VIEWS: 15 PAGES: 17 POSTED ON: 11/30/2011 Public Domain
Finite Sums Where we have been: rates of change, slopes of tangent lines, derivatives -differential calculus How can all those rates of change accumulate - areas under curves -integral calculus Newton and Leibniz proved connection between the two The Fundamental THM of Calculus -one of the most important insights ever!! Estimating with Finite Sums -remember problems from physics traveling constant velocity at a given time interval (distance = rate x time) -area under the curve was the distance This should work with varying velocity as well -just break up the velocity into small time intervals -then get a bunch of small rectangles - you can add up Ex. A particle starts at x=0 and moves along the x-axis with velocity For time greater than zero. Where is the particle at 3 seconds? Break into rectangles -more used - more accurate 3 ways to determine top of rectangle RAM - Rectangular Approximation Method MRAM - Midpoint RAM LRAM - Left hand RAM RRAM - Right hand RAM Sub Interval 0,1 1,2 2,3 Midpoint Definite Integral Page 1 Midpoint Height Area Sub 0,1 1,2 2,3 Interval Left Height Area Sub 0,1 1,2 2,3 Interval Right Height Area Not always least to greatest in this order - depends on curve (increasing or decreasing) -you can choose which to use or are told -sometimes called circumscribed, or inscribed rectangles Programs on Calculator RSUM N = number of rectangles desired Definite Integral Page 2 Definite Integrals -we looked at sums of areas of rectangles -leads to definite integrals -added area of rectangles k = number of rectangles Ck = midpoint, left or right -what part of rectangle is top DeltaXk = width of rectangle 1. Rectangles (# partitions) can be different sizes -tend toward same size, easier to read and keep track of 2. If this sum exits the function can be integrated -the value is the definite integral Riemann Sum - the limit of the sum, making smaller widths Or 3. This brings us to notation - Greek to Roman letters Definite Integral Page 3 Ex. The interval [-1,3] is partitioned into n subintervals of equal length. Let mk denote the midpoint of those intervals. Express the limit as an integral Ex. Evaluate If f(x) is negative integrals are more exciting Area is supposed to be positive -yet all height of rectangles would be negative -giving negative area When f(x)<0 Some functions include both positive and negative values -Therefore will need to do a combination of both ideas Definite Integral Page 4 -Therefore will need to do a combination of both ideas Area above x-axis - Area below x-axis Exploration: We know Find the values of the following. Definite Integral Page 5 If f(x)=c is constant on [a,b], then Integrals on Calculator - just does a sum of rectangles fnInt( f(x), x, a, b ) On AP test must see mathematical notation, not calc notation Need to watch for discontinuous functions - problems insue Definite Integral Page 6 General Rules of Definite Integrals - Thinking of the rectangles, these rules make sense - Similar to limit and derivative rules Ex. Suppose Find the following: Definite Integral Page 7 Ex. Show that the value of Is less than 3/2 Average (Mean) Value of a function on [a,b] Ex. Find the average value of On [0,3] Definite Integral Page 8 This is also the Mean Value THM for Definite Integrals - the integral will take on this value at some point in the interval Definite Integral Page 9 Differential to Integral Calculus Exploration 2 (book): Suppose we are given the following function y=f(x) 1. Copy the graph of f onto your own paper. Chose any x greater than a in the interval [a,b] and mark it. 2. Using only vertical line segments, shade in the region Between the graph of f and the x-axis from a to x. 3. Your shaded region represents a definite integral. Explain why this integral can be written as Why not 4. Compare your picture with others. Notice how your integral Depends on where you chose x to be. The integral is therefore A function of x. Call it F. 5. Recall the F'(x) is the limit of ΔF/Δx as Δx gets smaller. Represent ΔF in your picture by drawing one more vertical shading segment to the right of the last one you drew in step 2. ΔF is the (signed) area of your segment. 6. Represent Δx in your picture by moving x to beneath your newly drawn segment. That small change in Δx is the thickness of your segment. 7. What is now the height of your vertical segment? 8. Can you see why Newton and Leibniz could concluded that F'(x) = f(x)? Since we let Then Definite Integral Page 10 Then So the integral is the anti-derivative!! If F is any anti-derivative of f, then Assume x = a then Therefore HUGE!! Ex. Find Definite Integral Page 11 Fundamental Theorem of Calculus -we discovered both parts of this in the exploration Part 1: If f is continuous on [a,b] then Has a derivative at every point s in [a,b] and Find Find Find dy/dx if Definite Integral Page 12 Find dy/dx if Part 2: If f is continuous at every point in [a,b] and if F is any anti-derivative of f on [a,b], then Ex. Definite Integral Page 13 Net Area vs. Area The definite integral is the net area between the fn and x-axis - if asking for area we need to process what is below the x-axis so it can get a sign flip!! **Graphing the curve is a great idea Ex. Find the area between And the x-axis on [0,3] Ex. The fixed cost of starting a manufacturing run and producing the first 10 units is $200. After that, the marginal cost at x units output is Find the total cost of producing the first 100 units. Ex. Suppose a wholesaler receives a shipment of 1200 cases of boxes of chocolates every 30 days. The chocolate is sold to retailers at a steady rate, and x days after the shipment arrives, the inventory of cases still on hand is Definite Integral Page 14 Find the average daily inventory. Also find the average daily holding cost if the cost of holding one case is 3 cents per day. Definite Integral Page 15 Trapezoidal Approximations Area of a Trapezoid Where Ex. Use the trapezoid rule with n = 4 to estimate Definite Integral Page 16 Aside -- Simpson's Rule uses parabolic arcs on top of rectangles to approximate integrals Most Accurate Method --Don't need to know how to do Definite Integral Page 17