Review Problems for Exam 2
(Covers chapter 3)
1. Find the derivative of the function by the limit process. a) f ( x) x 2 2 x 3 2x 2 1 x 1 b) f ( x) 2 x 2. Find an equation of the line that is tangent to the graph of f ( x) ( x 2 8)( x 2 7) at (3, 2)
Math 2253 - FALL 2007 Secrions 001 and 002 Calculus I
Instructor: Nicolae R. Pascu Office: D151, Phone: (678) 915-4989 E-mail: [email protected] Web: http:/www2.spsu.edu/math/Npascu/index.html TEXTBOOK: Thomas' Calculus, Media Upgrade 11/e by George Th
Math 2253 sec. 004: June 5 Summer 2013
Derivative rules (so far)
d
c = 0,
dx
d
x = 1,
dx
(f g ) = f g ,
()
dn
x = nx n1 ,
dx
(fg ) = f g + fg ,
f
g
d
cf = cf
dx
=
f g fg
g2
June 4, 2013
1 / 25
Evaluate the derivative.
(a) y =
3
x+
(b) g (t ) =
()
1
3x 2
t
Math 2253 sec. 004: May 29 Summer 2013
Show that there exists a number that is exactly one less that its fth
power.
()
May 28, 2013
1 / 20
Section 2.1 Denition: The tangent line to the curve y = f (x ) at the
point P (a, f (a) is the line through P with s
Math 2253 sec. 004: June 7 Summer 2013
Chain Rule:
In Leibniz notation:
()
d
f (g (x ) = f (g (x )g (x )
dx
dy
dy du
=
dx
du dx
June 6, 2013
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Ex. Evaluate
()
d
dx
tan(x 2 )
June 6, 2013
2 / 27
Evaluate
d
dt
()
(t 3 + 2t 1)5
June 6, 2013
3 / 27
The p
Math 2253 sec. 004: June 10 Summer 2013
Brief overview of some of section 2.7: Applications of deriviatives in
various elds.
()
June 10, 2013
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Electro-Magnetics
When electrons move through a wire, a change in electric charge
occurs. Rate of change o
Math 2253 sec. 004: June 14 Summer 2013
A kite ies at a xed altitude of 100 ft above the ground, and travels
horizontally with a speed of 8 ft/sec. At what rate is the angle between
the kite and the horizontal decreasing when 200 ft of string has been
let
Math 2253 sec. 004: June 24 Summer 2013
Suppose f has the given derivative:
f (x ) = 2(x + 3)(x + 1)2 (x 2)(x 6)
We found that f was
Increasing on: (3, 1) (1, 2) (6, )
Decreasing on: (, 3)
()
(2, 6)
June 24, 2013
1 / 45
Figure: A function f with derivativ
Math 2253 sec. 004: June 21 Summer 2013
Section 3.2
Rolles Theorem: Let f be a function that is
i continuous on the closed interval [a, b],
ii differentiable on the open interval (a, b), and
iii such that f (a) = f (b).
Then there exists a number c in (a,
Math 2253 sec. 004: June 17 Summer 2013
Figure: We dened local (a.k.a. relative) and absolute (a.k.a. global) extrema.
Recall that a continuous function on a closed interval [a, b] is guaranteed to
attain both an absolute maximum and absolute minimum on [
Math 2253 sec. 004: June 26 Summer 2013
Analyze the function f (x ) = 5x 2/3 2x 5/3 , and use the results to
produce a rough plot of the graph y = f (x ).
()
June 26, 2013
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()
June 26, 2013
2 / 43
()
June 26, 2013
3 / 43
()
June 26, 2013
4 / 43
Figu
Math 2253 sec. 004: June 12 Summer 2013
Suppose t is an independent variable. If u is a function of t , and y is a
function of u , then y is in turn a function of t .
When these functions are differentiable
dy
dy du
=
.
dt
du dt
()
June 11, 2013
1 / 28
Se
Math 2253 sec. 004: June 3 Summer 2013
(From before:)
Section 2.2: If f (x ) is a function, then the set of numbers f (a) for
various values of a should dene a new function.
Let f (x ) be a function. Dene the new function
f (x ) = lim
h 0
f (x + h) f (x )
Math 2253 sec. 004: 5-20 Summer 2013
Suppose a ball is dropped from the top of the Space Needle 605 feet high. According to Galileos law, the distance s(t) feet the ball has fallen after t seconds
is (neglecting wind drag)
s(t) = 16t2 .
If average velocit
http:/www.spsu.edu/math/txu/math2253/2253f07/
Home Page of MATH 2253: Calculus I
4 credit hours Dr. Taixi Xu, Office: D-122, (678)915-7410, email: [email protected] Section 003: 10:00am-10:50am MTRF in D219 Meetings: Section 006: 1:00pm- 1:50pm MTRF in D
Math 2253 Review Problems for Exam 3 (Chapter 4)
1. Find the absolute maximum and minimum values of each function on the given interval. a) f ( x) x 2 1, 1 x 2 b) f ( x) 2 x 4 cos x , [0, ] 2. Find the extreme values of the function and where they oc
Review Problems for Exam 2
(Covers chapter 3) 1. Find the derivative of the function by the limit process. a) f ( x) x 2 2 x 3 x 1 b) f ( x) 2. Find an equation of the line that is tangent to the graph of f ( x) ( x 2 8)( x 2 7) at (3, 2). 3. Find th
Review Problems for Exam 4
(Covers Chapter 5) 1. ( A calculator can be used for this problem) Use finite approximation to estimate the area under the graphs of the function using i) A lower sum with four rectangles of equal width. ii) An upper sum wi
Math2253 Review Problems for Exam 1
(Covers chapter 2)
1. If f ( x) x 2 2 x 2 , find (a) f (3) (b) f (3 h) f (3 h) f (3) (c) h
(5) ( h2
4h 5 )
( h 4)
2. Complete the table and use the result to find the limit. x 2 3x 2 lim x 1 a) (0) x 2
x f (x)
Review Problems for Exam 3 (Chapter 4)
1. Find the absolute maximum and minimum values of each function on the given interval. a) f ( x) x 2 1, (abs. Max = 3, abs. min = -1) 1 x 2 b) f ( x) 2 x 4 cos x , [0, ] (abs. Max = 2 , abs. min = 4) 2. Find th
Review Problems for Exam 4
(Covers Chapter 5) 1. ( A calculator can be used for this problem) Use finite approximation to estimate the area under the graphs of the function using i) A lower sum with four rectangles of equal width. ii) An upper sum wi
Fall 2007 Math2253 Fianl Exam Review
Dr. Taixi Xu
Department of Mathematics, SPSU
Math2253 - Review Problems for Final Exam
Chpater 2: Limits and Continuity 1. If f ( x) x 2 2 x 2 , find (a) f (3) (5) (b) f (3 h) ( h 2 4h 5 ) f (3 h) f (3) (c) (h
Math 2253 sec. 004: 5-22 Summer 2013
Consider H (x ) =
0, x < 0
. Evaluate if possible
1, x 0
lim H (x ),
x 1
()
lim H (x )
x 0
May 21, 2013
1 / 17
We write
lim f (x ) = L
x a
and say that the left hand limit (or limit from below) of f (x ) as x
approache
Math 2253 sec. 004: 5-24 Summer 2013
Limit Taking: If f(x) is made up of polynomials, roots, products,
quotients, and/or trigonometric functions, try evaluating the limit by
substitution (i.e. evaluate f(a) if possible). If this fails, try using factoring
Math 2253 sec. 004: June 28 Summer 2013
Evaluate the limit
7x 2 + 3x 15
x 2x 4 x 3 + 2x + 1
lim
()
June 28, 2013
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Denition: Horizontal Asymptote
The line y = L is a horizontal asymptote to the graph of f if
lim f (x ) = L
x
or
lim f (x ) = L.
x
If