business calculus ii
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Business Calculus II. 5.1 Accumulating Change: Introduction to results of change. Accumulated Change. - PowerPoint PPT PresentationTRANSCRIPT
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Business Calculus II
5.1Accumulating Change: Introduction
to results of change
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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
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5.1 – Accumulated Distance (PAGE 319)
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Accumulated Change involving Increase and decrease
• Calculate positive region (A)• Calculate negative region (B)• Then combine the two for overall change
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Rate of Change (ROC)
Function Behavior
Negative Slope Positive SlopePositive Slope
ZeroZero
Minimum
Maximum
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Rate of Change (ROC)
Function Behavior
Concave UpIncreasing
Concave DownDecreasing
Inflection Point
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• Problems 2, 6, 7, 12 (pages 324-328)
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Business Calculus II
5.2 Limits of Sums and the Definite
Integral
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Approximating Accumulated Change
• Not always graphs are linear!– Left Rectangle approximation– Right Rectangle approximation– Midpoint Rectangle approximation
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Left Rectangle approximation
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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
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Right Rectangle approximation
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Mid-Point Rectangle approximation
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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.
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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
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Left rectangle
• Calculator Notation :Sum(seq(function * x, x, Start, End, Increment)
• Start: a • End: b - x• Increment: x
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Right Rectangle
• Calculator Notation:Sum(seq(function * x, x, Start, End, Increment)
• Start: a + x• End: b • Increment: x
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Related Accumulated Change to signed area
• Net Change in Quantity– Calculate each region and then combine the area.
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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.
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Problems 2, 8 (pages 338-342)
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Business Calculus II
5.3 Accumulation Functions
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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.
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2. Velocity (page 350)
x 0 1 2 3 4 5 6 7 8 9 10
Area
Acc. Area
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4. Rainfall (page 351)x 0 1 2 3 4 5 6AreaAcc. Area
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Using Concavity to refine the sketch of an accumulation Function (Page 348)
Increase
Increasedecrease
decrease
Slower
Slower
Faster
Faster
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Graphing Accumulation Function using F’f(x)=.05(x-1)(x+3)(x-5)^2
-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
-8
-6
-4
-2
2
4
6
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x
f'
When F’ Graph has x-intercept, then you have Max/Min/inflection point in accumulation graphHow 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 positiveInflection point in accumulation graph:When F’ touches the x-axis OrYou have MAX/MIN in F’ graph
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Graphing Accumulation Function using F’f(x)=.05(x-1)(x+3)(x-5)^2
-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
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f'
Max: Positive to negative Positive F’ x-intercept, MAX – in Accumulation graph Negative F’
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Graphing Accumulation Function using F’f(x)=.05(x-1)(x+3)(x-5)^2
-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
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x
f'
Min: negative to Positive Positive F’ x-intercept, MIN – in Accumulation graph Negative F’
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Graphing Accumulation Function using F’f(x)=.05(x-1)(x+3)(x-5)^2
-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
-8
-6
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-2
2
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x
f'
Inflection Point: F’ Touches the x-axis x-intercept, MIN – in Accumulation graph
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Graphing Accumulation Function using F’f(x)=.05(x-1)(x+3)(x-5)^2
-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
-8
-6
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-2
2
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x
f'
Inflection Point: inflection point in F’, also appears as inflection point in accumulation graph Inflection Points in F’
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WHAT WE HAVE COMBINE
f(x)=.05(x-1)(x+3)(x-5)^2
-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
-8
-6
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-2
2
4
6
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x
f'
MAXINF
INF
MIN
INF
INF
INF
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f(x)=.05(x-1)(x+3)(x-5)^2
-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
-8
-6
-4
-2
2
4
6
8
x
f'
f(x)=0.05(x^5/5-2x^4+2x^3/3+40x^2-75x)-9
-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
-10
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2
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x
f'
Positive area
Start at zero
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10-Sketch
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12-sketch
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14-sketch
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Business Calculus II
5.4 Fundamental Theorem
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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.
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Anti-derivativeReversal 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.
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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.
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Simple Power Rule for Anti-Derivative
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More Examples:
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Constant Multiplier Rule for Anti-Derivative
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Sum Rule and Difference Rule for Anti-Derivative
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Example:
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Connection between Derivative and Integrals
• For a continuous differentiable function fwith input variable x,
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Example:
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Problem: 2,12,14,16,20,22,24,37
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Business Calculus II
5.5 Anti-derivative formulas for Exponential, LN
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1/x(or x-1) Rule for Anti-derivative
ex Rule for Anti-derivative
ekx Rule for Anti-derivative
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Exponential Rule for Anti-derivative
Natural Log Rule for Anti-derivative
Please note we are skipping Sine and Cosine Models
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Example
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Example (16 – page 373):
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Problems: 2, 6, 8, 10, 20, 24 (page 373-374)
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Business Calculus II
5.6 The definite Integral - Algebraically
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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
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Sum Property of Integrals
• Where b is a number between a and c
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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
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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 x-axis from a to b
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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)
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Problems: 10, 14, 18, 20, 22
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Business Calculus II
5.7 Difference of accumulation change
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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
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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
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Problems: 2, 6, 10, 12, 14
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Business Calculus II
5.8 Average Value and Average rate of change
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Average Value
• If f is continuous function from a to b, the average value of f from a to b is
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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’.
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Problems: 2, 6, 10, 18