# Math 116

## Section 023 Fall 2016 (TuWeFr 11:30 AM - 1:00 PM 3448, 3314, 3448 MH)

Announcements (12/12, Mon, 2016)

• Final Exam: 12/19 MORNING! 8:00 AM

Web HW (all due at 11:00 pm)

• 10.3 due 12/13

• Chapter 10.1 ~ 10.3
• Review everything we have done by doing Past Exam problems

In-class lecture outline:

• 9/6 Tue: Computing area under curves (e.g., y = x vs. y = x^2); 1st Fundamental Theorem of Calculus: int_a,b F'dt = F(b) - F(a) (heuristics: F'dt = dF, so sum these up) (e.g., y = x^2); Left sum and Rigt sum vs. Upper sum and Lower sum (e.g., y = x^2 on [0, 1], [-1, 0], [-1, 1]); Integration of odd/even functions; Linearity of the integral operator (e.g., int_{-1, 1} x^777 + x^101 + 1 dx, sin x on [0, 2pi])
• 9/7 Wed: Definition of average value (e.g., c(t) = t^2 number of customers at time t); Compute int_{-1, 1} sqrt(1 - x^2) dx; Error estimate (e.g., c(t) = t^2 on [0, 24] (M-m)dt < error in picture); One can connect integrals; Anti-derivatives vs integrals (e.g., f(x) = x^2 gives a family of functions);
• 9/9 Fri: BLACKOUT
• 9/13 Tue: Quiz 1 on the materials including - Area between curves (e.g., sin x and cos x on [0, 2pi]); Anti-derivative of usual functions (x^n where n not -1, 1/x, e^x, sin(x), cos(x)); How to check -ln(|cos(x)|) is an antiderivative of tan(x); Notation for anti-derivative
• 9/14 Wed: 2nd Fundamental Theorem of Calculus (constructing anti-derivative by integral) e.g., d/dx int_{0, x}e^{-t^{2}}dt = e^{-x^{2}}; Computing d/dx int_{cos(x), sin(x)}e^{-t^{2}}dt ; Substitution method for integrals e.g., int_{0,pi/4}tan(x)dx; We don't cover 6.3 (Diff eq) for Exam 1
• 9/16 Fri: Review of Quiz 1
• 9/20 Tue: 7.3 Integration by parts: compute int ln(x) dx and int cos^2(x) dx; 7.4 skipped; 7.5 LEFT(n), Right(n)
• 9/21 Wed: Mid(n); Trap(n) = (Left(n) + Right(n))/2: Again, pictures! Concavity and comapring Trap(n) and Mid(n); Skip Simpson's rule; Past exam question; Skip 7.6 etc; Chapter 8: Compute the area of a circle with a given radius
• 9/23 Fri: Compute the volue of a sphere; Compute the volue of a cone with base radius 5 and height 5; Voluemes of revolution: y = e^{-x} from 0 to 1 (Answer: pi(1 - e^{-2})/2; Compute the volume of the solid given by rotating the curve y = tan(x) on [0, pi/4] with respect to x axis; the volume obtained by area between curves y = x and y = x^2 about the line y = 3 (approx 2.72); Arc length of curves given by y = f(x): consider y = sqrt(25 - x^2); Skip 8.3; 8.4 Desnity: Approximating total population using density f(x) people/mile as integeral; Find the mass of the column of air if the density of air at hight h is given by f(h) = 1.28e^{-0.000124h} kg/m^3 in a cylinder or bottom radius 1m and hight 25,000m
• 9/27 Tue: Review of Team HW 1; Write integral using density to compute the quantity
• 9/28 Wed: Center of mass: discrete - using seesaw example; c.o.m = sum x_i m_i / sum m_i; continuous definition: replace sum with integral - using mass density d(x) so that mass = d(x)dx; Find the total mass from x=0 to x=2 if the density mass is given by d(x) = 15x^2; and find the center of mass
• 9/30 Fri: Work = Force (in the direction of movement!) x Distance = integral Force(x) dx ; Free fall 5 kg of 20 m: the gravity worked (9.8)(5)(20) kg*m/sec^2 (joules); Example 5 in p.452
• 10/4 Tue: Shell method (find volume given by rotating y = (x-1)^2 w.r.t y-axis from x = 0 to x = 2); Quiz 2
• 10/5 Wed: Bring your Team HW 2 day
• 10/7 Fri: Review of Quiz 2
• 10/11 Tue: x = cos(theta), y = sin(theta) -> change r -> get plane! In general, we have x = rcos(theta), y = rsin(theta), r = sqrt(x^2 + y^2), tan(theta) = y/x; Computing Area using polar coordinates: draw r = 3 sin(2 theta) which gives a four leaves clover; dy/dx = (dy/dt)/(dx/dt); general arclength;
• 10/12 Wed: 11.1
• 10/14 Fri: 11.2, 11.3, start 11.4
• 10/18 Tue : FALL BREAK
• 10/19 Wed: Quiz 3
• 10/21 Fri: 11.5
• 10/25 Tue: 11.6
• 10/26 Wed: 4.7, 7.6
• 10/28 Fri: 7.7 (end of Exam 2 materials; now we mostly cover past exam problems)
• 11/1 Wed: Quiz 4
• 11/4 Fri: Review of Quiz 4, #2 of Team HW 4
• 11/8 Tue: #2, 3, 4 of Team HW 4
• 11/9 Wed: More problems for Exam 2
• 11/11 Fri: More problems for Exam 2
• 11/15 Tue: 9.1, 9.2
• 11/16 Wed: 9.3, 9.4
• 11/18 Fri: 9.4, 9.5
• 11/22 Tue: Reading Quiz 9.1 ~ 9.5
• 11/23 Wed: Past Exam Problems
• 11/23 Wed: Past Exam Problems
• 11/29 Tue: 10.1, 10.2
• 11/30 Wed: 10.3, Past Exam Problems
• 12/2 Fri: Reading Quiz 10.1 ~ 10.3
• 12/6, 7, 9, 13: Practice Problems