5.1 Accumulating Change: Introduction to results of change 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

Rate of Change (ROC) Function Behavior Maximum 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

Problems 2, 6, 7, 12 (pages 324-328)

5.2 Limits of Sums and the Definite Integral 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)

5.3 Accumulation Functions 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 1 2 3 4 5 6 7 8 9 10 Area   Acc. Area

4. Rainfall (page 351)

Using Concavity to refine the sketch of an accumulation Function (Page 348) Faster Slower Increase decrease decrease Increase Slower Faster

Graphing Accumulation Function using F’ 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’ Max: Positive to negative   Positive F’ x-intercept, MAX – in Accumulation graph Negative F’

Graphing Accumulation Function using F’ Min: negative to Positive   Positive F’ x-intercept, MIN – in Accumulation graph   Negative F’

Graphing Accumulation Function using F’ Inflection Point: F’ Touches the x-axis   x-intercept, MIN – in Accumulation graph

Graphing Accumulation Function using F’ Inflection Point: inflection point in F’, also appears as inflection point in accumulation graph   Inflection Points in F’

WHAT WE HAVE COMBINE INF INF MAX MIN INF INF INF

Positive area Start at zero

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 Anti-Derivative

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

5.5 Anti-derivative formulas for Exponential, LN 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)

5.6 The definite Integral - Algebraically 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 x-axis 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

5.7 Difference of accumulation change 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

5.8 Average Value and Average rate of change 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