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