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Calculus 1 Ms. Konnick Name ________________________________
Spring 2014
The following are the daily homework assignments for Chapter 4A β Graphing (Sections 4.2-4.4)
Section Pages Topics Assignment
4.2
4/9
p.209-216 Extreme values of functions (max and min),
First Derivative Theorem, critical points,
increasing and decreasing
p.216-218: # 1 β 21odd
4.2
4/10
p.209-216 Increasing and decreasing, 1st derivative test,
finding max/min values
p.216-218: # 27 β 33odd, 41 β 43,
49, 50, 55, 56
4.3
4/11
p.218-224 Concavity and the 2nd derivative test, points of
inflection, Intro Special Cases of Second
Derivative Test
p.224-227: # 1-2, 9-11, 13-15
4.3
4/14
p.218-224 Graphing Using First and Second Derivative
Tests
Worksheet
4.3
4/15
p.218-224 Review of First and Second Derivative Tests
Graphing using fβ and fβ, Special Cases: Cusps
and Vertical Tangents
Worksheet
p.216 #βs 3 β 5, 7, 29, 31, 35
4.3
4/16
p.218-224 Sketching Graphs when given properties
Sketching Graphs given derivatives
Worksheet
4.2-4.3
4/17
p.209-224 Review and Quiz on Sections 4.2-4.3 NONE
Enjoy your long weekend!
4.4
4/21
p.227-232 Vertical, horizontal, and slant asymptotes,
curve sketching using derivatives
p.233-234: # 1-7 odd, 9, 13, 15,
17, 21, 27, 28
4.4
4/22
p.227-232 Vertical, horizontal, and slant asymptotes,
curve sketching using derivatives
p.233-234: # 1-7 odd, 9, 13, 15,
17, 21, 27, 28
Review
4/23
Review for Chapter 4A Test Complete study guide problems
Test
4/24
Chapter 4A Test ~ Part 1 Continue to Study for Part 2 of
Unit 4A Assessment
Test
4/25
Chapter 4A Test ~ Part 2 Print Notes on Optimization for
class tomorrow!
CALCULUS I -NOTES Name __________________________________
SECTION 4.2-1 Date ______________ Block _______
4.2 Extreme Values of Functions (Day 1)
Let c be a point of the domain D of a function f. The number f(c) is:
Absolute maximum: of f if f(x) f(c) for all x in D; highest point on graph
(no greater value anywhere)
Absolute minimum: of f if f(x) f(c) for all x in D; smallest point on graph
(no smaller value anywhere)
Local maximum: of f if f(x) f(c) for all domain points in some open interval
(no greater value βnearbyβ; part of the graph that βpeaksβ)
Local minimum: of f if f(x) f(c) for all domain points in some open interval
(no smaller value βnearbyβ; part of the graph that βvalleysβ)
A functions maximum values (maxima) and minimum values (minima) are the functionβs extreme values
(extrema). They can occur at interior points and at the endpoints of the domain D.
Does f have a max or min value on [a,b]?
Max-Min Existence Theorem:
If f is continuous on closed interval [a,b] then f attains both an absolute maximum and minimum value.
How do we find Extreme Values (not looking at a graph)?
1) Find critical points
endpoints of the function (closed)
interior points of the domain where fβ(x) = 0 (stationary points)
interior points of the domain where fβ(x) DNE (singular points)
(sharp corners, jumps, etc.)
2) Plug critical points into f(x) (find the y-values)
3) Largest value = max Smallest value = min
Examples: Find the coordinates of the absolute maximum and absolute minimum on the given interval.
Then graph the function.
1) f(x) = 4 β x2 on [-1, 3]
2) f(x) = 3
2
x3 on [-1,1]
3) f(x) = -x2 + 4x β 1 on [0, 3]
4) f(x) = 51 (2x
3 + 3x
2 β 12x) on [-3, 3]
5) f(x) = sin x on [-, 2]
CALCULUS 1 Name: _______________________________
WORKSHEET 4.2-1 Date: ________________ Block: _______
Find the coordinates of the absolute maximum and absolute minimum on the given interval.
Then graph the function.
1. 1x6x3xf 2 , [0, 3]
2. 2x4xxf 4 , [β2, 1]
3. osxc2xf ,
2
3,0
4. 2x9xf , [β3, 2]
CALCULUS I -NOTES Name __________________________________
SECTION 4.2-2 Date ______________ Block _______
4.2 Extreme Values of Functions (Day 2)
First Derivative Test for Increasing and Decreasing
If f is continuous on [a,b] and differentiable (no cusps, breaks, asymptotes) on (a,b), then
1) If fβ(x) 0 (positive) for all x in (a,b), then f is increasing on [a,b]
2) If fβ(x) 0 (negative) for all x in (a,b), then f is decreasing on [a,b]
Where fβ(x) = 0 we have local max or min
The First Derivative Test for Local Extreme Values
1) If fβ changes from positive to negative at c, then f has a local max at c
2) If fβchanges from negative to positive at c, then f has a local min at c
3) If fβ does not change signs at c, then f has no local extreme value at c
4) At a left endpoint a: 5) At a right endpoint b:
c c
c c
c c
a a b b
fβ(c) = 0
fβ(c) = 0
fβ(c) = 0
fβ(c) is und.
fβ(c) is und.
fβ(c) is und.
Locate all extrema.
Examples: Find the intervals where the function is increasing and decreasing. Find the extreme values.
1) f(x) = 2x3 β 3x
2 β 12x + 7
2) f(x) = -x3 + 12x + 5 on [-3,3]
3) f(x) = x2 β 4x + 2 on [-3, 3]
4) f(x) = (x + 1)3
5) f(x) = 2x
5x 2
CALCULUS I Name __________________________________
WORKSHEET 4.2- 2 Date ______________ Block _______
Find the intervals where the function is increasing and decreasing. Find the extreme values.
1) f(x) = -4x2 β 7x + 5 2) f(x) = x
4 β 8x
2 + 1
3) f(x) = x3 β 4x 4) f(x) = 4x
3 β 6x
2
CALCULUS I -NOTES Name __________________________________
SECTION 4.3-1 Date ______________ Block _______
4.3 How yand y Determine the Shape of a Graph (Day 1)
How do we know if our graph looks like or ?
The difference between the two has to do with concavity (in very simple terms βopening upβ or βdownβ)
Definition:
The graph of a differentiable function y = f(x) is concave up on an interval where y is increasing and
concave down where y is decreasing.
The Second Derivative Test for Concavity:
The graph of y = f(x) is concave down on any interval where f 0 and concave up on any interval
where f 0.
Concave up: Concave down:
CU: f 0 CD: f 0
Consider the curve: We have critical points c1, c2 and c3.
c1 is a max
c3 is a min
the curve is concave down from (-, c2)
the curve is concave up from (c2, )
The place where the concavity changes is at c2.
We call c2 a point of inflection (also a critical point).
Definition:
A point where the graph of a function has a tangent line and where the concavity changes is called a
point of inflection.
Thus a point of inflection on a curve is a point where y is positive on one side and negative on the other.
At such a point, y is either zero or undefined. So y = 0 at a point of inflection.
c c
c1 c2 c3
CALCULUS I -NOTES Name __________________________________
SECTION 4.3-3 Date ______________ Block _______
4.3 How yand y Determine the Shape of a Graph (Day 3)
How to use yand y to Graph a Function
1) Compute the derivative )x(f
Find the first-order critical numbers of f (where )x(f = 0 and )x(f DNE)
Find where the graph is increasing or decreasing
Identify local extrema
2) Compute the second derivate )x(f
Find the second-order critical numbers of f (where )x(f = 0 and )x(f DNE0
Find where the graph is concave up or concave down
Identify the point(s) of inflection
Shapes of Graphs
)x(f 0
graph rises from left to
right (may be wavy)
)x(f 0
graph falls from left to
right (may be wavy)
)x(f 0, )x(f 0
increasing, concave up
)x(f 0, )x(f 0
increasing, concave down
)x(f 0, )x(f 0
decreasing, concave up
)x(f 0, )x(f 0
decreasing, concave down
)x(f = 0, )x(f 0
local maximum
)x(f = 0, )x(f 0
local minimum
)x(f = 0
point of inflection
f(x) = 0
x-intercepts
Find the intervals on which the graph of f is increasing, decreasing, concave upward, and concave downward.
Find the coordinates of any extreme points and point of inflection. Sketch the graph.
1) π π₯ = π₯2 β 6π₯ + 8
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
2) π π₯ = π₯3 β 12π₯ β 5
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
3) π π₯ = π₯4 β 4π₯3 + 5
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
4) π π₯ = π₯4 + 8π₯3 + 18π₯2 β 8
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
5) π π₯ = 1 β (π₯ + 3)3
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
6) π π₯ = π₯2
3 β 1
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
Calculus 1- Homework Name:
Section 4.3: How yβ and yβ Determine the Shape of a Graph Date:
Find the intervals on which the graph of f is increasing, decreasing, concave upward, and concave downward.
Find the coordinates of any extreme points and point of inflection. Sketch the graph.
1) π π₯ = π₯2 β 8π₯ β 9
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
2) π π₯ = π₯4 β 4π₯3 + 2
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
3) π π₯ = βπ₯4 + 6π₯2 β 4
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
4) π π₯ = π₯2
3 + 3
π β² π₯ = π" π₯ =
Critical Points: Critical Points:
Increasing: Decreasing:
Local Max(s): pt(s) of inflection:
Local Min(s):
CALCULUS I -NOTES Name __________________________________
SECTION 4.3-2 Date ______________ Block _______
4.3 How yand y Determine the Shape of a Graph (Day 2)
Vertical tangents
The graph of a continuous function has a vertical tangent at a point where the tangent is vertical and
the concavity is different on both sides.
Example: Identify the intervals where the function is increasing and decreasing and is concave upward
and concave downward. Find the coordinates of any extreme points and points of inflection.
1) f(x) = 31
x
Cusps
The graph of a continuous function has a cusp at a point where the tangent is vertical and the
concavity is the same on both sides. A cusp can be either a local maximum or a local minimum.
Example: Identify the intervals where the function is increasing and decreasing and is concave upward
and concave downward. Find the coordinates of any extreme points and points of inflection.
2) f(x) = 32
x
c c
fβ(c) is und fβ(c) is und
c
fβ(c) is und. fβ(c) is und.
c
CALCULUS 1- NOTES NAME:
SECTION 4.3: HOW Yβ AND Yβ DETERMINE THE SHAPE OF A GRAPH DATE:
1.) Use the graph to determine where the function meets the criteria
a) π π₯ = 0 b) πβ² π₯ = 0
c) πβ²β² π₯ = 0 d) π π₯ < 0
e) π β² π₯ < 0 f) π β²β² (π₯) < 0
g) π β²(π₯) > 0 h) π β²β² (π₯) > 0
i) π β²(π₯) > 0 & j) π β²(π₯) < 0 &
π β²β² π₯ < 0 π β²β² π₯ > 0
2.) Draw a function with the following properties:
π β3 = β4
π β2 = β1
π β1 = 2
π β²(π₯) = 0 π€βππ π₯ = β1,β3
π β² π₯ < 0 π€βππ π₯ < β3 πππ π₯ > β1
π β² π₯ > 0 π€βππ β 3 < π₯ < β1
π β² β² π₯ = 0 π€βππ π₯ = β2
π β² β² π₯ < 0 π€βππ π₯ > β2
π β² β² π₯ > 0 π€βππ π₯ < β2
3.) Sketch f(x)
π β7 = β1
π β4 = 2
π 0 = 8
π 4 = 2
π 7 = β1
4.) Draw a function with the following properties
π β5 = 0 π β² β² π₯ = 0 π€βππ π₯ = β5
π β² π₯ < 0 πππ πππ π₯ π β π β² β²(π₯) < 0 π€βπππ₯ < β5
π β² β² π₯ > 0 π€βπππ₯ > β5
5.) Draw a function with the following properties
π β2 = 8 π β² π₯ > 0 πππ π₯ < β2 π β² β² π₯ < 0 πππ π₯ < 0
π 0 = 4 π β² π₯ > 0 πππ π₯ > 2 π β² β² π₯ > 0 πππ π₯ > 0
π 2 = 0 π β² π₯ < 0 πππ β 2 < π₯ < 2
π β² β2 = π β² 2 = 0
CALCULUS 1- HOMEWORK NAME:
SECTION 4.3: HOW Yβ AND Yβ DETERMINE THE SHAPE OF A GRAPH DATE:
1.) Use the graph to determine where the function meets the criteria
a) π π₯ = 0 b) πβ² π₯ = 0
c) πβ²β² π₯ = 0 d) π π₯ < 0
e) π β² π₯ < 0 f) π β²β² (π₯) < 0
g) π β²(π₯) > 0 h) π β²β² (π₯) > 0
i) π β²(π₯) > 0 & j) π β²(π₯) < 0 &
π β²β² π₯ < 0 π β²β² π₯ > 0
2.) Draw a function with the following properties:
π β2 = β4 π β²(π₯) = 0 π€βππ π₯ = β2, 5 π β² β² π₯ = 0 π€βππ π₯ = 1.5
π 5 = 7 π β² π₯ < 0 π€βππ π₯ < β2 ππ π₯ > 5 π β² β² π₯ < 0 π€βππ π₯ > 1.5
π 1.5 = 3 π β² π₯ > 0 π€βππ β 2 < π₯ < 5 π β² β² π₯ > 0 π€βππ π₯ < 1.5
3.) Sketch f(x)
π 3 = β2
π β3 = β2
π β6 = 4
π 0 = β6
π 6 = 4
4.) Draw a function with the following properties
π 3 = 4 π β² β² π₯ = 0 π€βππ π₯ = 3
π β² π₯ > 0 πππ πππ π₯ π β π β² β² π₯ < 0 π€βπππ₯ < 3
π β² β² π₯ > 0 π€βπππ₯ > 3
5.) Sketch f(x) if f(0) = 0
6.) Sketch a continuous curve with the following properties. Label coordinates where possible. Give the coordinates
of any maximums or minimums.
x y Curve
2x Falling, concave up
2 1 Horizontal tangent
42 x Rising, concave up
4 4 Point of Inflection
64 x Rising, concave down
6 7 Horizontal tangent
6x Falling, concave down
CALCULUS 1 - NOTES Name _______________________________
SECTION 4.4 Date ___________________ Block ____
4.4: Graphs of Rational Functions
rational function: f(x) = h(x)
g(x); h(x) 0 (quotient of 2 polynomials)
f(x) = x
1 * undefined at ______ because _________________
* undefined at ______ because _________________
The graph approaches lines called ______________
vertical asymptote: the line _____ is a VA if
)x(flimax
or
)x(flimax
VA: f(x) = h(x)
g(x) VA occurs where x is undefined or h(x) = 0!
set denominator = 0 and solve for x
horizontal asymptote: the line _____ is a HA if b)x(flimx
order of polynomial: (compare highest powers of x)
same
same HA at y = ratio of coefficients
big
small HA at y = 0
small
big no HA
1. 3x
1y
VA: ________
HA: ________
CALCULUS 1 Name: _____________________________
WORKSHEET 4.4 Date: ______________ Block: ______
Write the equation(s) of each of the asymptote(s). Then, using the first derivative, sketch the graph.
1. f(x) = 2x
2
VA: _____
HA: _____
2. f(x) = 6x2
1x4
VA: _____
HA: _____
3. f(x) = 16x
22
VA: _____
HA: _____
slant (oblique) asymptote: the line _____ is a SA if b)x(flimx
if the degree of the numerator is one
more than the degree of the denominator
small
big no HA, may have an SA if given .etc,
x
x,
...x
x,
x...
x
3
4
2
...3
...2
*A function will have a HA or a SA but never both.
1. 1-x
4xy
2
VA: ________
HA: ________
SA: ________
2. 2x
4xy
2
VA: ________
HA: ________
SA: ________
ASYMPTOTES AND HOLES
One VA; One HA
Two VAs; One HA (just the middle)
One VA; one SA
c.p.
+ + c.p.
+ c.p.
+ c.p.
+ c.p. c.p. c.p.
+ c.p. c.p. c.p.
c.p. c.p.
+ c.p. c.p.
+ + c.p. c.p. c.p.
+ + c.p. c.p. c.p.
+ + c.p.
c.p.
max
min
max max
min min
Calculus 1 Name ___________________________________
Chapter 4A Study Guide
Chapter 4A review problems are located on p.267 in your textbook. You should be able to do the followingβ¦..
1. Use the first and second derivative test to find intervals of increasing/decreasing, points of maxs/mins, intervals
of concave up/down, points of inflection, and sketch an accurate graph of a polynomial function.
o # 22, 23, 26
2. Use the first and second derivative test to find intervals of increasing/decreasing, points of maxs/mins, intervals
of concave up/down, points of inflection, and sketch an accurate graph when given the first derivative.
o # 32, 33, 36
3. Use the first and second derivative test to find intervals of increasing/decreasing, points of maxs/mins, intervals
of concave up/down, points of inflection, and sketch an accurate graph of a function with fractional exponents.
o # 27, 28
4. Find asymptotes rational functions and use the first derivative test to find intervals of increasing/decreasing and
points of maxs/mins.
o # 46, 48, 52, plus other problems from section 4.4
5. Use a graph to determine whereβ¦
a.) absolute maximum:
b.) 0)( xf
c.) 0)(' xf
d.) 0)('' xf
e.) 0)(' xf
f.) 0)('' xf
g.) 0)(' xf
h.) 0)('' xf
i.) 0)('',0)(" xfxf
j.) 0)('',0)(' xfxf
6. When given criteria, sketch the graph of a function.
o Given the criteria below, sketch the graph of a continuous function, f(x), on the interval [-4,6] only. At
what points are there maximum and minimum values? Why?
510)(510)(
310)(310)(
0)5(0)1(0)3(0)1(
2)6(1)5(3)3(1)1(1)1(2)3(8)4(
xwhenxfxandxwhenxf
xwhenxfxandxwhenxf
ffff
fffffff
o Draw a function with the following properties
52 4)1(
11 0)('' 20 ;20)(3)0(
1 ;1 0)('' 2 x 0;x2- 0)('4)1(
1,1 0)('' 2 0, ,2 0)('5)2(
) f(f
xwhenxfxxwhenxff
xxwhenxfwhenxff
xwhenxfxwhenxff
7. Understand how the 1st and 2
nd derivatives are helpful.
o When is there a point of inflection?
o Can a function have both a local and absolute minimum? If yes, draw a graph. If no, explain why.
o Name the three places where a function can have extreme values? (maximums or mimimums)