Problem 1. Let f (x) = . Using the denition of the derivative prove that
1
Jd (x) @
Solution. The function f (x) is only dened when x 2 0, so we will assume that x 2 0 for the
remainder of the solution.
By the denition of the derivative,
f(X+h)f(x)
x) 2
10/15/2015 Sample Exam
Sample Exam
Math 31A
There will be eight questions, each worth 20 points.
1, Jana/E
. Using the definition of the derivative prove that
1
30:7
.1:
2. Given that
sin(h) :
11m 1
a Ii
prove, using the definition of the derivative, that
Math 31A Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
N0 due date, but the quiz in Week 2 in the discussion section (on October 6th or 8th) will
be based upon this homework.
Assignment 2
Exercise 1. Dene
1,1
Math 31A Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
No due date, but the quiz in Week 3 in the discussion section (on October 13th or 15th) will
be based upon this homework.
Assignment 3
Exercise 1. Comput
Math 31A Steven Heilrnan
Please provide complete and wellwritten solutions to the following exercises.
Due October 2, at the beginning of class.
(Remember to also read the syllabus by noon PST, October 2.)
Assignment 1
Exercise 1. Find the equation for th
MATH 31A: DIFFERENTIAL AND INTEGRAL CALCULUS
FALL 2016 LECTURE 5
GENERAL GUIDELINES
IN SHORT
You need to know everything that was covered in lectures 18, and everything
from the textbook that corresponds to what was covered in those lectures. The officia
MATH 31A (Butler)
Practice for Final (A)
Try to answer the following questions without the use of book, notes or calculator;
but you can use the equation sheet posted on the course website. Time yourself and
try to finish the questions in less than three
MATH 31A (Butler)
Practice for Final (C)
Try to answer the following questions without the use of book, notes or calculator;
but you can use the equation sheet posted on the course website. Time yourself and
try to finish the questions in less than three
MATH 31A (Butler)
Practice for Final (B)
Try to answer the following questions without the use of book, notes or calculator;
but you can use the equation sheet posted on the course website. Time yourself and
try to finish the questions in less than three
1. Let
f(a:)m;1;m4  $3  gm? + 1.
(a) Find the critical numbers of f.
(0) Find the intervais of increase and decrease of f.
_/
(LAW 61
bi/ 7c Xx? XH
(Vega!)  "
(vi, 0] ~ ' +
(big) ~+ ~ +
[300/ + f "f 2. Calculate the limit, if it exists.
limtanzix
j
MATH 31A (Butler)
Practice for Final (A)
Try to answer the following questions without the use of book, notes or calculator;
but you can use the equation sheet posted on the course website. Time yourself and
try to finish the questions in less than three
Final practice, Math 31A  Lec 1, Fall 2013
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
Total:
90
Score
1. (a) (4 points) Find all points x at which the function f (x) = x2
4x + 3 + x1/3 is not dierentiable.
First 
MATH 31A (Butler)
Practice for Final (C)
Try to answer the following questions without the use of book, notes or calculator;
but you can use the equation sheet posted on the course website. Time yourself and
try to finish the questions in less than three
,
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Final practice, Math 31A  Lec 1, Fall 2013
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
Total:
90
Score
1. (a) (4 points) Find all points x at which the function f (x) = x2 4x + 3 + x1/3 is not dierentiable.
First 
Final practice, Math 31A  Lec 1, Fall 2013
Instructor: Toni Antunovi
c
c
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
Total:
90
Score
1. (a) (4 points) Find all points x at which the function f (x) = x2 4x + 3 + x1/
Math 31A Lecture 2: Homework 9 part II, Due 12/6 in class
5.6: 72, 79.
6.1: 15, 22, 29, 38, 48.
6.2: 6, 8, 14, 20, 22.
6.3: 8, 16, 18, 26, 30, 46, 56.
and the following problems:
1. Find c such that the area between the curves x4 c and c x4 is 1.
2. Compu
Math 31A Lecture 2: Homework 8, Due 11/22 in class
Section 5.2: 14, 24, 40, 64, 74, 79
Section 5.3: 4, 18, 32, 38, 44, 47, 50.
and the following problems:
1. Evaluate
2
f (x)dx,
0
where f (x) = x2 if 0 x 1, and f (x) = 2 x if 1 x 2.
2. Find f and a value
Practice problems for Midterm II
Try to answer the following questions without the use of book, notes or calculator. Each problems
count for 10 points.
1. Find the area of the largest rectangle that you can make where the bottom edge is on the xaxis
and
Math 31A Lecture 2: Homework 6, Due 11/8 in class
Section 4.5: 10, 28, 42,50,62.
Section 4.6: 10, 16 , 38, 34, 48, 56
Section 4.8: 8, 22, 32,40, 50, 60, 74.
and the following problems:
1. Find the maximum volume among all cylinders that can t inside a uni
Math 31A Lecture 2: Homework 4, Due 10/25 in class
Section 3.8: 22, 36, 42, 48.
Section 3.9: 10,18, 28,30,38.
Review Exercises: 78.
Section 4.1: 10, 24, 38, 42, 48.
and the following problems:
1. Find all (x, y) on the curve y 4 + y + x + 3 = 0 such that
Practice Midterm I
Try to answer the following questions without the use of book, notes or calculator. Each problems
count for 10 points.
1.
(a) Find
lim (
t0
1
1
).
3t t(t + 3)
(b) Find
sin 2
.
0 tan 3
lim
2. Let
g(x) =
1
x2
if
if
ax + b
0 < x 1;
x 1.
(a
Math 31A Homework 3: Due 10/18 in class
Section 3.3: 16, 22, 40, 56.
Section 3.4: 16, 22
Section 3.5:12,16, 26, 40, 41.
Section 3.6: 20, 34, 42
Section 3.7: 2,16, 42, 62.
and the following problems:
1. Compute the second derivative of f (x)g(x) in terms o