# Math 115

## If you are using any material here (e.g., Quizzes) for your class/presentation etc, you are supposed to let me know. Of course, even if you use it without telling me, I am not going to do anything about it, but just note that that's not how you are supposed to use other people's work.

Announcements (4/11, Mon, 2016)

• Post Test due 4/13 Wednesday (4% of Final Exam)
• Place: Gateway lab - East Hall B069
• Time: 7PM-10PM, 4/10 or 11AM - 4PM / 7PM - 10PM, 4/11, 4/12, 4/13
• Post Survey due 4/13 Wednesday (1% of Final Exam)
• Team HW 7 due 4/11 Monday
• Final Exam Thursday 4/21 10:30 AM - 12:30 AM / Locations: TBA
• Final Exam Material: check here.
• Web HW
• Web HW 5.3 due: 4/10 11:59 PM
• Web HW 5.4 due: 4/11 11:59 PM
• Web HW 6.1 due: 4/17 11:59 PM
• Web HW 6.2 due: 4/18 11:59 PM
• Read HW 6.2 due: 4/11 08:00 AM

• (Jan 11, MUST) "Properties of Logarithms" on p.30.
• (Jan 18, OPTIONAL) A proof of some trig sum identities. (i.e., sin(a + b) = sin(a)cos(b) + cos(a)sin(b); cos(a + b) = cos(a)cos(b) - sin(a)sin(b))

Other remarks

• WebWork for Chapter 1 is not counted as Uniform component, but I will count it as a quiz grade that counts for the Section component.
• Do not worry about Web HW 0, Read HW 1-1, and Read HW 1-2, because they will not be counted.

In-class lecture outline:

• 1/7 Thu: Definition of functions, Number e, Linear functions, Concavity.
• 1/11 Mon: Ad-hoc definition of concavity, How to compose functions, How to shift/shirnk/expand functions, Exponential functions, Trigonometric functions, Natural log and log, Invertibility of a function and its inverse function
• 1/12 Tue: Half-life, Shrinking/expanding amplitude of sine function, Quiz 1
• 1/14 Thu: Review of Quiz 1, General "sinusoidal functions", Power function vs Exponential function, How to "invert" sine functions
• 1/18 Mon: MLK
• 1/19 Tue: Rational functions and their possible asymptodes, Limit of a sequence (e.g. 0.99999... = 1), Limit of a function (by graph and then definition), Computing limits using some intuitions, Definition of continuity of a function, 5 classes of continuous functions (poly, exp, log, sin, cos), Properties of limit.
• 1/21 Thu: Odd/even: sine, cosine, tangent, How to "invert" tangent functions. (I won't be teaching the class on this day, but there will be a more-than-qualified substitude.) Quiz 2
• 1/25 Mon: Review of Quiz 1, Intermediate value theorem, Extreme value theorem
• 1/26 Tue: Differentiable functions and their derivatives, Two meaninngs of derivatives: slope and velocity (limit of average rate of change and distance/time when time goes to 0), Examples (3, 50, x, x^2, x^3, x^4, x^3 + 10x, 100x, 2x - 4, (x - 1)^2.), Derivative of x^n, Linearity of differentiation, Specific derivatives (constants, mx + b, polynomials), Positive derivative: strictly increasing, Negative derivative: strictly decreasing, Applications to optimizations for polynomial functions
• 1/27 Thur: Review of Quiz 2, Plans for Exam 1
• 2/1 Mon: Interpretations of derivatives,
• 1/27 Thur: Review of Review for Exam 1
• 2/2 Tue: Quiz 3 - Mock Exam 1 (Winter 2015)
• 2/3 Tue: Review of Quiz 3
• 2/8 Mon: Review of Team HW 2 and Team HW 3
• 2/9 Tue: Last minute questions for Exam 1 (which occurs at 6:00 pm)
• 2/11 Thu: Revisiting Concavity: f'' > 0 concave up / f'' < 0 concave down (e.g., x^3 - 1, x^3, x^2 + 1), Concavity has nothing to do with increading, deacreaing (e.g., e^x, e^{-x}), Velocity and accelation (with notations dy/dt, d^2y/dt^2 and example y = sin(t) + 1), Propertites of the operator d/dx on functions f(x): 1. linear and 2. product rule
• 2/15 Mon: Exam 1 claim registration, Differentiability implies continuity (intuitively and formally), Continuous functions that are not differentiable (|x|, gluing x with 1/x at x = 1, x^{1/3}), generalization of d(x^n)/dx = nx^{n-1}
• 2/16 Tue: Propertites of the operator d/dx on functions f(x): 3. (d/dx)(x^a) = ax^(a-1), 4. chain rule, 5. quotient rule, 6. Inverse function theorem: (d/dx)f^{-1}(x) = 1/f'(f^{-1}(x)); (d/dx)ln(x) = 1/x; Concrete definition of e = lim_{h -> 0}(1 + h)^{1/h} (after plugging h = 0.1, 0.01, 0.001, ...); (d/dx)e^x = e^x; (d/dx)a^x = ln(a)a^x
• 2/18 Thu: (d/dx)log_a(x) = 1/(ln(a)x), (d/dx)e^x = e^x; (d/dx)a^x = ln(a)a^x, Meaning of lim_{x -> 0}sin(x)/x = 1 in graph, Review: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) and cos(A + B) = cos(A)cos(B)- sin(A)sin(B), (d/dx)sin(x) = cos(x), (d/dx)cos(x) = -sin(x), (d/dx)tan(x) = sec^2(x), (d/dx)arcsin(x) = 1/sqrt(1 - x^2), (d/dx)arctan(x) = 1/(1 + x^2)
• 2/22 Mon: Quiz 4
• 2/23 Tue: Meaning of implicit functions with exampls such as y^2 = x, xy = 1, x^2 + y^2 = 1; taking derivative of implicit function requires two values prescribed (x AND y); a hint of higher dimension x^2 + y^2 = z by varing z; Mean Value Theorem with one picture and and arrow-shooting example
• 2/25 Thu: Review of Quiz 4; linear approximation vs quadratic approximation with example e^x at x = 0, ; general quadratic approximation f(x) ~ f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2; 4arctan(x) at x = 1 and approximate pi; sin(x) at x = 0, cos(x) at x = 0.
• (Special topic for 2/25 Thu - NOT REQUIRED TO ATTEND): "Infinite approximation" - heruistics: try to obtain "(d/dx)e^x = e^x". Infinite approximations of cos(x) and sin(x): notice that (d/dx) cos(x) = - sin(x) and (d/dx) sin(x) = cos(x). Imaginary number i: i^2 = -1. Euler's identity: e^(ix) = cos(x) + isin(x). Applications to compute: cos(2x), sin(2x), cos(3x), ...
• 3/7 Mon: Team HW 5 help
• 3/8 Tue: EVT -> Actual min max (e.g. y = sin x, cos x, x^3 - 1); local extrme values and global extreme values (y = x^3 - 1); critical point (confusing term) - critical points are all posible candidates for MAX/MIN; First derivative is enough to tell extreme values, but sometimes second derivatives can be used; inflection point - f'' where changes sign ; critical points/inflection points are necessary but NOT sufficient to check extreme values/concavity; find all the critical points of f(x) = 2x^4(9 - x)^5 with local maxima and minima
• 3/10 Thu: Some working problems
• 3/14 Mon: Quiz 5
• 3/15 Tue: Team HW 6 help
• 3/17 Thur: Quiz 6
• 3/21 Thur: Review Quiz 6 #8, Examples in 4.3
• 3/28 Mon: Example 1 - 5 in Section 4.6 / Section 5.1, 5.2: Definite integral is Area under y = f(x) with sign; Riemann sum (approximation by rectengles, what it means to be Left or Right sum)
• 3/29 Tue: Computing area under the curve y = x^2 from x = 0 to x = 1 by using 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6; Hints on general formula; First Fundamental Theorem
• 3/31 Thur: Quiz 7
• 4/2 Mon: Properties of integral - int_a,b + int_b,c = int_a,c; int_b,a = - int_a,b; int is a linear operator in functions; int_a,a = 0; int_-a,a odd = 0; int_-a,a even = 2 int_0,a; f leq g => int f leq int g; Average value of f from a,b = (int_a,b f)/(b-a) (do this for linear f first!); Problems if time permits
• 4/3 Tue: Antiderivatives (F(x) + c)' = F'(x) = f(x) and how to construct them in general, more computations
• 4/7 Thur: Quiz 8
• 4/11 Mon: Final Exam Practice
• 4/12 Tue: Final Exam Practice, Team HW 8
• 4/14 Thur: Final Exam Practice
• 4/18 Mon: Q & A