4.7
Optimization Problems
In solving such practical problems the greatest challenge is often to convert
the word problem into a mathematical optimization problem by setting up the
function that is to be maximized or minimized.
Lets recall the problem-solv
1. (a) Find the differential dy of f (x) =
p
1+
1 x.
(b) Find the linearization of f (x) in part (a) at a = 0.
Winter 2016 - MATH 114 - Worksheet 6 - due Mar. 17 (11 p.m.)
Page 1 of 4
2. Show that the equation x3 + ex = 0 has exactly one real root.
Winter
1. Evaluate the limit. If it does not exist, explain why. Show all steps. No marks will be given
if lHospitals rule is used.
2 x+14
|5 x|
(a) (5 pts) lim
(b) (5 pts) lim 3
2
x3 2x x 15
x3 x 2x2 15x
Winter 2016 - MATH 114 - Worksheet 3 - due Feb. 3 (11 p.m
x
. Find (a) the domain of f , the x- and y-intercepts; (b) the critical points
1
(numbers) and the intervals on which f is increasing or decreasing; (c) the local maximum
and minimum values; (d) the intervals on which f is concave upward or concave downw
1. Consider the function f (x) =
2x 3 + 1.
(a) (7 points) Use the definition of the derivative to find f 0 (x). No marks will be
given if the definition is not used.
(b) (3 points) Find the equation of the line tangent to f (x) at x = 2.
Winter 2016 - MAT
1. Find the general antiderivative of each of the following functions.
(a) (5 points)
f (x) =
(b) (5 points)
2 6 cos2 x + cos x tan x
cos2 x
4x3 3 x + 10x4
g(x) =
2x4
Winter 2016 - MATH 114 - Worksheet 8 - due Mar. 31 (11 p.m.)
Page 1 of 3
2. (10 points)
MATH 114, WORKSHEET #1
FALL 2015
Due: See the schedule for your section on eC'lass
SOLUTIONS
1. S(,)lvo the inequality. Express your solution in the interval notation and illustrate the solution
on the real number line.
(at)
(b)
2. Find all I in the int
MATH 114, WORKSHEET #1
WINTER 2016
Due: Thursday, January 21, by 16:00
Please mark Q#1, Q#2, (b), and Q#3
1. (a) Consider the piecewise function:
f (x) =
sin x
, 1 < x < 0
2
1+x , 0 x<2
(i) ( 4 marks) Find f (1), f ( 2 ), f ( 4 ). Do f (2) and f (5) exist
1. Find the general antiderivative of each of the following functions.
(a) (5 points)
f (x) =
2 6 cos2 x + cos x tan x
cos2 x
f (x) = 2 sec2 x 6 + sec x tan x
and has antiderivative
F (x) = 2 tan x 6x + sec x + C.
(b) (5 points)
4x3 3 x + 10x4
g(x) =
2x4
1. (a) Consider the piecewise function:
f (x) =
sin x
, 1 < x < 0
2
1+x , 0x<2
(i) (4 marks) Find f (1), f ( 2 ), f ( 4 ). Do f (2) and f (5) exist?
(ii) (6 marks) Sketch the graph of f .
Winter 2016 - MATH 114 - Worksheet #1
Page 1 of 6
(b) (10 marks) So
MATH 114
WORKSHEET #2
COURSE SECTION:
LAST NAME:
FIRST NAME:
INSTRUCTIONS
Fill in all the information above.
Write your ID number on each INSIDE
page of your assignment.
DO NOT write your ID number on the
cover page.
This cover page must accompany you
Chapter 5 Integrals
Appendix E Sigma
The Greek letter
(capital sigma) is called sigma notation.
Definition: If f is a function defined on (0, ) and m and n are integers such
that m n, then
n
f (i) f (m) f (m 1) f (n 1) f (n)
i m
Example:
n
a) i
i 1
5
b)
4.3
How Derivatives Affect the Shape of a Graph
What does f say about f ?
To see how the derivative of f can tell us where a function is increasing or
decreasing, look at Figure 1.
Between A and B and between C and D, the tangent lines have positive slope
Ch 4 Application of Differentiations
4.1
Maximum and Minimum Values
Some of the most important applications of differential calculus are
optimization problems, in which we are required to find the optimal (best)
way of doing something. These can be done b
4.9
Antiderivatives
A physicist who knows the velocity of a particle might wish to know its
position at a given time.
An engineer who can measure the variable rate at which water is leaking from
a tank wants to know the amount leaked over a certain time p
What is a NARRATIVE?
Narrative refers to the idea that events fit together in a meaningful sequence-a sequence
that produces meaning from otherwise random events.
Narrative creates the sense that events have a beginning and an end and that as you move
fro
MATH 113/114, WORKSHEET #1
FALL 2014
1. Consider a quadratic expiaition
12 a 201+ 1).]: + 3a + 2 2 0,
where a E R. Find all values of a so that the equation has two distinct. real roots.
2 Solve the inequality. Express your sohuion in the interval netmi
MATH 113/114, WORKSHEET #2
FALL 2011
S OLVTXQf
1. Solve the equation.
(a) .
(125) :41? (1512 )T
(b)
(3) 5 I 17
521'? : (025)128W 4
2. First find the (1011111111 and then solve the (squation/inequality:
/ \
Kai! ' , -
i ' 111(v.I:+3) )+ 111(x/4JZI+ 3):11
MATH 113/114, WORKSHEET #3
FALL 2014
3 SLUTl SIM?
1. Evaluate the lilllii or explain why it does not exist. Show all work. No marks will he givon
if lHospitals Rule is used.
1 71; M
(a) hill W
.I' ~> () ,If
(c) 11111 2.172 + 5.]? + 25
W: 12.1: L 3|
(1)
MATH 113/114, WORKSHEET #4
FALL 2014
5&CUWO/US
1. Evaluate the limit or explain why it, does not exist. Show all work, No Illll'kh will he givtn
if 1Hospita18 Rule is used.
(a) lini (V122 + 1 ~
.r ,1, 1 x
1
3 *1
(1) lim
.rn-atl 3m+1
2. Find the ve
MATH 114
WORKSHEET #1
COURSE SECTION:
LAST NAME:
FIRST NAME:
INSTRUCTIONS
Fill in all the information above.
Write your ID number on each INSIDE page
of your assignment.
DO NOT write your ID number on the cover
page.
This cover page must accompany you
QW'EMthAIWLI.
WWI: W57 — Tommyan 35915
=22) 1. Let f(:c) = $3 _ 1. Find (a) the domain of f, the w— and y-lntercepts, (b) the crltlcal pomts
(numbers) and the intervals on which f is increasing or decreasing; (C) the local maximum
and minimum values; (