculus Keesling
11/27/07
Test on 12/4/07
1
1.
Evaluate: (a)
!e
sin x dx
(b)
"
2.
Evaluate: (a)
! sin( x ) dx
(b)
! sin( x ) cos( x ) dx
(c)
! sin ( x ) dx
(e)
x2
! ( x 3 + 1)5 dx
(f)
! tan x dx
3
52
! ( x + 1) x dx
(d)
3.
E
Study Sheet for Test 23
Honors Calculus
Keesling
Test on 10/26/07
1.
2.
3.
Suppose that F ( x ) and G ( x ) are differentiable functions on the interval [ a, b ] .
Suppose that F !( x ) " G !( x ) on the interval [ a, b ] . Show that there is a constant C
Study Sheet for Test 2
Honors Calculus
Keesling
Test on 10/9/07
1.
State the Mean Value Theorem.
2.
Suppose that f !( x ) < 1 for all x in [ x0 ! " , x0 + " ] and that f ( x0 ) = x0 . Let
x1 ![ x0 " # , x0 + # ] with x1 ! x0 . Show that f ( x1 ) ! x0 < x1
Honors Calculus I
Quiz 1
1.
Let R be the real numbers. Determine whether the following sets S ! R have a
least upper bound. If so, what is the least upper bound of the set.
(a) S = [ 0, 5 ]
cfw_
cfw_
(b) S = 1 ! 1 n n = 1, 2,
(c) S = 1 + 1 n n = 1, 2,
(
Honors Calculus I
Quiz 2
!
1.
Show that
1
"n = !.
n =1
!
2.
Give a geometric argument that
"x
n=0
n
=
1
for 0 < x < 1 .
1# x
(!1)n +1
# n converges.
n =1
"
3.
Show that
4.
Suppose that
!
"a
n =1
n
= b . That is, the series converges to b. What if the term
Honors Calculus I
Quiz 6
Name _
1.
Determine the following derivatives.
(a)
d ( 3x10 + 5 x )
dx
d ( x 3 tan( x )
(b)
dx
(c)
d ! sin x $
#
&
dx " x %
(d) x 2 x
(e)
d
tan( x 2 )
dx
Honors Calculus I
Quiz 5
1.
2.
Determine the line passing through the point (1, 2 ) and tangent to the graph of the
function f ( x ) = 2 x 2 .
Find the lines that pass through the point ( 3,1) and are tangent to the graph of
f (x) = 2 x2
Honors Calculus I
Quiz 5
1.
The point (3,15) is on the graph of the function f ( x ) = x 3 ! 5 x + 3 . Find the
tangent to the graph of the function f ( x ) = x 3 ! 5 x + 3 at this point.
2.
The point (3,10) is not on the graph of the function f ( x ) = x
Honors Calculus I
Quiz 4
1.
Let f ( x ) = 4 ! x ! (1 " x ) . Find a point that has period 5.
2.
How many points are there that have period five for this function?
Quiz 4
Name _
Honors Calculus I
df
= 3x 2 .
dx
1.
Show that if f ( x ) = x 3 , then
2.
Find an equation for the tangent line to the graph of the function
f ( x ) = x 2 + 3x + 5 passing through the point (2,15 ) .
Honors Calculus I
Quiz 3
1.
What is lim( x # 2x ) ?
2.
What is lim x # sin( 1 ) ?
x
3.
What is lim
4.
What is lim
5.
What is lim
6.
Explain why the decimal expansion .99 9 equals 1 as a limit.
3
x" 2
x" 0
x3 # 1
?
2
x"1 x # 1
x" 0
x +1 #1
?
x
tan(x )
?
x"
Honors Calculus I
Quiz 3
1.
What is lim( x " 2x ) ?
2.
What is lim x " sin( 1 ) ?
x
3.
What is lim
4.
What is lim
5.
What is lim
6.
Explain why the decimal expansion .99 9 equals 1 as a limit.
7.
Show that
3
x! 2
x! 0
x3 " 1
?
2
x!1 x " 1
x +1 "1
?
x
x! 0
Honors Calculus I
Quiz 2
!
1.
Show that
"x
n=0
n
=
1
for x < 1 .
1# x
2.
Show that the Harmonic Series has infinite sum. The Harmonic Series is given by
!
1
the formula " .
n =1 n
3.
Show that the Alternating Series converges to a number. The Alternating
Logarithm and Exponential Functions
We want to give a precise definition for the logarithm and derive its properties.
The exponential function is the inverse function for the logarithm. Based on properties of
the logarithm, the properties of the exponenti
Lagrange interpolating Polynomials
James Keesling
1
Determining the Coecients of the Lagrange Interpolating Polynomial by Linear Equations
It is frequently the case that we will have certain data points, cfw_(x0 , y0 ), (x1 , y1 ), . . . , (xn , yn ),
and
Inadequacy of Newton-Cotes Integration
The example below shows that Newton-Cotes Integration is not a
very reliable method for approximating an integral. As the number
of interpolation points grows large, there is no guarantee that the
estimate gets close
Name
MAC 3472 Honors Calculus I
Keesling
Test 2 10/27/10
Do all problems. Explain your answers and show all work. Each problem is 20 points.
1.
Determine the following integrals.
(a) ! tan 3 x dx
(b) ! cos 3 x dx
2.
Determine the following integrals.
(a)
Name
MAC 3472, Honors Calculus 1
Keesling Test 2 10/23/09
Do all problems. Show your work and explain your answers. Each problem is 20 points.
1.
Suppose that F ( x ) and G ( x ) are differential functions such that F !( x ) " G !( x ) on an
interval. Sho
Name
MAC 3472 Honors Calculus I
Keesling
Test 1 9/20/10
Do all problems. Explain your answers and show all work. Each problem is 20 points.
1.
Compute the following limits.
(a) lim( x 5 + 2 x 3 " 4 )
x!2
x5 " 1
x !1 x 4 " 1
(b) lim
!
2.
Prove that
"x
n=0
Name
MAC 3472 Honors Calculus I
Keesling
Test 1 9/23/09
Do all problems. Show your work and explain your answers. Each problem is 20 points.
1.
Compute the following.
(
)
(a)
(b)
lim x 2 " sin
(c)
d ! x+3$
"
%
dx # sin( x ) &
(d)
2.
lim x 4 " 3x 2 + 5
d3
Study Sheet for Test 3
Honors Calculus
Keesling
Test on 12/3/10
1
dt . Show that ln( x ) with this definition has the
1t
property that ln(a ! b ) = ln a + ln b .
!
x
1.
Define a function ln x =
2.
Show that ln( x ) as defined above has the following prope
Study Sheet for Test 3
Honors Calculus
Keesling
Test on 12/8/09
1.
Determine the following integrals. Be able to use the following methods and
combinations of them to determine the integrals: (1) Substitution, (2) Integration
by Parts, (3) Trigonometric I
Study Sheet for Test 2
Honors Calculus
Keesling
Test on 10/27/10
1.
Suppose that f is an increasing function on the interval [ a, b ] . Subdivide [ a, b ] into
n equal subintervals. Let Sn ( f , a, b ) be the upper sum and let Sn ( f , a, b ) be the lower
Study Sheet for Test 2
Honors Calculus
Keesling
Test on 10/20/09
1.
Suppose that f is an increasing function on the interval [ a, b ] . Subdivide [ a, b ] into
n equal subintervals. Let Sn ( f , a, b ) be the upper sum and let Sn ( f , a, b ) be the lower