Unit 8: Limits and
Continuity
Lesson 2: Limits and Piecewise Functions
Right-hand and Left-hand Limits
Observed last time that can take limit with
smaller or bigger numbers
Each choice has a name
Either right-hand or left-hand
Right-hand and Left-hand Lim
HPC Limits Name:
[Dav 4]
Evaluate each limit or state that it does not exist.
1- lim # :LTH. , 3 2. lim x+4 - E:
x>1 x2+2x+1 (I')u|)+| 6 x41 x2+2x+1 l+Z.-+l '1'
underied means you are approachirg .
a veracai astph'Jte- '
Sice nds 15 a Mo sidcd litm'L-i ,
Lesson 6:
Applications of
Derivatives
Unit 8: Limits and Continuity
Last Time
The derivative of a function is defined as
f ( x + h) f ( x)
f ( x ) = lim
h 0
h
Tells us instantaneous change
Graphically: Slope of the tangent line
Long, time-consuming proces
Lesson 5: Application of
Limits - Rates of Change
and the Derivative
Unit 8: Limits and Continuity
Average Rate of
Change
Change in y divided by change in x
y y 2 y1
m=
=
x x 2 x1
Also known as slope
Gives an idea of what happens over a given
period of
Pre-Calculus
Final Exam Review
Name:_
May June 2015
Use the following schedule to complete the final exam review.
Homework will be checked in every day.
Late work will NOT be accepted.
Homework answers will be provided at the beginning of each class p
HPC Limits Review Name:
For the function 1" whose graph is given, state the value of the given quantity, if it exists, or state it does not
exist.
1
li.mx_,1f(x)i limx_,1+f(x)i emf. our 4* 1+ 9.2
does not
hunk.1 f(x) ELM f (1)_.2-_
2
Ii'mxssxJL limxsax) L
Unit 8: Limits and Continuity
Lesson 1: Introduction to Limits
What is a limit?
Lets look at an example first
Calculating the area of a square
4
Square the length of the side and you get:
42 = 16
What is a limit?
Lets change the problem a just little
Lesson 3: Evaluating Limits
Unit 8: Limits and Continuity
Quick Review
How would you evaluate lim( 3x 4 ) ?
x3
Make a table
Come from both sides
See if both sides go to same value
What if there were an easier way?
There is an Easier Way
Notice that
Lesson 4:
Discontinuities
and Limits at
Infinity
Unit 8: Limits and Continuity
Last Time
We found that
lim f ( x ) = f (when
c)
f is defined
xc
at c
when denominator 0
Made finding limits much easier
What happens when denominator = 0?
Well get there, dont
Limits and Continuity
Quiz Review
For #1-2, use a table to find the limit.
1)
lim
x3
x 3 3x 2 2 x 6
x 4 4x3 6x 9
2)
lim
x0
cos x 1
sin x
For #3-9, find the limit. You may need to use algebraic techniques to aid you.
3)
5)
lim
( 2 x 2 4 x 7)
x5
lim
x 4
7)
Winter 2017 CS 32
Homework 1 FAQ
For problem 3 (and 5), what's an easy way to flip my test routine between testing a Sequence of std:strings and a Sequence of unsigned longs?
Here's one technique that lets you flip by commenting out or uncommenting just o
Homework 1 Test Data
As the syllabus said, not every problem of every homework will necessarily be graded. For this homework, we primarily graded problems 3 and 5, although you lost 20 points if your ScoreList in Problem 4 was not implemented using a Sequ
Winter 2017 CS 32
Homework 1 Solution
Problems 1, 2, and 3
Problem 4
Problem 5
Problems 1, 2, and 3:
In this solution, the functions with small, fast implementations are inlined. Alternatively, the inline keyword can be removed and the function definition
UCLA Computer Science Department
TA: Kung-Hua Chang
Student Name and ID _
CS 32, WINTER 2015, PRACTICE FINAL EXAM.
Problem #
1
2
3
4
5
6
7
8
9
10
11
12
13
Total
Maximal Possible Points
6
9
4
12
4
12
3
6
4
3
3
14
20
100
Received
*Every problem below has on
UCLA Computer Science Department
TA: Kung-Hua Chang
Student Name and ID _
CS 32, WINTER 2015, PRACTICE FINAL EXAM.
Problem #
1
2
3
4
5
6
7
8
9
10
11
12
13
Total
Maximal Possible Points
6
9
4
12
4
12
3
6
4
3
3
14
20
100
Received
*Every problem below has on
Speech
What is a smart car?
Cars that have all kinds of smart functions and features.
The ultimate goal for smart car is to achieve self-driving car.
Explain evolution of car
What do we need to create a smart car?
360-degree cameras, to view all sides of
MAE 94
Spring 2014
Introduction to Computer-Aided Design and Drafting
Lecture: Tuesday 10:00-11:50 AM
Instructor: Professor Rajit Gadh
Office Hours: Tuesday, 1 30 p.m. 3 30 p.m.
Lab Sections: Thursday and Friday, Engineering IV 38-138 (CAD Lab)
Teaching a
Problem 1. A researcher sends a questionnaire to 9,000
individuals. She gets back 5,000 responses. The percentage of
women among the respondents is 90%.
Because of nonresponse, the fraction of women among recipients
of the survey could be anything between
TEST #1
1.
(5) Set UP integrals to find the centroid of the region bunded by
Y = x2 and y a x + 2.
co W
2- (10) Evaluate: a. S; dx/(xz _ 2x + 2) b. Sgdx/(x + 1) 2/3
130 -
4~ utJnstrd. do ,3_
3 (5) Let f(x) X7 + x3 + 1. Explain why f is 1-1 and find the sl
Math260 Trig Review Handout
Sketch an angle q in standard position such that q has the
smallest positive measure and the given point is on the
terminal side of q.
Use the appropriate identity to find the indicated function
value. Rationalize the denominat
MachGO GpTest#1 C2toC4 Name
Show work for full Credit. Box in nal answer if appropriate. Member
If Graph, label as instructed in class. Member
Graph the function Compute and simplify the difference quotient
_ 4 f(x + h) x)
boon1 h ,h0.3
3) 100 = 11 - 6x