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### Business Calculus II Function (Page 348)

### Business Calculus II Function (Page 348)

### Business Calculus II Function (Page 348)

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

Rate of Change (ROC) Function Behavior

Minimum

Positive Slope

Negative Slope

Positive Slope

Zero

Zero

Rate of Change (ROC)Function Behavior

Inflection Point

Concave Down

Decreasing

Concave Up

Increasing

Approximating Accumulated Change

- Not always graphs are linear!
- Left Rectangle approximation
- Right Rectangle approximation
- Midpoint 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

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.

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.

Using Concavity to refine the sketch of an accumulation Function (Page 348)

Faster

Slower

Increase

decrease

Increase

decrease

Slower

Faster

Graphing Accumulation Function using F’ Function (Page 348)

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’ Function (Page 348)

Max: Positive to negative

Positive F’

x-intercept, MAX – in Accumulation graph

Negative F’

Graphing Accumulation Function using F’ Function (Page 348)

Min: negative to Positive

Positive F’

x-intercept, MIN – in Accumulation graph

Negative F’

Graphing Accumulation Function using F’ Function (Page 348)

Inflection Point: F’ Touches the x-axis

x-intercept, MIN – in Accumulation graph

Graphing Accumulation Function using F’ Function (Page 348)

Inflection Point: inflection point in F’, also appears as inflection point in accumulation graph

Inflection Points in F’

Positive area Function (Page 348)

Start at zero

10-Sketch Function (Page 348)

12-sketch Function (Page 348)

14-sketch Function (Page 348)

Fundamental Theorem of Calculus (Part I) Function (Page 348)

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 Function (Page 348)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 Function (Page 348)

- 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 Function (Page 348)

More Examples: Function (Page 348)

Constant Multiplier Rule for Anti-Derivative Function (Page 348)

Sum Rule and Difference Function (Page 348)Rule for Anti-Derivative

Example: Function (Page 348)

Connection between Function (Page 348)Derivative and Integrals

- For a continuous differentiable function fwith input variable x,

Example: Function (Page 348)

Problem: 2,12,14,16,20,22,24,37 Function (Page 348)

1/x(or x Function (Page 348)-1) Rule for Anti-derivative

ex Rule for Anti-derivative

ekx Rule for Anti-derivative

Exponential Rule for Anti-derivative Function (Page 348)

Natural Log Rule for Anti-derivative

Please note we are skipping Sine and Cosine Models

Example Function (Page 348)

Example (16 – page 373): Function (Page 348)

Problems: 2, 6, 8, 10, 20, 24 Function (Page 348) (page 373-374)

5.6 The definite Integral - Algebraically

The fundamental theorem of Calculus Function (Page 348)(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 Function (Page 348)

- Where b is a number between a and c

Definite Integrals as Areas Function (Page 348)

- 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 Function (Page 348)

- 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

Definite Integrals as Areas Function (Page 348)

- 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 Function (Page 348)

5.7 Difference of accumulation change

Area of the region between two curves Function (Page 348)

- 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 Function (Page 348)

- 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 Function (Page 348)

5.8 Average Value and Average rate of change

Average Value Function (Page 348)

- 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 Function (Page 348)

- 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 Function (Page 348)

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