Business Calculus II. Accumulated Change

Transcription

1 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 1

2 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 2

3 Rate of Change (ROC) Function Behavior Maximum Minimum Positive Slope Negative Slope Positive Slope Zero Zero Rate of Change (ROC) Function Behavior Concave Down Decreasing Inflection Point Concave Up Increasing 3

4 Problems 2, 6, 7, 12 (pages ) Business Calculus II 5.2 Limits of Sums and the Definite Integral 4

5 Approximating Accumulated Change Not always graphs are linear! Left Rectangle approximation Right Rectangle approximation Midpoint Rectangle approximation Left Rectangle approximation 5

6 Sigma Notation When x m, x m+1,, x n are input values for a function f and m and n are integers when m<n, the sum f(x m )+f(x m+1 )+.f(x n ) can be written using the greekcapital letter sigma (Σ) as Right Rectangle approximation 6

7 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 x i is the midpoint of the ithsubinterval of length x= (b-a)/n between a and b. 7

8 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 8

9 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. 9

10 Definite Integral Let fbe a continuous function defined on interval from a to b. the accumulated change (or definite Integral) of ffrom a to b is Where x i is the midpoint of the ithsubinterval of length x= (b-a)/n between a and b. Problems 2, 8 (pages ) 10

11 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. 11

12 2. Velocity (page 350) x Area Acc. Area 4. Rainfall (page 351) x Area Acc. Area 12

13 Using Concavity to refine the sketch of an accumulation Function (Page 348) Slower Faster Slower Faster 13

14 Graphing Accumulation Function using F f' f(x)=.05(x-1)(x+3)(x-5)^ x Max: Positive to negative Positive F x-intercept, MAX in Accumulation graph Negative F Graphing Accumulation Function using F f' f(x)=.05(x-1)(x+3)(x-5)^ x Min: negative to Positive Positive F x-intercept, MIN in Accumulation graph Negative F 14

15 Graphing Accumulation Function using F f' f(x)=.05(x-1)(x+3)(x-5)^ x Inflection Point: F Touches the x-axis x-intercept, MIN in Accumulation graph Graphing Accumulation Function using F f' f(x)=.05(x-1)(x+3)(x-5)^ x Inflection Point: inflection point in F, also appears as inflection point in accumulation graph Inflection Points in F 15

16 WHAT WE HAVE COMBINE f' f(x)=.05(x-1)(x+3)(x-5)^2 MAX MIN INF INF 2 x INF INF INF -8 f' f(x)=.05(x-1)(x+3)(x-5)^2 Positive area x f' f(x)=0.05(x^5/5-2x^4+2x^3/3+40x^2-75x) Start at zero x

17 10-Sketch 12-sketch 17

18 14-sketch Business Calculus II 5.4 Fundamental Theorem 18

19 Fundamental Theorem of Calculus (Part I) For any continuous function fwith 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 fbe a function of x. A function F is called an anti-derivative of f if That is, F is an anti-derivative of fif the derivative of F is f. 19

20 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 20

21 More Examples: Constant Multiplier Rule for Anti- Derivative 21

22 Sum Rule and Difference Rule for Anti-Derivative Example: 22

23 Connection between Derivative and Integrals For a continuous differentiable function fwith input variable x, Example: 23

24 Problem: 2,12,14,16,20,22,24,37 Business Calculus II 5.5 Anti-derivative formulas for Exponential, LN 24

25 1/x(or x -1 ) Rule for Anti-derivative e x Rule for Anti-derivative e kx Rule for Anti-derivative Exponential Rule for Anti-derivative Natural Log Rule for Anti-derivative Please note we are skipping Sine and Cosine Models 25

26 Example Example (16 page 373): 26

27 Problems: 2, 6, 8, 10, 20, 24 (page ) Business Calculus II 5.6 The definite Integral - Algebraically 27

28 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 28

29 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 29

30 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 30

31 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 31

32 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 32

33 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 33

34 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 34