Section 3.11 Related Rates
Example 1:
Let A be the area of a square whose sides have length x, and assume that x varies with time t.
a. Draw a picture of the square with the labels A and x placed appr
Math 271 Lab 1 on Limits
1.
Name: _
10
8
6
4
2
x
-10
-5
0
-2
0
5
10
15
-4
-6
-8
-10
y
a)
_
lim f ( x)
b)
=_
e)
lim
x ( )
f ( x)
c)
_
lim f ( x)
f)
x 6
x ( 5 )
d)
_
lim f ( x)
x ( 7)
f (0) _
_
lim f
Test 1A MAT 272 SP16 Name: S0 M1 on 5
There are 100 minutes for the entire test. A Tl36X-Pro is allowed. You must show your integralsI workI
etc. for credit.
if
1. Set up and evaluate the integral tha
'3)
MAT 271 - Lab at Related Rates Name ‘L. :2, q.
1. A ladder 10 feet long leans against a vertical building. If the bottom of the-”ladder slides away from t_h_e
lyﬂdhmizo—ntalllatﬂtsof 2 ft/s, how f
Interpreting the Instantaneous Rate of Change Key
Exercise 1: According to data found on the-numbers.com website the weekend gross in millions of
dollars, G, for the movie Zombieland can be modeled by
BIOL 1201 Principles of Biology II Lab
2015 Spring
Science and Tech 333
Instructor:
Adam Stuckert
Stuckerta10@students.ecu.edu
Office: Howell N313
Office hours to be determined
BIOL 1201 is designed t
MAT 271 Lab 2: Interpreting the Instantaneous Rate of Change SU16
In the following exercises, four interpretations are given. Determine which interpretations are correct and
which ones are incorrect a
Limits
MAT 271 Lab 1
Name_
6
-3
1. Use the graph above to find the following limits (note: y-axis scale is 1).
lim f ( x) _
x5
lim f(x) _
x 10
lim f ( x) _
x5
lim f(x) _
x
lim f ( x) _
lim f ( x) _
Section 2.2
Definitions of Limits
In the previous section, we looked at using
calculations of secant slopes to approximate the
tangent slope. We did this by looking at what
value the secant slope appr
Section 2.3
Techniques for Computing Limits
Limits of Linear Functions
Let a, b and m be real numbers. For linear
functions f(x) = mx + b,
lim f ( x) f (a) m(a) b
x a
Limit Laws
If the limits lim f (
Introduction to Calculus
Section 2.1
The Idea of Limits
What is Calculus?
Calculus is the name given to a group of
systematic methods of calculation,
computation, and analysis in mathematics
which use
Section 3.1
Intro to Derivative
Recall,
In section 2.1, we looked at the slope of a
tangent line to the curve as the limit of the
slopes of secant lines by using a numerical
approach we created tables
ST 311
Evening Problem Session - Solutions
Week 5
1. True or false: (Modules 5.2, 5.3) [Learning Objective F2]
A 95% confidence interval means that 95% of all sample statistics will be in our interval
MAT 271
Unit 2 Review/Practice (sections 3.1-3.10); study labs, quizzes, notes, &HW, too!
#1
Differentiate each function below. Do not simplify your derivative.
a) f (x) 3 ln(3x 6) * sin(e x )
b) f (
Section 2.4
Infinite Limits
Figure 2.23
Given f(x)
f(x)
a. lim
x 0
find:
f(x)
b. lim
x 1
Given f(x)
f ( x) 4
a. lim
x 0
find:
b. lim f ( x)
x 1
Figure 2.24 (a)
Figure 2.24 (b)
One-Sided Infinite Limi
Limits
MAT 271 Lab 1
Name: KEY
6
-3
1. Use the graph above to find the following limits (note: y-axis scale is 1).
lim f ( x) 1
lim f ( x) DNE
lim f ( x)
x 5
x 3
lim f(x) 4.5
lim f(x) 7
lim f(x) 0
x
MAT 271
2.1
2.1 and 2.2
The Idea of Limits Finding the slope at a point on a curve
1. Draw a tangent line on the curve at the specified point.
For example, find the slope on the curve at x = 3.
2. Cre
MAT 271 PreCalculus Skills Review Quiz
Name _
Fall 2015 50 minutes total, no calculator allowed
Please show your work neatly to receive credit. Leave answers in exact form.
1.
Find k so that the line
Finding Limits Algebraically - Classwork
We are going to now determine limits without benefit of looking at a graph, that is lim f ( x ) .
x !a
There are three steps to remember: 1) plug in a
2) Facto
Section 2.6
Continuity
Most physical phenomena can be modeled by
continuous functions, therefore many of the
ideas we will explore in calculus require
continuous functions.
There are two types of cont
Section 2.4
Infinite Limits
Figure 2.23
Given f(x)
a. lim f ( x )
x 0
find:
b.
lim f ( x )
x 1
Given f(x)
a. lim f ( x)b. 4
x 0
find:
lim f ( x)
x 1
Figure 2.24 (a)
Figure 2.24 (b)
One-Sided Infinite
Section 2.5
Limits at Infinity
Figure 2.31
What about limits at infinity?
What purpose would these serve?
Many real-world situations can be modeled
with mathematical functions, so we can use
limits t
Section 3.1
Intro to Derivative
Recall,
In section 2.1, we looked at the slope of a
tangent line to the curve as the limit of the
slopes of secant lines by using a numerical
approach we created tables
MAT 171 Review: Rational Functions
What is a rational function?
A rational function is a function that can be written in the form _ where P(x) and Q(x)
are _. Also Q(x)0.
Recall from MAT 171, you lear
Section 2.3: Techniques for Computing Limits
Law
Linear
Functions
Explanation
For linear functions f(x) = mx +b,
Example
(3 + 8)
2
lim f ( x) f (a) m(a) b
x a
Polynomial
Functions
If p(x) is a polyno
Section 2.1: The Idea of Limits
Types of Velocity can sometimes be thought of as _ or _
Type # 1: _
This is velocity that is calculated over an interval of time.
To find average velocity, we use a com
Section 2.5: Limits at Infinity
If f(x) become arbitrarily large as x becomes arbitrarily large, when we write:
lim () =
Following limits are defined similarly:
lim () =
lim () =
lim () =
A few he
Section 2.2: Definitions of Limits
Quick Review from MAT 171
Using the graph to the left, find
the following:
h(1)
h(2)
h(4)
h(5)
Example: Use the graph of g(x) in the figure to find the
following val
HAT2?1 Lab #5 Name 151;!
There are several formulas from geometry that allow us to calculate areas of certain plane regions. We will develop
some methods that use calculus to nd areas of plane regions