Warm up: Recall we can approximate R b

Transcription

1 Warm up: Recall we can approximate R b a f(x) dx using rectangles as follows: i. Pick a number n and divide [a, b] into n equal intervals. Note that x =(b a)/n is the length of each of these intervals. ii. Choose a point c in each of the intervals (usually either the left-most point, the right-most point, or the mid point). iii. Use a rectangle with base (b a)/n and height f(c) to model the area under the curve y = f(x) over each of the intervals. iv. Add up the area of the rectangles. Now consider I = R 4 x dx. Approximate I using the given n and c, and draw a picture to go with that shows (a) y = x, (b) the n intervals on the x-axis, (c) the point c in each of the intervals, and (d) the rectangle that approximates the area under the curve. () n =3and c being the left-most point of each of the intervals. () n =6and c being the right-most point of each of the intervals. (3) n =and c being the midpoint of each of the intervals.

2 Approximating I = R 4 x dx with n =3intervals using left endpoints: I = 4 Approximating I = R 4 x dx with n =6intervals using right endpoints: I = 4.875

3 Approximating I = R 4 x dx with n =intervals using midpoints: I = Review from Section 4. Approximating I = R b a f(x) dx using n intervals: The intervals are of length x =(b a)/n and I nx i= x f(c i ), where c i is... a+ x a+ 3 x a a+ x a+ x b = a+n x Left-hand endpoints: c i = a +(i ) x Right-hand endpoints: c i = a + i x Midpoints: c i = a +(i ) x

4 Section 6.5: Approximate integration Why would we need approximations now that we have a bunch of fancy new integration techniques? Example: What is R e x dx? From Wikipedia: In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial di erential equations. Approximating R e x dx using rectangles Let n =4(so that x =( ( ))/4 =). Left endpoints: - - Right endpoints: I e 4 + e + e 0 + e = I e + e 0 + e + e 4 =

5 Approximating R e x dx using rectangles Let n =4(so that x =( ( ))/4 =). Midpoints: - - I e (.5) + e (.5) + e (.5) + e (.5) = Error Let L n, R n,andm n be the estimates of a definite integral with n intervals, using left, right, and midpoints, respectively. For example, for the definite integral R e x dx, L 4 = , R 4 = , and M 4 = In reality, Z e x dx = So the errors for L 4 and R 4 were about 0.0, andtheerrorform 4 was about The reason I can tell you, definitively, to 4 digits, the value of this integral is because we also have formulas for upper bounds on the error using various approximation methods. This bound depends on things like the number of intervals (the more the better), the length of the total interval we integrate over (the smaller the better), and the curvature (second derivative) of the function (the flatter the better).

6 Error for the midpoint rule Suppose we approximate R b a f(x) dx using n intervals and midpoints. The error of the approximation M n is exactly E M = Z b a f(x) dx M n. Calculate f 00 (x). Findasmallestvalue0 apple K where you can calculate that f 00 (x) apple K over the interval [a, b]. Example: Calculating R 4 x dx. WecalculatedM = Now, f 00 (x) =.SoletK =. Then E M apple Example continued: K =, b K(b a)3 4n. a =3and n =.So E M apple (3)3 =9/6 = Comparing against the exact value, R 4 x dx =. So E M = = So our bound was exact! Error for the midpoint rule E M apple K(b a)3 4n where f 00 (x) apple K over [a, b]. Another example: Calculating R e x.we showed M Here, f 00 (x) =e x (4x ): - - Max value: 4/e 5/4 = , Min value:. So let K =.46. Then since b a =4and n =4, E M apple (.46)43 =.90 apple Checking against exact values: - E M apple.9x.

7 You try: M n = nx i= x f(c i ),where x =(b a)/n and c i = a +(i ) x E M apple K(b a)3 4n where f 00 (x) apple K over [a, b].. Use the midpoint rule to approximate R x4 dx using n =3. Draw a picture to help yourself.. Calculate d x 4 and maximize d x 4 over [ dx dx that maximum value., ]. LetK be 3. Calculate an upper bound on E M using the formula above. 4. Calculate R x4 dx exactly, and use that to calculate E M exactly. Compare to your bound.

8 Approximations using other shapes: Trapezoids! Instead of picking one height over each interval (approximating the function as a constant) we can pick a sloped line over each interval (approximating the function as a line) and use a trapezoid to approximate the area under the curve. Use the line that intersects the function at both endpoints of each interval. Example: Approximate R e x dx using trapezoids with n = h h - - b Area ( trapezoid ) = b h +h For example: b =, h = f( ), h = f(0), so A = f( )+f(0) Approximations using other shapes: Trapezoids! Example: Approximate R e x dx using trapezoids with n = Area ( trapezoid ) = b h +h For example: b =, h = f( ), h = f(0), so A = f( )+f(0) f( ) + f( ) f( ) + f(0) f(0) + f() A = = f( ) + f() + f( ) + f(0) + f() In general, f() + f() T n = x(f(c 0 )+f(c n ) + (f(c )+f(c )+ + f(c n ))) where x =(b a)/n and c i = a + i x.

9 You try:. Draw a graph of f(x) =x 4 over [, ].. Let n =3and calculate x and c i = a + i x for i =0,,, and 3. Mark the c i s on the x-axis. 3. Mark the 4 points on the graph corresponding to f(c i ). 4. Draw the three trapezoids whose tops are the line segments joining f(c i ) to f(c i ). 5. Calculate the areas of the three trapezoids. 6. Add the areas together to get T n. 7. Use the formula T n = x(f(c 0 )+f(c n ) + (f(c )+f(c )+ + f(c n ))) and compare to your previous answer (you should get the same thing). 8. Compare your answer to the exact value of R x4 dx.

10 Trapezoid error Let K be such that f 00 (x) apple K over [a, b] as before. Then the error E T = is bounded above by Z b a f(x) dx T n E T apple K(b a)3 n. (Recall E M apple K(b a)3 4n.) You try: Give an upper bound for E T for our estimate T 3 of R x4 dx. Simpson s rule Rectangles are like approximating f(x) as a constant. (Needed one point over each interval.) Trapezoids are like approximating f(x) as a line. (Needed two points over each interval.) Simpson s rule is approximating f(x) as a parabola. A generic parabola is given by a 0 + a x + a x. So we need three points to define a parabola. So over each interval, take () the left endpoint, () the midpoint, and (3) the right endpoint, and find the parabola that passes through f(x) above those three points. Actually, caution!! The book s convention is to call this n intervals, and pick one parabola for every two intervals.

11 Simpson s rule So over each n intervals, take () the left endpoint, () the midpoint, and (3) the right endpoint, and find the parabola that passes through f(x) above those three points. Actually, caution!! The book s convention is to call this n intervals, and pick one parabola for every two intervals. Example: R e x dx. CalculateS 8. (This has 4 parabolas for n =8.) - - Simpson s rule So over each n intervals, take () the left endpoint, () the midpoint, and (3) the right endpoint, and find the parabola that passes through f(x) above those three points. Actually, caution!! The book s convention is to call this n intervals, and pick one parabola for every two intervals. Example: R e x dx. CalculateS 8. (This has 4 parabolas for n =8.) - - Whatever a 0, a,anda are, we can calculate Z ci c i a 0 + a x + a x dx = a 0 x + a x + 3 a x 3 c i c i

12 Simpson s rule Let n be even. The resulting approximation, once the curves are fit and the integrals are taken, gives S n = 3 x f(c 0 )+4f(c )+f(c )+4f(c 3 ) + +f(c n )+4f(c n )+f(c n ) where x =(b a)/n and c i = a + i x. (Read pp in the book) You try: Approximate R x4 dx using S 6. Error: For E S = R b a f(x) dx S n and K f (4) (x) over [a, b] (new K!!), K(b a)5 E S apple 80n 4. You try: Calculate an upper bound for E S for R x4 dx and n =6. Compare to the exact value of E S.

13 Consider R 0 sin(x).. Calculate the maximum value of d dx sin(x) over [0, ]. Let this be K.. For each of M 4, T 4 and S 4, do the following: (a) Draw a picture of the approximation, with y =sin(x) overlaid. (b) Calculate the approximation. (c) Calculate an upper bound of the error of the approximation. (d) Compare your upper bound against the actual value of the error.