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 ( x)
x 3
2. For the function h(x) below, find each sp
BIOL 1201 Principles of Biology II Lab
2015 Spring
Science and Tech 333
Instructor:
Adam Stuckert
[email protected]
Office: Howell N313
Office hours to be determined
BIOL 1201 is designed to complement BIOL1200 by providing hands-on experience
Quiz 2.1 F16 Tsai
1. Consider the graph above showing the relationship between x and 3!. Use it to answer the
following questions.
a] Find the average rate of change in y with respect to X over the interval [-1,2].
Raj43017: o-J_ I
an "T Gi?
b) Estimate
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. Create a right triangle using your tangent line:
3. Using
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 through
2.
Given
y log( x 5)
( k , 2)
and
is perpendicu
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) Factor/cancel and go back to step 1
3) !, -!, or DNE
2x " 6
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 continuity:
1. continuity at a point
2. continuity over an
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 Limits
Suppose f is defined for all x near a with
x>a
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 to look at the long-term behavior.
Find:
lim f ( x )
x
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 of values for
analysis.
We also looked at the instanta
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 learned many different characteristics of a rational functi
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 polynomial and a is in the domain of p(x), then
lim p( x) p(a
MAT 271 Continuity Discovery Activity
Names_
Using each of the graphs for f(x) below, find
lim f ( x )
&
x c
f(c)
, then answer the following
questions:
lim f ( x )
x c
f(c)
y2
y1
1.
_
o
_
c
y2
y1
2.
_
o
_
c
y2
y1
3.
_
o
_
c
y2
y1
4.
_
_
c
y2
y1
5.
_
_
c
Practice 2.1 Average and Instantaneous Rates of Change
The position of a car is given by the following data:
t
s(t)
0
0
1
10
2
32
3
70
4
119
Plot the data on the grid below:
Find the average velocity for the time period beginning at t = 2 and lasting.
i)
Trig Graphing Basics
1. List the coordinates of the ordered pairs, degrees, and radians for each of the 5 locations shown below in Quadrant I of the Unit Circle.
e
Coordinates
d
Degrees
Radians
a
c
b
b
c
d
a
2. Draw one complete standard rotation of
e
y=s
1. Given the graphs of f(x)and g(x) below, estimate the value of each indicated derivative:
a. (f g)(-6) = _
Y1 =
f(x)
b. Let u(x) = g(f(x). Find u(3) _
c. (f +g)(-3) = _
y 2= g(x)
'
'
2. Suppose that f ( 2 )=1, g ( 2 )=2, f ( 2 )=3,g ( 2 )=4,
a
( f g )'
Derivative Practice
Use the shortcut formulas to find
1.
for each function below: (Do not simplify!)
f x
f x x 3e cos3 x
2
6
2.
f x x x2 5
3.
f x sin 3 x 4 2 x 2 5 tan 5 x 1 6 x
4.
3
2e 2 x 1
f x
x
3
4
5.
f x
4x
3
3 x
2
9x5
4
MAT 271 Lab 2 Derivatives and Tangent Lines
For each function below in problem 1 and 2,
I. Symbolically:
a. Find the derivative at any x using
derivative rules,
b. Evaluate the derivative at the specified
points,
c. Specify all x-values where the derivati
Wake Technical
Community College
Course Syllabus
MAT 271 Calculus I Section
Mathematics & Physics Department
Mathematics & Sciences Division
Fall
2016
LECTURE
3
LAB
2
CREDIT
4
Description
This course is designed to develop the topics of differential and i
Derivative Practice Take 2
Use the shortcut formulas to find
1.
for each function below: (Do not simplify!)
f x
f x (ln( 3x 2 4 x) 3 e tan
1
x
1
2 6 x 4 tan 1 x
1
2
f ' x (ln( 3 x 2 4 x) 3 e tan x
3
ln
3
x
4
x
2
e
2
1 x
3x 4 x
2.
f x (2 x 3 ln
Derivative Practice
Use the shortcut formulas to find
1.
f x
for each function below: (Do not simplify!)
f x (ln( 3x 2 4 x) 3 e tan
2.
f x (2 x 3 ln x)3 (5 x 3 8)
3.
f x sin x 3 e 6 tan x
4.
arccos( 2 x)
f x
4
3x 5
4
1
x
5.
f x
3x 2
5
6 x2
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 common formula you learned in MAT 171: _
The position of a
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 helpful rules:
Examples: Determine the following limits
l
MAT 271 Test #lA . Name ~6N’
Fall 2015 . 190 minutes
Show all work for full credit. Cl‘Early mark final answers. |nc|ude units where appropriate. Calculators are not allowed.
_ 2x2 + 5x —3
1. Given f(x) = 7 , (12 pts)
x“ + 2x — 3i
JHA, ‘I/ i’i\
[/ L
MAT 271 Test #3A Name
Fall 2015 2100 minutes
Show all work for full credit. Clearly mark final answers. include units where appropriate. Calculators are not allowed.
1. The graph shown below is the graph @Txythe derivative of a continuous function fix)
MAT 271 Test #4A
Fall 2015
Name
l Qae,
q 00 minutes
Show all work for full credit. Clearly mark final answers. Include units where appropriate. Calculators are not allowed.
1. Determine the most general antiderivative of each function below. Do not simp
Section 2.6: Continuity
There are two types of continuity:
Type # 1: _
Type # 2: _
Type # 1: _
In other words: f(x) is continuous at a point where x = a if there are no holes, gaps, or jumps in the graph of
f(x) at x = a (you can trace the curve without l
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 values or state that they do not exist.
Use the graph of f