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Session |
Date |
Read & Study |
Discussion Topics |
Practice HW Problems |
Technology Tips/Other Info |
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28 |
5 - 9 |
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Final Exam Tuesday, May 9, 10
am - 12:25 pm
Bring a scientific calculator
Course Grades |
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27 |
4 - 27 |
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Test 3 (All topics since test 2)
Bring a scientific calculator |
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26 |
4 - 25 |
class notes 12.1 12.5 |
Section 12.1 Just like for derivatives, sometimes a formula
must be rewritten before an antiderivative is found: reciprocals are
rewritten using negative exponents, 1/x3 = x-3
, roots are rewritten using fractional exponents, √x = x1/2
, the reciprocal of a root is rewritten using a negative fractional exponent,
1/√x = x-1/2 , etc. There are rules for finding
antiderivatives of exponential functions: f(x)=ex
is F(x)=ex , of f(x)=aex
is F(x)=aex where a is any
constant. For example, f(x)=ex + 2ex
has an antiderivative F(x)= ex
+ 2ex . Remember how to check an
antiderivative: F'(x)=f(x). After an
antiderivative is found, it can be used to evaluate a definite integral using
the fundamental theorem of calculus. See the previous session for a description
of this rule. |
Section 12.1 Text Page 789
Find an antiderivative of an exponential function # 13, 14
Section 12.5
Text Page 830
Find an antiderivative of an exponential function and then use it to evaluate
a definite integral : example 2. Also consider the "Explore and Discuss" problem
1 on page 832 - find the correct answer. |
Note: There is a typing mistake in problem 11, page 732. The
function formula should be y=4ex-3xe
. |
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25 |
4 - 20 |
12.1 12.5 |
Section 12.1 The antiderivative of a constant is the
constant times x: for f(x)=3 an
antiderivative is
F (x) = 3x. The process of going from a function to an
antiderivative of the function is called integration. To find the antiderivative of a sum,
you just "integrate" one term at a time: for f(x)=9x2
+ 12x - 1 an antiderivative is
F (x) = 9(x3/3) + 12(x2/2) -
1∙x = 3x3 + 6x2 - x .
Section 12.5 The Fundamental Theorem of Calculus is a
rule that helps us evaluate definite integrals. Let
F(x) be an antiderivative of f(x) then
Study the examples done in class. |
Section 12.5 Text Page 837
Find an antiderivative then evaluate the definite integral #
5 - 14 all, 23-26 all |
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24 |
4 - 18 |
9.7 12.1 |
Section 9.7 Another application of marginal analysis. Since C(x)
is the total cost to produce x items, C(101)-C(100) is the exact cost to produce
the 101st item. It can be shown that C '(100) ≈ C(101) - C(100, that is, C
'(100) gives the approximate cost of the next item, the 101st item. The general
rule for any problem: C '(x) gives the approximate cost of producing the next
item, the (x+1)st item. This is also true for the other marginal functions:
R '(x) gives the approximate revenue from selling the next item, the (x+1)st
item; P '(x) gives the approximate profit from selling the next item, the
(x+1)st item.
Handout # 9 When the graph of a density function is symmetric,
the median is the middle value that divides the area of the graph
equally: hence 50% of the area is to the left and 50% of the area is to the
right. When the graph is not symmetric, the median can be estimated by moving
from the middle until there appears to be 50% to the left and right.
Section 12.1 A function F(x) is an
antiderivative of a function f(x) if F '(x) =
f(x). The formula for the antiderivative of xn is
xn+1 / (n+1) "add one to the exponent and
divide by that exponent" , provided n is not equal to -1. Hence for f(x)=x2
, an antiderivative is
F (x) = x3/3. The formula for the antiderivative
of cxn is cxn+1 / (n+1)
. |
Section 12.1 Text Page 789
Find a formula for the antiderivative # 1 - 8 all |
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23 |
4 - 13 |
class notes 11.2 |
Practice with your calculator on evaluating an exponential
function: evaluate f(x) = 350e-0.001x
when x = 1000.
The inverse of differentiation is integration. Geometrically,
the definite integral of f(x) from x=a to
x=b is the net area between the curve f(x), the x-axis,
x=a and x=b. The notation for this is
see page 822. An application of
areas involves density functions. A density function: (1) is never
negative and so its graph is never below the x-axis; (2) the total
area between the graph of the function and the x-axis is 1 ; (3) the area
between the function curve, the x-axis, x=a and x=b
gives the percent or proportion of the population that is between x=a
and x=b .
Each student received a copy of HW Handout #9. |
Homework Handout #9
Evaluate and find the rate of change for an exponential function # 1
Set up the definite integral for a specified area. # 2, 3, 4
Find the net area # 5
Density functions # 6, 7, 8, 9
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22 |
4 - 11 |
9.7 |
In economics, the word marginal refers to a rate of
change: C '(x) is the marginal cost
function; R '(x) is the marginal revenue function; etc.
Review the interpretation of the rate of change value f '(a) from
session 8: When the input is a, the function output is changing by f '(a)
output units per input unit, and in particular, the function output is
increasing if f '(a) is positive and the function output is
decreasing if f '(a) is negative. Study the examples in section
9.7 and the examples done in class. |
Section 9.7 Text Page 620
# 1, 3, 5, 11(abc), 13(abcdefghi) |
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21 |
4 - 6 |
9.6 11.2
class notes |
Section 9.6 The function f(x)=(2x+3)4
is "a function to a power" . To find the rate of change function
f '(x), use the general power rule: we get that f(x) = [
u(x) ]n has rate of change function
f '(x) = n[ u(x) ]n-1
∙ u '(x), see page
605. (This is a special case of the chain rule that is used to find the
rate of change of a function that has a formula with a function inside of a function.) Study
example 1, page 605, and the examples done in class.
Section 11.2 In exponential functions, the input variable appears in the
exponent of the function formula. The most important base in exponential
functions is the irrational number e ≈ 2.718281.... named to honor
the Swiss mathematician Leonhard Euler. The exponential function f(x)=ex
has an amazing property "it is its own derivative" that is f '(x)=ex.
Study example 3a on page 726. The general exponential derivative rule
says that a function f(x)=eu(x)
has rate of change function f '(x)=eu(x)
∙ u '(x). Study example 4
on page 727 and the examples done in class. |
Section 9.6
Text Page 610
Use the general
power rule
# 7, 11, 15, 17, 19, 27, 29, 31, 37, 39, 43
Section 11.2
Text Page 732
Rate of change of an exponential function # 9, 11, 13, 17, 19, 21, 23, 27,
29, 35, 57ab |
Online lectures on some of our course topics, including the
chain rule, are available. Click
here to view the
lecture topics. |
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20 |
4 - 4 |
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Test 2 (All topics discussed beginning with session 11, study class
notes and homework handouts # 5, 6, 7, 8) Bring a scientific calculator
(a clear straight edge is also recommended) |
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19 |
3 - 30 |
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Questions were answered in review for the test. Study the
rules and strategies described below. |
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18 |
3 - 28 |
class notes 10.2 |
The second derivative of the function f
(x), denoted f
"(x), is the derivative of the first derivative. Study the
examples done in class and in section 10.2. The second derivative gives
information about concavity: when f "(x)>0, the function is
concave up, and when f "(x)<0, the function is concave down.
At an inflection point f "(x)=0 or f "(x)
does not exist. Given f(x), the strategy to find the
inflection points of f(x) is: (1) solve f
"(x)=0 to get the possible candidates for inflection points; (2)
check the concavity at test points on each side of the candidate by
substituting into f"(x); (3) if the concavity changes from one
side to the other, then you may conclude that the point is a true inflection
point. Remember in application problems that at an inflection point:
the tangent line has greatest or smallest slope compared to nearby points;
hence, the output is increasing or decreasing at the fastest or slowest rate
compared to nearby points.
Each student received a copy of HW Handout #8. |
See below and
Text Page 665
Find the second derivative
# 7 - 12 all
All of Homework Handout #7
Find the coordinates of all possible inflection points #
4, 5All of Homework Handout #8
Is the function concave up or concave down? Is the point an
inflection point? Find the coordinates of possible inflection points. #1,2,3,4 |
Online lectures on some of our course topics are available. Click
here to view the
lecture topics. |
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17 |
3 - 23 |
class notes 10.1 |
In the graphing window [a,b] x [c,d]
the x-axis extends from x=a to x=b and
the y-axis extends from y=c to y=d.
Different persons may choose different graphing windows; what is important
is whether or not the graphing window that you choose shows the important
properties of the function like the intercepts, turning points, inflection
points, etc. Review the steps outlined in session 16
for using the rate of change function f
'(x) to find the possible turning points of f. Note in step 3
that if f
'(x) does not change sign from the left side to the right side of
the possible turning point, then the point is not a turning point of
f:
the graph of f is leveling off at the point but it is not changing direction.
After you find the intercepts and turning points of the given function, you
can set up a graphing window that will contain these important points. Study
the examples completed in class.
Note: peak points are also called local maximum points;
valley points are also called local minimum points; the x-coordinates
of possible turning points are also called critical values. |
See below and
Homework Handout #7
Write a possible graphing window # 2a(v), 2c(v), 4a(v), 4b(v), 5(v), 6a(v),
7a(v), 7b(iii)
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16 |
3 - 21 |
class notes
10.1 |
More examples were done to find the rate of change
function f
'(x) by using the four shortcut rules (power, constant multiple,
constant and sum rules).
Problem: Use the rate of change function f
'(x) to find the turning points and to determine if it is a
peak point or a valley point on the graph.
Solution: (step 1) solve f
'(x) = 0 to find the x-coordinate of possible turning
points, then substitute the x-value into the original function f(x)
to find the y-coordinate;
(step 2) calculate the rate of change f
'(x) at a test point on the left side and on the right side of
the possible turning point;
(step 3) if rate of change f
'(x) changes from negative to positive, then the function changes
from decreasing to increasing and you can conclude
that the point is a turning point and that it is a valley point; if f
'(x) changes from positive to negative, then the function changes
from increasing to decreasing and you can conclude
that the point is a turning point and that it is a peak point; if f
'(x) does not change sign, then the function does not change
direction and you can conclude that the point
is not a turning point. |
Homework Handout #7 Find the coordinates of all possible turning points # 1a,
1b, 2a, 2b, 2c, 4a, 4b, 5, 6a, 7a, 7b
Determine which of the points are actually turning points. If yes, is the
turning point a peak point or is it a valley point? # 3, 8
Text Page 649
(a) Find f
'(x); (b) Find the (x,y) coordinates of possible
turning points; (c) Which of the points in part b are actually turning points?
If the point is a turning point, is it a peak point or is it a valley point?
Justify your answers. # 19 - 22 all |
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15 |
3 - 9 |
9.4 |
You should become familiar with and be able to use the
following four shortcut rules for finding the rate of change
function f
'(x), which is also called the derivative of f. (1) Power rule,
see page 588. (2) Constant multiple rule , if f
(x) = c∙g(x), then f
'(x) = c∙g
'(x), see other versions on page 589. (3) Constant rule ,
if f(x) = c, then f
'(x) = 0, see page 587 for other versions. (4) Sum rule,
if f
(x) = [ g(x) + h (x) ] then f
'(x) = g
'(x) + h
'(x), "you differentiate a term at a time", see page 591 for
other versions. Note (a) a special case of the power rule is when f(x)
= x then f
'(x) = 1; (b) sometimes you must rewrite a function formula
before you can apply a shortcut rule. Review special exponents: 1/xn
= x-n , and "nth root of x" = x1/n
, and x0
=1 for nonzero x. Some standard
terminology: The process of going from f
(x) to f '(x) is called differentiation
or differentiating f
(x).
Each student received a copy of HW Handout #7 which we
will begin discussing in our next class session. |
See below. |
Online lectures on some of our course topics are available. Click
here to view the
lecture topics. |
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14 |
3 - 7 |
class notes 9.3
9.4 |
Note: When we use the limit definition to find the
rate of change function f
'(x), we are finding exact answers, not approximate
answers.
Review the relationships: (1) f
'(x) = instantaneous rate of change of f at x; (2) f
'(x) = slope of tangent line at x; (3) "use the limit
definition to find f
'(x)" is the same as "use the 4-step method to find f
'(x)".
There are many shortcut rules for computing f
'(x): the power rule states that f
(x)=xn has rate of change f '(x)=nxn-1
where n is any real number. See page 588. |
Text, Page 593 Find the rate of
change function using shortcut rules (power rule, constant multiple rule, sum
rule, constant rule)
# 1 - 17 odd, 25 - 47 odd |
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13 |
3 - 2 |
class notes 9.3 |
The class practiced with more examples to approximate the
rate of change f
'(x) by using a table of average of rates of change of f
between x and x + h. By studying the table, we can
complete the statement: As the increment h approaches 0, the average
rates of change approaches what particular number? Then f
'(x) is approximately equal to this particular number. Each
student received a copy of HW handout #6 to practice this method.
The limit definition of the rate of change function f
'(x) generalizes the method of approximating f
'(x) from a table of average rates of change. See the formula on
page 577. Review the relationships: f
'(x) = rate of change of f at x and f
'(x) = slope of tangent line at x. You may find it helpful
to use the four-step process to find f
'(x) using the limit definition as shown in example 4, page 577.
Since the graph and tangent line coincide at the tangent point: when the
slope of the tangent line f
'(x) > 0, the function is increasing at x; when the
slope of the tangent line f
'(x) < 0, the function is decreasing at x. Review
constant functions: if f
(x) = 3, then no matter the input, the output is 3. |
Homework Handout #6 Estimate f
'(x) using a table of average rates of change # 1, 2, 3, 4, 5
Text, Page 583
Find f
'(x) using the limit definition
# 5, 7, 9, 11, 31 - 38 all, 59bc, 60bc |
Note: I made a mistake on the answer key to test 1, problem
9a. The correct answer is $150. To make up for this, four points will be
added to each student's test 1 grade. |
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12 |
2 - 28 |
class notes |
There was a review of determining where f
'(a) is not defined when you are given a function graph - see
problem 12, page 648 in the text. There was a review
of concavity when you are given a function graph: approximating the
coordinates of an inflection point; finding (the inputs) where the
function is concave up; finding (the inputs) where the function is
concave down.
We went through our first example of approximating the
rate of change f
'(x) by using the 3rd method described at the last class session.
In this method, a table is created: (1) in each row, the value of f[x,x+h]
is computed; (2) the value of x does not change from row to row; (3)
but, from row to row, the value of the increment h gets smaller and smaller, "h
approaches 0"; (4) last, look for a pattern in the f[x,x+h]
column, what number do the f[x,x+h] values approach? This
answer is approximately is the rate of change of the function f
at x, f '(x). More examples will be studied in class next time. |
See below. |
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11 |
2 - 23 |
class notes |
Sometimes it is not possible to calculate the instantaneous
rate of change f
'(a) and we say f
'(a) is not defined: (1) x=a is not in the
domain of the function; (2) the graph is broken at x=a; (3)
the graph has a sharp turning point at x=a; (4) the graph has
a vertical tangent line at x=a.
The graph of a function may have different types of
curvature: intuitively, we say the function is concave up for the
inputs where the graph "holds water" and we say the function is concave
down for the inputs where the graph "spills water". An inflection
point is a point where the graph changes concavity. Geometrically: the
tangent line has the greatest or smallest slope at an inflection
point. The practical meaning of an inflection point: the function is
increasing or decreasing at the fastest or slowest rate at the
inflection point.
There is a 3rd method of estimating the slope of
the tangent line from a graph: 1st, keep one point fixed (the tangent
point), 2nd, choose a second point on the graph, and draw the secant
line connecting these two points. Keep repeating this process: choose a new
second point that is closer to the tangent point, and draw the secant
line connecting the points. As the second point on the graph gets closer and
closer to the tangent point, the secant line will look more and more like
the tangent line. More importantly, as the second point gets closer and
closer to the tangent point, the slope of the secant line gets closer and
closer to the slope of the tangent line. This method can be quantified by
using the idea of an increment h. The increment h
may be positive or negative. For example, if we increment x=2 by h=0.5,
we get x+h=2+0.5=2.5. Geometrically, this means we are focusing on
the secant line through the tangent point with x-coordinate 2
and the second point with x-coordinate 2.5. Next, we calculate
the slope of this line. We will go through a complete example in our next
class.
Homework Handout #5 was given to each member of the class. |
Text, Page 648 Follow given directions
for #6
For each of #12, 13, 15, 16, find any x coordinates in
the domain of the function where f '(x) does not exist.
Text, Page 664
Follow given directions for #1(abg), #2(abg)
For each of #43-46, determine (i) where the function is
concave up; (ii) where the function is concave down; (iii) the coordinates of
inflection points Homework Handout #5
Find the inflection point # 1 - 7 |
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10 |
2 - 21 |
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Test 1 (All topics discussed since session 1, study
class notes and homework
handouts # 1, 2, 3, 4) Bring a scientific calculator
(a clear straight edge is also recommended) |
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2 - 16 |
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All UHD classes are canceled until 4 p.m. A fire in the
downtown area has compromised power to the university.
Test 1 is rescheduled for Tuesday, February 21. |
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9 |
2 - 14 |
class notes |
Remember the instantaneous rate of change of f at
x=a, denoted f
'(a), is defined to be the slope of the
tangent line at the point (x,y)=(a,
f(a)). Hence many properties of instantaneous rate of change
can be determined by visual inspection of the graph of f(x).
For a function graph: (1) if the tangent line at
x=a is rising then the slope is positive and so the
instantaneous rate of change f '(a) > 0; (2) if the
tangent line at
x=a is falling then the slope is negative and so the
instantaneous rate of change f '(a) < 0; (3) if the
tangent line at
x=a is horizontal then the slope is zero and so the
instantaneous rate of change f '(a) = 0. |
See below |
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8 |
2 - 9 |
class notes |
Remember that the instantaneous rate of change of f at
x=a, denoted f
'(a), is defined to be the slope of the
tangent line at the point (x,y)=(a,
f(a)). This gives us a second method for approximating f
'(a): (1) draw a tangent line at the point (x,y)=(a,
f(a)); (2) choose any two points on the tangent line and
estimate the (x,y) coordinates of each point; (3) use the two
points from step 2 to calculate the slope of the tangent line; (4) then f
'(a) is approximated by the slope from step 3. When drawing a
tangent line, keep in mind that: (1) the line should contain the tangent
point (x,y)=(a,
f(a)); (2) the line should have the same tilt as the curve at
the point (x,y)=(a,
f(a)); (3) the line and the curve should appear to coincide
near the point (x,y)=(a,
f(a)). Just as for average rate of change, the units on the
instantaneous rate of change are "output units" per "input unit". When
interpreting f
'(a) and this is a positive number, a template that may be
helpful is "At the input x=a, the output quantity is
increasing by f
'(a) output units per input unit." |
Homework Handout # 4 Draw a tangent line,
estimate the slope of the tangent line, estimate the instantaneous rate of
change, interpret an instantaneous rate of change # 1, 2, 3, 4, 5, 6, 7, 8, 9 |
For the answers to the odd-numbered problems in the homework
handouts, just click on the column heading "Practice HW Problems." |
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7 |
2 - 7 |
class notes and 9.3 (figures
with tangent lines) |
Motivating example: think of John with his cruise
control set at 60 mph; (1) the graph relating distance traveled to elapsed
time is a straight line; (2) his average velocity is 60 mph between any two
times; and (3) his velocity at any instant is 60 mph. More generally, when a
graph relates distance traveled to elapsed time and the graph is a straight
line then : (1) the average velocity or average rate of change between any
two inputs equals the slope of the line; (2) the velocity or instantaneous
rate of change at any single input also equals the slope of the line. We can
generalize this to any function as follows. On a smooth unbroken curve,
when you zoom-in at the
point where x = a, it will appear to be a straight line and we will assume the curve has the same
properties as the line. Therefore, the instantaneous rate of change
of the function at x = a is the slope of this line,
approximately, and this value is denoted f '(a) "f
prime of a." As it turns out, if you could repeatedly zoom-in indefinitely,
the graph will look more and more like a particular line called the
tangent line at the point x = a. Formally, f
'(a) is defined to be exactly equal to the slope of the
tangent line of the function f at the point (x,y)=(a,
f(a)).
Homework Handout #4 was given to each member of the class.
This will be the cut-off for material for Test 1. |
Study the examples done in class Here is
a practice zoom-in problem. |
For the answers to the odd-numbered problems in the homework
handouts, just click on the column heading "Practice HW Problems." |
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6 |
2 - 2 |
Class notes and section 1.3 |
Note that the average rate of change of the function f(x)
between x=a and x=b is the same as the slope
of the line that goes through the graph of f(x) at the points
(x1,y1)=(a, f(a)) and (x2,y2)=(b, f(b));
this line is called a secant line. Hence many properties of average
rate of change can be determined by visual inspection of the graph of f(x):
(1) a line that rises from left to right has positive slope; (2) a line that
falls from left to right has negative slope; (3) a horizontal line has zero
slope; (4) a vertical line has no slope. The slope of a line measures the
steepness or tilt of the line - the more steeply a line rises, the larger
the slope, the more steeply a line falls, the more negative the slope. If a
line relates the distance traveled (y-values) to elapsed
time (x-values), then "the slope of the line" equals "the average
rate of change of the function between any two inputs" which equals "the average
velocity between any two input times" which equals "the velocity at any particular
time". This is a unique property of lines. When you zoom-in on a
point on a graph which is a smooth unbroken curve, it will
appear to be a straight line - hence we will assume the curve has the same
properties as the line. This observation will help us switch from
calculating the "average velocity between two times" to calculating the "instantaneous
velocity at a particular time". |
Text, Page 49 Which lines have slope that
is negative? positive? # 1, 2
Write an equation of the line # 13, 14
Find the slope of the line # 43a, 46a, 49a, 50a |
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5 |
1 - 31 |
Class notes and section 9.3 (only average rate of change) |
Given the graph of a function: (1) to find where the
function is increasing, find the interval of inputs where the graph is
rising from left to right; (2) to find where the function is decreasing,
find the interval of inputs where the graph is falling from left to right;
(3) a turning point is a point on the graph where the direction of
the function changes, that is, on one side of the turning point the graph is
rising and on the other side the graph is falling. The maximum output
of a function is the y-coordinate of the highest point on the graph,
if there is one. The minimum output of a function is the y-coordinate
of the lowest point on the graph, if there is one.
The average rate of change of the function f(x)
between x=a and x=b is denoted by f[a,b]
and the formula is f[a,b] = [change in
outputs]/[change in inputs] = [ f(b) - f(a) ] /
[b - a ]. The units of measure for the average rate of
change are "output units per input unit". The interpretation of a
positive average rate of change is: "On average, from x=a to
x=b, the function output increases f[a,b]
output units per unit increase in the input." If the average rate of change
is negative, then replace "increases" by "decreases."
Homework Handout #3 was given to each member of the class. |
Text, Page 583
Find and interpret the average rate of change # 1a, 3a, 4a,
29a, 59a, 60a
Homework Handout # 3
Find and interpret the average rate of change # 1, 2, 3, 4
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The number of absences that you have in the course will be
counted starting today since the official day of record was January 30.
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4 |
1 - 26 |
Class notes and section 1.1 |
Business applications of functions: For a price-demand
function, x is the number of items that can be sold at p
dollars per item; revenue is (number of items sold)(price per item)
so R(x)=xp; the cost for manufacturing and
selling x items is C(x); the break even point is
where revenue equals cost; the profit P(x) for
manufacturing and selling x items is revenue less cost, and so P(x)=R(x)-C(x).
What is the practical meaning if the profit P(x)=0? if P(x)<0?
P(x)>0? Study pages 13 - 16 in section 1.1.
Determining the turning points of a function and the
intervals where the function is increasing and where the function is
decreasing is often important in application problems.
Homework Handout #2 was given to each member of the class. |
Text, Page 19 Finding
demand/revenue/profit
# 81(ab), 83(ab)
Homework Handout # 2
Finding: turning points; intervals where the function is
increasing; intervals where the function is decreasing # 1, 2, 3, 4 |
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3 |
1 - 24 |
Class notes and section 1.1 |
Algebra review: (a) if each term of an expression has a
common variable factor, then the smallest power of the variable should
be factored out from the expression. For example, 3x2-5x=x(3x-5); (b) to solve an equation where one side is zero, and
the other side is a product, you should set each variable factor in the
product to 0. Quadratic equations may sometimes be solved by
factoring, but if factoring is not possible, then the quadratic formula
should be used. For a function formula: (1) to find
any x-intercepts you should set the
output to 0 and solve for the matching inputs; to find the y-intercept, you should set the input
to 0 and find the matching output. Note that a function may not have any x-intercepts
and it may not have a y-intercept. In our course, if a function formula does not have a
denominator containing a variable and does not have a variable in the
radicand of an even root, then usually the domain is the set of all
real numbers. For a function, it is often important to find the intervals of
inputs where the function is increasing (the graph is rising) and the
intervals of inputs where the function is decreasing (the graph is
falling). A point on the graph where the function changes direction (on one
side the function is increasing, and on the other side the function is
decreasing) is called a turning point. |
Homework Handout # 1 Finding x and y-intercepts
# 1, 2, 3, 4
And see below |
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2 |
1 - 19 |
Class notes and section 1.1 |
Section 1.1 (1) Note that repeated outputs are OK for
a function. (2) If a point (x,y) is contained in the
graph of the function f , then x is the input and y
is the matching output, that is, f(x)=y. See problems
19-26 on page 18. (3) For function graphs, it is helpful to use vertical
lines to find the domain, and it is helpful to use horizontal lines
to find the range. (4) To find the domain of a function represented by a
formula, you must remember that division by zero is not defined
and that an even root of a negative number is not a real number.
Hence, if a function formula has a denominator containing a variable, set
the denominator to zero to help find the domain; and if the formula has a
variable in the radicand of an even root, set the radicand > 0 to
help find the domain. (5) In application problems, f(x) and
y=f(x) have practical meaning in the context of the
problem. |
Text,
Page 17
Finding the domain # 39, 42,
43,
Writing function formulas #47,
48,
Homework Handout 1
Finding the domain #1, 2, 3, 5 - 8
Interpreting function notation # 5 - 8
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The number of absences that you have in the course will be
counted starting with the official day of record, January 30.
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1 |
1 - 17 |
Class notes and section 1.1 |
Section 1.1 A function assigns exactly one output
to each input. A function can take the form of a table, graph
or a formula. Each point (x,y) contained in a graph
gives a matching input-output pair, that is, the input is x and the
matching output is y. Formulas are often written using function
notation: x is the input variable and in total f(x),
read "f of x", stands for the matching output. Hence, f(2) "f of 2"
stands for the output at the input 2. The set of all inputs is called the
domain of the function. The set of all outputs is called the range
of the function. The x-intercepts are the inputs that have
output 0, if there are any. The y-intercept is the output for
the input 0, if there is one. A copy of Homework Handout #1 was given
to each student. |
Text, Page 17
Basics of functions
# 1, 4, 5, 8, 9,
17a, 19, 22, 23, 27, 32, 33, 35, 38,
Function
notation # 69, 71 |
You are expected to have your own calculator (at least
scientific) for this course. |