CALCULUS OCTOBER 7 # Day Date Assignment Description 6 M / p. - #,, 8,, 9abc, abc. For problems 9 and, graph the function and approximate the integral using the left-hand endpoints, right-hand endpoints, and midpoints for finding the heights of the rectangles. A look at sigma notation in expressing a sum.. Approximating an integral using rectangles (Riemann Sum). 7 Tu / p. #abc, abc. Show graphs and calculate the approximation for both AND p. 5 #,,, For problem #,, : (a) Estimate the integral using the trapezoidal rule with n = 4, and (b) Evaluate the integral directly. For problem : Do parts (a) and (c) but do not find the error, E T. 8 W /4 p. #9, and # 6 part b only and also the following: Approximate to decimal places the integral 4 (6 x ) dx with 4 equal intervals using: a) rectangles whose height is the right-hand endpoint b) rectangles whose height is the left-hand endpoint Approximating area under a curve using trapezoids. Finding the exact area under a curve. c) rectangles whose height is the midpoint of the interval d) trapezoids (trapezoidal rule) e) Evaluate the integral directly. 9 Th /5 Worksheet #6 Finding area under a curve 4 F /6 p. 7-7 #, 6, 9,, 5, 8,, 4 Finding the area between and also do the following: two curves. Approximate using the Trapezoid Rule: 4 x dx with n = 6 and compare it with the actual integral. 4 M /9 Worksheet #7 Review area under and between curves and find average value. 4 Tu / Worksheet #8 Test Review 4 W / TEST 5 TODAY!!!!! Test on area and the other Worksheet #9 material covered; review 44 Th / p.85-86 #,,, 5, 6,, 4 Finding volumes of solids of revolution using disks. 45 F / p. 85-86 #9,,,, 5, 6 Finding the volume using p.7 # disks and washers. 46 M /6 p.9 #, 4, 8,8, 6ab, 8abcd Finding volumes using disks, washers, and shells.
CALCULUS OCTOBER 7 48 Tu /7 Do the following 5 problems: ) Find the volume of the solid whose base is the circle x + y = 9 and whose cross sections perpendicular to the x-axis are squares. ) Find the volume of the solid whose base is bounded by the lines y = x, y = x, and x = and whose cross sections are semicircles perpendicular to the x-axis. ) Find the volume of the solid whose base is the region bounded between the curves y = x and y = x and whose cross sections perpendicular to the x-axis are squares. 4) Find the volume between y = x and y = 4x in the st quadrant if it is revolved around the x axis. 5) Find the area between y = 6x and y = x. 49 W /8 p. 98 #9,,, p.749 #8 AND THE FOLLOWING 7 Problems: ) The base of a solid in the x-y plane is a right triangle bounded by the axes and x + y =. Cross sections of the solid perpendicular to the x-axis are squares. Find the volume. ) The cross sections of a solid cut by planes perpendicular to the y-axis are equilateral triangles whose base is bounded by the parabola x = 8y and the line y = 4. Find the volume. ) Find the volume of the solid whose base is the circle x + y = 9 and whose cross sections perpendicular to the x-axis are equilateral triangles with one side across the base. Find the volume of a solid with a given cross section. Finding the length of a curve. Bring Worksheet # to class tomorrow Find the volumes as the given regions are revolved about the given axes: 4) y = x, y = x x axis disc/washer 5) y = x, y = x x axis shell 6) y = x, y =, x = line y = 7) y = x, y = 4x (st quad) line y = 8 47 Th /9 Worksheet # Review finding volumes. EARTHQUAKE DRILL @ :9 5 F / Worksheet # Review for test 5 M / Worksheet # Review for easy test. 5 Tu /4 Worksheet # LATE START DAY 5 W /5 TEST 6 TODAY!!! Worksheet #4 Review for easy test tomorrow. Easy test on applications & Curve analysis and sketching f(x)
CALCULUS OCTOBER 7 54 Th /6 p. 7 #,, 7,, 6, 79 Find intervals where curve is increasing, decreasing, concave up, concave down, critical Use the second derivative to determine concavity of a curve and to find inflection points. points, points of inflection, and graph the curve. 55 F /7 Worksheet #5 Solving word problems that involve maximizing or minimizing a function. 56 M / Worksheet #6 Determine the absolute maximum and minimum of a function. 59 T / Worksheet #7 Trick or Treat Abs max/ mins Assignment #6 ) + 4 = 7 ) + / + / = 7/6 a) b) c) 8a) yes b) yes c) no ANSWERS 4 ) k= k 9a).5 b).75 c).65 or 5/8 Assignment #7 a).875 or 7/ b).46875 or 5/ c).85 or /64 a) 6.8 or π b) 6.8 or π c) 6.8 or π a) / b) / a) 6 b) 6 a) /4 b) 8/ a).99 c) / Assignment #8 9) 6/ ) 6 ) 9/ ) 8/ ) / or 7. 4) 5) 8/ 6) Calendar Problem a) 4 b) 5 c) 4 d) 4 e) 4.667 or 8/ Assignment #4 Calendar Problem ) / 6) /5 9) 8/ ) 9/4 Trapezoid: 65.778 Actual: 64 5) 48/5 8) 4 ) 8 4) 9/ Assignment #44 ) π/ ) 6π ) 4 π 5) π/5 6) 8π/7 ) π 4) 4π Assignment #45 9) π ) π(π 8) ) π / π ) π 5) 7π/5 6) 8π/5 ) 7/ Assignment #46 ) 6π 4) 9π/ 8) π 8) π/6 6a) 4π/5 6b) 7π/ 8a) 6π 8b) π/ 8c) 64π/ 8d) 48π
CALCULUS OCTOBER 7 Assignment #48 ) ( 9 x ) dx = 44 units ) π ( x) dx = π 6 units ) 8 4) π y y / y 4 dy = 5π 5) 4 Assignment #49 p.98: 9) p.749: 8) 7 4) π [( x) (x ) ] dx = π 8 ) 7 ( ) = 9.7 Calendar Problems ) 8 5) π y( y y ) dy = π ) 5 6 = 8.8 ) ) 64 ) 6 6) π ( x ) dx = 8π 5 (disk) =.667 7) π [(8 x ) (8 4x) ] dx = 8π (disk) Assignment #54 ) max (,7), inc x<, dec x>, conc down all x ) rel min (,), max (,5), inf pt (,), inc x<,x>, dec <x<, conc up x>, dn x< 7) min: (,) and (, ), max: (,), dec: x <, (,), inc: (,), x >, inf pt: ( /, 5/9) and (/, 5/9), conc up: x < / and x > /, conc dn: ( /,/ ) ) min: ( /, 7/6), terrace: (,), inc: x > /, except, dec: x < /, inf pt: (,) and (, ), conc up: x< and x>, conc dn: (,) 6) inc: all Reals except, conc up: x< conc dn: x>, inf (terrace) pt: (,) 79. b = when y"() =
CALCULUS OCTOBER 7 EXTRA CREDIT Extra Credit is due at the BEGINNING OF THE PERIOD, on a SEPARATE, FULL SHEET OF PAPER. Turn into the basket in the back of the room before the bell rings. Make sure to include the full heading: NAME, DATE, PERIOD, ROW-SEAT, EC # in ink. Show ALL work, SIGN INTEGRITY. You may NOT receive any help from ANYONE or ANYTHING. Work must match the answer or you will NOT receive any credit for the problem. Correct work + Correct Answer + Integrity = 5 points each 6 Boota is crusing down the space highway when he sees parsec marker. To help relieve some of the boredom of driving, he decides to do some math. Let A equal the number of positive integral factors of, and let B equal the sum of the positive integral factors of. Find AB. (no calculator) 7 Boota has accidentally caused a rip in time and space which has sucked him into the 7 th dimension. In order to get back he must pass the 7D guardian who refuses to allow him to leave unless he can find sine of the smaller angle between two 7 dimensional vectors (,,-7,,5,,-) and (,- 7,6,,7,4,). Boota thinks for a few minutes, tells the guardian his answer, and is allowed to leave. What was his answer? (no calculator) 8 Amazed at Boota s abilities, the 7D guardian asks Boota to help find the value of his long lost brother, 7x. If log x log log (4x 49) log, find the value of 7x. (no calculator) 9 9 When Boota leaves the 7 th dimension, he quickly realizes that he has been transported through time far into the future. He asks one of the locals, a Sigmoid, what year it is, but Sigmoids can only talk by using summations. If the Sigmoid tells him that the year is x x 6x 8x 6, what year is it, expressed as one number? (no calculator) 4 Boota arrives at a time machine, but finds that the line is a lightyear long. One of his future descendants offers to let Boota skip to the front of the line, but only if he can find a certain result. If C = 4x x, x A = The total distance in feet a ball travels if it drops from feet and bounces back up 4 of the way each bounce, and B = The sum of the first 5 terms in an arithmetic series whose rd term is 7 and whose 7 th term is 9, find AB ( C). (no calculator) 4 Simplify to one trigonometric ratio: sin sin cos sin cos cot cot ( cos ) cos sin 4 None 4 Boota s polynomial, x x 5x, has roots a, b, and c. Find ab bc ac. (no calculator) 44 p.86 #4 45 p.86-87 #4 46 If log (x + x + 4) log (x + 7) =, the x =?? 47 p.78 # 48 p.9 #4b Note: Correction area is bounded by x = /4 (not y = /4 as book says) 49 p.749 #
5 p.446 #4 5 p.445 # 5 4 Find 4 6 x dx without using an integral. 5 None 54 55 56 CALCULUS OCTOBER 7