HW2
assignment
1.
2.
3.
A fair coin is tossed 5 times. Find a probability of obtaining (a) exactly 3
heads; (b) at most 3 heads
Solution:
Binomial distribution.
(a) P(n=3) = C5,3 0.530.52 = Binomial[5,3]*0.55 = 0.31
(b) P(n<=3) = 1-P(4)-P(5)= 0.81
A hat c
Math 20A. Lecture 10.
Theorem 1 (LHopitals Rule) Suppose that f (x) and g (x) exist with g (x) 0 for x x0 in an open
interval containing x0 , that either f (x) and g(x) both tend to 0 or both tend to as x x0 ,
and that f (x)/g (x) L as x x0 . Then
lim
xx
Math 20A. Lecture 8.
The First-Derivative Test at a point
A function f is increasing at the point x0 if there are numbers a and b with a < x0 < b such that
f (x) < f (x0 ) for a < x < x0 and f (x0 ) < f (x) for x0 < x < b. Similarly, f is decreasing at th
Math 20A. Lecture 4.
Leibniz notation and the differentiation operator
Isaac Newton (16421727) and Gottfried Leibniz (16461716), who are considered to be the founders of
calculus, each introduced notation for the derivative. Prime notation f (a) is simila
Math 20A
Lecture 1, p. 2
The natural exponential function y = ex is dened by
e = lim
n
1
1+
n
n
x
1+
n
or e = lim
x
n
n
.
Its graph and the graph of its inverse, the natural logarithm y = ln x, are in Figure 4.
The number e has the decimal value e = 2.718
Math 20A. Lecture 9.
Maxima and minima on finite, closed intervals
In the last lecture we found local maxima and minima of functions by determining where their derivatives
were positive and where they were negative. In cases where we need to find maxima a
Math 20A. Lecture 7.
This lecture deals with related rate problems which differ from those in earlier lectures in that the
necessary equations relating the functions are harder to find. We will use similar triangles and the
Pythagorean Theorem.
Using simi
Math 20A. Lecture 5 (corrected).
The Chain Rule for powers
We start with the rule for differentiating powers of functions.
Theorem 1 (The Chain Rule for powers)
df
of a function f exists at a point x and that n is a constant such
dx
that the value f (x) o
Math 20A. Lecture 6.
Compound interest
Theorem 1 Suppose that deposit of B0 dollars is made at at time t = 0 (years) in a bank account that pays P%
P
annual interest compounded n times a year. Let r =
denote the fraction that corresponds to
100
1 2 3
P%.
Math 20A, Lecture 2
One-sided and two-sided limits
If y = f (x) is dened on an open interval (a, x0 ) to the left of x = x0 and the number f (x) approaches
a number L as x approaches x0 from the left (Figure 1), we say that L is the limit of f (x) as x
te
Name: ._ PID:
TA: Sec. No: Sec. Time:
Math 20A.
Final Examination
December 8, 2010
Turn ofjr and put away your cell phone.
No calculators or any other electronic devices are allowed during this ezam.
You may use one page of notes, but no books or othe
MATH 20A
PRACTICE
FINAL
Answers
1. (a) cos2 () tan() sin2 () tan() + sin() sec()
(b) cos(cos(tan(x) sin(tan(x) sec2 (x)
(c) 2100 sin(2t )
1
6
1
6
2. ( 3 , 3 ), ( 3 , 3 )
3. (a) If f (x) = cos x x, then f (0) = 1 and f (/2) = /2, which
implies by the IVT t
0")
Name:
L/ ('
r;-.
U\ c
PID: - - - - - -
TA: _ _ _ _ _ _ _ _ _ _ _ Sec. No:
Sec. Time: _ _
Math 20A.
Final Exam
March 23, 2012
Tum off and put away your cell phone.
No electronic devices may be used during this exam.
You may use one page of notes, but n
Math 20A Final Review Outline
Prepared By Will Garner
Compiled on 10-13-2008
Math 20A Final Exam Review Outline
1
Chapter 1: Precalculus Review
Section 1.1: Real Numbers, Functions, and Graphs
Know the dierent types of shifts and what they do to a graph
Math 20A
Final Exam. December 9, 2002
VERSION 1
Instructions:
No books or notes; graphing calculators without symbolic manipulation
programs are permitted. Do all 10 problems in your blue book. Show all work; unsubstantiated
answers will not receive credi
Name: TA Name: Math 20A. Final Examination December 11, 2003
Section Number: Section Time:
You may use one page of notes, but no other assistance on this exam. Read each question carefully, answer each question completely, and show all of your work. Write
Math 20A
28 November 2006
Final Exam Review!
The following problems should provide a good review of the material being covered in the
exam. Once you understand how to do the homework problems, see if you can do these.
The Problems
1 If f (4) = 7 and f ( x
Math 20A. Final Examination December 9, 2008 1. Find the following limits: (a) (2 points) lim 3x - 1 , x 0 x 3x - 1 H ln(3)3x - 0 = lim = ln(3) . x 0 x 0 x 1
By L'Hopital's Rule, lim (b) (2 points) lim+ x log x ,
x 0
1 log x H lim x log x = lim = lim x =
Name: TA: Math 20A. Final Examination December 11, 2007 Sec. No:
PID: Sec. Time:
Turn o and put away your cell phone. No calculators or any other electronic devices are allowed during this exam. You may use one page of notes, but no books or other assista
Final for MATH 20A, Fall 2009
1. Use the intermediate value theorem to show that the equation cos(2x) = x has a solution in the
interval (0, /4).
Ans. Note the intermediate value themrem : If f (x) is continuous on [a, b] and f (a) = f (b),
then for any M
Fall 2011
Math 20A
Exam 1 vAR
University of California, San Diego
Department of Mathematics
Instructions
1. Write your Name, PID, Section, and Exam Version on the front of your Blue Book.
2. No calculators or other electronic devices are allowed during th
Fall 2011
Math 20A
Final vCR
University of California, San Diego
Department of Mathematics
Instructions
1. Write your Name, PID, Section, and Exam Version on the front of your Blue Book.
2. No calculators or other electronic devices are allowed during thi
Fall 2011
Math 20A
Exam 2 vBR
University of California, San Diego
Department of Mathematics
Instructions
1. Write your Name, PID, Section, and Exam Version on the front of your Blue Book.
2. No calculators or other electronic devices are allowed during th