Business Calculus II
5.1
Accumulating Change: Introduction
to results of change
Accumulated Change
• If the rate-of-change function f’ of a quantity is
continuous over an interval a<x<b, the accumulated
change in the quantity between input values of a and b
is the area of the region between the graph and
horizontal axis, provided the graph does not crosses
the horizontal axis between a and b.
• If the rate of change is negative, then the accumulated
change will be negative.
• Example:
– Positive- distance travel
– Negative-water draining from the pool
5.1 – Accumulated Distance
(PAGE 319)
Accumulated Change involving
Increase and decrease
• Calculate positive region (A)
• Calculate negative region (B)
• Then combine the two for overall change
Maximum
Rate of Change
(ROC)
Function
Behavior
Minimum
Positive Slope
Zero
Negative Slope
Positive Slope
Zero
Rate of
Change (ROC)
Function
Behavior
Inflection Point
Concave Down
Decreasing
Concave Up
Increasing
• Problems 2, 6, 7, 12 (pages 324-328)
Business Calculus II
5.2
Limits of Sums and the Definite
Integral
Approximating Accumulated Change
• Not always graphs are linear!
– Left Rectangle approximation
– Right Rectangle approximation
– Midpoint Rectangle approximation
Left Rectangle approximation
Sigma Notation
• When xm, xm+1, …, xn are input values for a
function f and m and n are integers when
m<n, the sum
f(xm)+f(xm+1)+….f(xn)
can be written using the greek capital letter
sigma () as
Right Rectangle approximation
Mid-Point Rectangle approximation
Area Beneath a Curve
• Area as a Limit of Sums
• Let f be a continuous nonnegative function
from a to b. The area of the region R between
the graph of f and x-axis from a to b is given by
the limit
Where xi is the midpoint of the ith subinterval
of length x= (b-a)/n between a and b.
Page 334- Quick Example
• Calculator Notation for midpoint approximation:
Sum(seq(function * x, x, Start, End, Increment)
• Start: a + ½ x
• End: b - ½ x
• Increment: x
Left rectangle
• Calculator Notation :
Sum(seq(function * x, x, Start, End, Increment)
• Start: a
• End: b - x
• Increment: x
Right Rectangle
• Calculator Notation:
Sum(seq(function * x, x, Start, End, Increment)
• Start: a + x
• End: b
• Increment: x
Related Accumulated Change to signed
area
• Net Change in Quantity
– Calculate each region and then combine the area.
Definite Integral
• Let f be a continuous function defined on
interval from a to b. the accumulated change
(or definite Integral) of f from a to b is
Where xi is the midpoint of the ith subinterval
of length x= (b-a)/n between a and b.
Problems 2, 8 (pages 338-342)
Business Calculus II
5.3
Accumulation Functions
Accumulation Function
• The accumulation function of a function f,
denoted by
gives the accumulation of the signed area
between the horizontal axis and the graph of f
from a to x. The constant a is the input value
at which the accumulation is zero, the
constant a is called the initial input value.
2. Velocity (page 350)
x
Area
Acc. Area
0
1
2
3
4
5
6
7
8
9
10
4. Rainfall (page 351)
x
Area
Acc. Area
0
1
2
3
4
5
6
Using Concavity to refine the sketch of
an accumulation Function (Page 348)
Faster
Slower
Slower
Faster
Graphing Accumulation Function using F’
f'
f(x)=.05(x-1)(x+3)(x-5)^2
8
6
4
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
-2
-4
-6
-8
When F’ Graph has x-intercept, then you have Max/Min/inflection point in
accumulation graph
How to identify the critical value(s):
MAX in Accumulation graph:
When F’ graph changes from Positive to negative
MIN in Accumulation graph:
When f’ graph changes from negative to positive
Inflection point in accumulation graph:
When F’ touches the x-axis
Or
You have MAX/MIN in F’ graph
Graphing Accumulation Function using F’
f'
f(x)=.05(x-1)(x+3)(x-5)^2
8
6
4
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
-2
-4
-6
-8
Max: Positive to negative
Positive F’
x-intercept, MAX – in Accumulation graph
Negative F’
7.5
Graphing Accumulation Function using F’
f'
f(x)=.05(x-1)(x+3)(x-5)^2
8
6
4
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
-2
-4
-6
-8
Min: negative to Positive
Positive F’
x-intercept, MIN – in Accumulation graph
Negative F’
7.5
Graphing Accumulation Function using F’
f'
f(x)=.05(x-1)(x+3)(x-5)^2
8
6
4
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
-2
-4
-6
-8
Inflection Point: F’ Touches the x-axis
x-intercept, MIN – in Accumulation graph
Graphing Accumulation Function using F’
f'
f(x)=.05(x-1)(x+3)(x-5)^2
8
6
4
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
-2
-4
-6
-8
Inflection Point: inflection point in F’, also
appears as inflection point in accumulation
graph
Inflection Points in F’
6.5
7
7.5
WHAT WE HAVE COMBINE
f'
f(x)=.05(x-1)(x+3)(x-5)^2
8
INF
INF
6
MAX
4
MIN
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
-2
-4
-6
INF
-8
INF
INF
6.5
7
7.5
f'
f(x)=.05(x-1)(x+3)(x-5)^2
8
Positive
area
6
4
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
-2
-4
-6
f'
f(x)=0.05(x^5/5-2x^4+2x^3/3+40x^2-75x)-9
10
-8
8
6
4
Start at
zero
2
x
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
-2
-4
-6
-8
-10
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
10-Sketch
12-sketch
14-sketch
Business Calculus II
5.4
Fundamental Theorem
Fundamental Theorem of Calculus
(Part I)
For any continuous function f with input x, the
derivative of
in term use of x:
FTC Part 2 appears in Section 5.6.
Anti-derivative
Reversal of the derivative process
Let f be a function of x . A function F is called
an anti-derivative of f if
That is, F is an anti-derivative of f if the
derivative of F is f.
General and Specific Anti-derivative
• For f, a function of x and C, an arbitrary constant,
is a general anti-derivative of f
When the constant C is known, F(x) + C is a specific
anti-derivative.
Simple Power Rule for Anti-Derivative
More Examples:
Constant Multiplier Rule for AntiDerivative
Sum Rule and Difference
Rule for Anti-Derivative
Example:
Connection between
Derivative and Integrals
• For a continuous differentiable function fwith
input variable x,
Example:
Problem: 2,12,14,16,20,22,24,37
Business Calculus II
5.5 Anti-derivative formulas for
Exponential, LN
1/x(or x-1) Rule for Anti-derivative
ex Rule for Anti-derivative
ekx Rule for Anti-derivative
Exponential Rule for Anti-derivative
Natural Log Rule for Anti-derivative
Please note we are skipping Sine and Cosine Models
Example
Example (16 – page 373):
Problems: 2, 6, 8, 10, 20, 24
(page 373-374)
Business Calculus II
5.6 The definite Integral Algebraically
The fundamental theorem of Calculus
(Part 2) – Calculating the Definite
Integral (Page 375)
• If f is continuous function from a to b and F is
any anti-derivative of f, then
• Is the definite integral of f from a to b.
• Alternative notation
Sum Property of Integrals
• Where b is a number between a and c
Definite Integrals as Areas
• For a function f that is non-negative from a to b
= the area of the region between f and the x-axis from a to b
Definite Integrals as Areas
• For a function f that is negative from a to b
= the negative of the area of the region between f and the xaxis from a to b
Definite Integrals as Areas
• For a general function f defined over an interval
from a to b
= the sum of the signed area of the region between f and the x-axis
from a to b
= ( the sum of the areas of the region above the a-axis) minus (the
sum of the area of the region below the x-axis)
Problems: 10, 14, 18, 20, 22
Business Calculus II
5.7 Difference of accumulation
change
Area of the region between two curves
• If the graph of f lies above the graph of g from
a to b, then the area of the region between
the two curves is given by
Difference between accumulated
Changes
• If f and g are two continuous rates of change
functions, the difference between the
accumulated change of f from a to b and the
accumulated change of g between a and b is
the accumulated change in the difference
between f-g
Problems: 2, 6, 10, 12, 14
Business Calculus II
5.8 Average Value and Average rate
of change
Average Value
• If f is continuous function from a to b, the
average value of f from a to b is
The average value of the rate of
change
• If f’ is a continues rate of change function
from a to b, the average value of f’ from a to b
is given as
• Where f is a anti-derivative of f’.
Problems: 2, 6, 10, 18