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## Business Calculus II

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**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**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 Negative Slope Positive Slope Zero Zero**Rate of Change (ROC)Function Behavior**Inflection Point Concave Down Decreasing Concave Up Increasing**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**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.**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.**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’**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**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-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.**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.**Connection between Derivative and Integrals**• For a continuous differentiable function fwith input variable x,**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