Math 3B: Lecture 1. Noah White. September 23, 2016
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1 Math 3B: Lecture 1 Noah White September 23, 2016
2 Syllabus Take a copy of the syllabus as you walk in or find it online at math.ucla.edu/~noah
3 Class website There are a few places where you will find/receive information about Math 3B: The class website: CCLE Piazza
4 Instructor and TAs Instructor office hours TA office hours Noah White MS 6304, M 3:30pm-5pm, R 9-10:30am Kyung Ha MS 6943, W 4-5pm Robert Houseden MS 3915B, T 3-4pm Dustan Levenstein MS 3965, R 3-4
5 Instructor and TAs Instructor office hours TA office hours Noah White MS 6304, M 3:30pm-5pm, R 9-10:30am Kyung Ha MS 6943, W 4-5pm Robert Houseden MS 3915B, T 3-4pm Dustan Levenstein MS 3965, R 3-4 Note! Change of room for Dicussion Section 2D (Thurdsay with Dustan). You are now in Boelter 5419.
6 Textbook S. J. Schreiber, Calculus for the Life Sciences, Wiley
7 Problem sets, homework, and quizzes Problem sets Assigned every week. Long list of problems. Not graded, but recommended!
8 Problem sets, homework, and quizzes Problem sets Assigned every week. Long list of problems. Not graded, but recommended! Homework Due every second week. A small number of questions drawn from the problem sets. There will be 5.
9 Problem sets, homework, and quizzes Problem sets Assigned every week. Long list of problems. Not graded, but recommended! Homework Due every second week. A small number of questions drawn from the problem sets. There will be 5. Quizzes Administerd every other week in discussion session. There will be 4.
10 Homework; collection and collaboration Homework will be collected at the start of the lecture on the dues date.
11 Homework; collection and collaboration Homework will be collected at the start of the lecture on the dues date. Late homework will not be accepted.
12 Homework; collection and collaboration Homework will be collected at the start of the lecture on the dues date. Late homework will not be accepted. You may collaborate in small groups if:
13 Homework; collection and collaboration Homework will be collected at the start of the lecture on the dues date. Late homework will not be accepted. You may collaborate in small groups if: you aknowledge your collaborators,
14 Homework; collection and collaboration Homework will be collected at the start of the lecture on the dues date. Late homework will not be accepted. You may collaborate in small groups if: you aknowledge your collaborators, write up your own solutions, in your own words.
15 Exams There will be two midterms and a final exam Midterm 1 2-2:50pm Monday, 17 October, 2016
16 Exams There will be two midterms and a final exam Midterm 1 2-2:50pm Monday, 17 October, 2016 Midterm 2 2-2:50pm Monday, 21 November, 2016
17 Exams There will be two midterms and a final exam Midterm 1 2-2:50pm Monday, 17 October, 2016 Midterm 2 2-2:50pm Monday, 21 November, 2016 Final 3-6pm Monday, 5 December, 2016
18 Exams There will be two midterms and a final exam Midterm 1 2-2:50pm Monday, 17 October, 2016 Midterm 2 2-2:50pm Monday, 21 November, 2016 Final 3-6pm Monday, 5 December, 2016 Cheatsheets and calculators You will be allowed a small cheatsheet in each exam. Must be self-written and one side, half a letter size piece of paper. You are also allowed to use non-programmable, non-graphing calculators.
19 Grading Your final grade will be calculated using the maximum of the following two grading schemes. 15% (best 7 quizzes/hw) + 40% (midterms) + 45% (final) or 15% (best 7 quizzes/hw) + 30% (best midterm) + 55% (final)
20 Schedule
21 Where to get help Piazza Here you can ask questions and answer others questions. Lets take a look...
22 Where to get help Piazza Here you can ask questions and answer others questions. Lets take a look... Office hours Use them wisely! Prepare and seek information first. If you learn something, post it to Piazza!
23 Where to get help Piazza Here you can ask questions and answer others questions. Lets take a look... Office hours Use them wisely! Prepare and seek information first. If you learn something, post it to Piazza! Student Math Center (SMC) Location: MS 3974, times: M-R 9am-3pm. The SMC offers free, individual and group tutoring for all lower division math courses. This service is available on a walk-in basis; no appointment is necessary. Students may ask any of the TAs in attendance for assistance with math problems.
24 Differentiation You should have a good feel for what the derivative means. I.e. derivative at a point = tangent slope. You need to understand differentiation algebraically as well as geometrically.
25 Differentiation You should have a good feel for what the derivative means. I.e. derivative at a point = tangent slope. You need to understand differentiation algebraically as well as geometrically. You should also know the definition of the derivative d f (x + h) f (x) f (x) = lim dx h 0 h
26 Differentiation You should be able to differentiate many of the standard functions we will see in this course. This includes: polynomials/power functions d dx (x n ) = nx n 1
27 Differentiation You should be able to differentiate many of the standard functions we will see in this course. This includes: polynomials/power functions d dx (x n ) = nx n 1 exponentials d dx (ex ) = e x
28 Differentiation You should be able to differentiate many of the standard functions we will see in this course. This includes: polynomials/power functions d dx (x n ) = nx n 1 exponentials logarithms d dx (ex ) = e x d dx (ln x) = 1 x
29 Differentiation You should be able to differentiate many of the standard functions we will see in this course. This includes: polynomials/power functions d dx (x n ) = nx n 1 exponentials logarithms trig functions d dx (ex ) = e x d dx (ln x) = 1 x d (sin x) = cos x dx
30 Product rule The product rule lets us differential functions of the form f (x) = g(x)h(x). It says d dx f (x) = g (x)h(x) + g(x)h (x)
31 Product rule The product rule lets us differential functions of the form f (x) = g(x)h(x). It says d dx f (x) = g (x)h(x) + g(x)h (x) Example Let s differentiate the function f (x) = e x sin x. ( ) ( ) d d d dx f (x) = dx ex sin x + e x dx sin x
32 Product rule The product rule lets us differential functions of the form f (x) = g(x)h(x). It says d dx f (x) = g (x)h(x) + g(x)h (x) Example Let s differentiate the function f (x) = e x sin x. ( ) ( ) d d d dx f (x) = dx ex sin x + e x dx sin x = e x sin x + e x cos x
33 Product rule The product rule lets us differential functions of the form f (x) = g(x)h(x). It says d dx f (x) = g (x)h(x) + g(x)h (x) Example Let s differentiate the function f (x) = e x sin x. ( ) ( ) d d d dx f (x) = dx ex sin x + e x dx sin x = e x sin x + e x cos x = e x (sin x + cos x)
34 The chain rule The chain rule is very important! It allows us to differentiate functions of the form f (x) = g(h(x)). It says d dx f (x) = h (x)g (h(x))
35 The chain rule The chain rule is very important! It allows us to differentiate functions of the form f (x) = g(h(x)). It says d dx f (x) = h (x)g (h(x)) Example Let s differentiate f (x) = sin(e x ). In this example h(x) = e x and g(x) = sin x
36 The chain rule The chain rule is very important! It allows us to differentiate functions of the form f (x) = g(h(x)). It says d dx f (x) = h (x)g (h(x)) Example Let s differentiate f (x) = sin(e x ). In this example h(x) = e x and g(x) = sin x
37 The chain rule The chain rule is very important! It allows us to differentiate functions of the form f (x) = g(h(x)). It says d dx f (x) = h (x)g (h(x)) Example Let s differentiate f (x) = sin(e x ). In this example h(x) = e x and g(x) = sin x so h (x) = e x and g (x) = cos x
38 The chain rule The chain rule is very important! It allows us to differentiate functions of the form f (x) = g(h(x)). It says d dx f (x) = h (x)g (h(x)) Example Let s differentiate f (x) = sin(e x ). In this example h(x) = e x and g(x) = sin x so so h (x) = e x and g (x) = cos x f (x) = e x cos(e x )
39 The quotient rule is stupid The quotient rule says d dx ( g(x) h(x) ) = g (x)h(x) g(x)h (x) h(x) 2
40 The quotient rule is stupid The quotient rule says d dx ( g(x) h(x) ) = g (x)h(x) g(x)h (x) h(x) 2 This is annoying to remember (where does that minus sign go again?). Luckily we can notice g(x) h(x) = g(x)k(x) where k(x) = (h(x)) 1 So we can just use the product rule!
41 Example Question Differentiate f (x) = sin 1 x
42 Example Question Differentiate f (x) = sin 1 x Solution We should use the chain rule. Notice f (x) = g(h(x)) where h(x) = x 1 and g(x) = sin x
43 Example Question Differentiate f (x) = sin 1 x Solution We should use the chain rule. Notice f (x) = g(h(x)) where h(x) = x 1 and g(x) = sin x
44 Example Question Differentiate f (x) = sin 1 x Solution We should use the chain rule. Notice f (x) = g(h(x)) where h(x) = x 1 and g(x) = sin x so h (x) = x 2 and g (x) = cos x
45 Example Question Differentiate f (x) = sin 1 x Solution We should use the chain rule. Notice f (x) = g(h(x)) where h(x) = x 1 and g(x) = sin x so so h (x) = x 2 and g (x) = cos x f (x) = h (x)g (h(x)) = 1 x 2 cos(x 1 )
46 Example Question Differentiate f (x) = x 1 x + 1
47 Example Question Differentiate f (x) = x 1 = (x 1)(x + 1) 1 x + 1 Solution We should use the product/quotient rule. Notice f (x) = g(x)h(x) where h(x) = (x + 1) 1 and g(x) = x 1 h (x) = (x + 1) 2 and g (x) = 1
48 Example Question Differentiate f (x) = x 1 = (x 1)(x + 1) 1 x + 1 Solution We should use the product/quotient rule. Notice f (x) = g(x)h(x) where h(x) = (x + 1) 1 and g(x) = x 1 h (x) = (x + 1) 2 and g (x) = 1
49 Example Question Differentiate f (x) = x 1 = (x 1)(x + 1) 1 x + 1 Solution We should use the product/quotient rule. Notice f (x) = g(x)h(x) where h(x) = (x + 1) 1 and g(x) = x 1 so h (x) = (x + 1) 2 and g (x) = 1 f (x) = g (x)h(x) + g(x)h (x)
50 Example Question Differentiate f (x) = x 1 = (x 1)(x + 1) 1 x + 1 Solution We should use the product/quotient rule. Notice f (x) = g(x)h(x) where h(x) = (x + 1) 1 and g(x) = x 1 so h (x) = (x + 1) 2 and g (x) = 1 f (x) = g (x)h(x) + g(x)h (x) = 1 x + 1 x 1 (x + 1) 2 = 2 (x + 1) 2
51 Example Question Differentiate F (x) = sin x 2 1 sin x 2 + 1
52 Example Question Differentiate F (x) = sin x 2 1 sin x Solution We should notice that F(x) = f(g(x)) so we can use the chain rule! f (x) = x 1 x + 1 and g(x) = sin x 2 f (x) = 2 (x + 1) 2 and g (x) = 2x cos x 2
53 Example Question Differentiate F (x) = sin x 2 1 sin x Solution We should notice that F(x) = f(g(x)) so we can use the chain rule! f (x) = x 1 x + 1 and g(x) = sin x 2 f (x) = 2 (x + 1) 2 and g (x) = 2x cos x 2
54 Example Question Differentiate F (x) = sin x 2 1 sin x Solution We should notice that F(x) = f(g(x)) so we can use the chain rule! f (x) = x 1 x + 1 and g(x) = sin x 2 so f (x) = 2 (x + 1) 2 and g (x) = 2x cos x 2 f (x) = g (x)f (g(x))
55 Example Question Differentiate F (x) = sin x 2 1 sin x Solution We should notice that F(x) = f(g(x)) so we can use the chain rule! f (x) = x 1 x + 1 and g(x) = sin x 2 so f (x) = 2 (x + 1) 2 and g (x) = 2x cos x 2 f (x) = g (x)f (g(x)) = 2x cos x 2 2 (sin x 2 + 1) 2
56 Example Question A bacterial colony is estimated to have a population of P thousand individuals, where 24t + 10 P(t) = t and t is the number of hours after a toxin is introduced.
57 Example Question A bacterial colony is estimated to have a population of P thousand individuals, where 24t + 10 P(t) = t and t is the number of hours after a toxin is introduced. 1. At what rate is the population changing when t = 1?
58 Example Question A bacterial colony is estimated to have a population of P thousand individuals, where 24t + 10 P(t) = t and t is the number of hours after a toxin is introduced. 1. At what rate is the population changing when t = 1? 2. Is the rate increasing or decreasing at this time?
59 Example Question A bacterial colony is estimated to have a population of P thousand individuals, where 24t + 10 P(t) = t and t is the number of hours after a toxin is introduced. 1. At what rate is the population changing when t = 1? 2. Is the rate increasing or decreasing at this time? 3. At what time does the population begin to decrease?
60 Example (cont.) 1. At what rate is the population changing when t = 1? We want to know how many individuals are being added per hour at t = 1. We need to find the derivative: P(t) = (24t + 10)(t 2 + 1) 1
61 Example (cont.) 1. At what rate is the population changing when t = 1? We want to know how many individuals are being added per hour at t = 1. We need to find the derivative: P(t) = (24t + 10)(t 2 + 1) 1 and d dx (t2 + 1) 1 = 2t(t 2 + 1) 2
62 Example (cont.) 1. At what rate is the population changing when t = 1? We want to know how many individuals are being added per hour at t = 1. We need to find the derivative: P(t) = (24t + 10)(t 2 + 1) 1 and d dx (t2 + 1) 1 = 2t(t 2 + 1) 2 P (t) = 24(t 2 + 1) 1 2t(24t + 10)(t 2 + 1) 2
63 Example (cont.) 1. At what rate is the population changing when t = 1? We want to know how many individuals are being added per hour at t = 1. We need to find the derivative: P(t) = (24t + 10)(t 2 + 1) 1 and d dx (t2 + 1) 1 = 2t(t 2 + 1) 2 P (t) = 24(t 2 + 1) 1 2t(24t + 10)(t 2 + 1) 2 = 4(3t 2)(2t + 3) (t 2 + 1) 2
64 Example (cont.) 1. At what rate is the population changing when t = 1? We want to know how many individuals are being added per hour at t = 1. We need to find the derivative: P(t) = (24t + 10)(t 2 + 1) 1 and d dx (t2 + 1) 1 = 2t(t 2 + 1) 2 P (t) = 24(t 2 + 1) 1 2t(24t + 10)(t 2 + 1) 2 4(3t 2)(2t + 3) = (t 2 + 1) 2 P (1) = = 5
65 Example (cont.) 1. Is the rate increasing or decreasing at this time? We need to figure out when the derivative P (t) goes from being positive to negative. First let s ask when does P (t) = 0
66 Example (cont.) 1. Is the rate increasing or decreasing at this time? 2. At what time does the population begin to decrease? We need to figure out when the derivative P (t) goes from being positive to negative. First let s ask when does P (t) = 0
67 Example (cont.) 1. Is the rate increasing or decreasing at this time? 2. At what time does the population begin to decrease? We need to figure out when the derivative P (t) goes from being positive to negative. First let s ask when does P (t) = 0
68 Example (cont.) 1. Is the rate increasing or decreasing at this time? 2. At what time does the population begin to decrease? We need to figure out when the derivative P (t) goes from being positive to negative. First let s ask when does P (t) = 0 4(3t 2)(2t + 3) (t 2 + 1) 2 = 0
69 Example (cont.) 1. Is the rate increasing or decreasing at this time? 2. At what time does the population begin to decrease? We need to figure out when the derivative P (t) goes from being positive to negative. First let s ask when does P (t) = 0 4(3t 2)(2t + 3) (t 2 + 1) 2 = 0 t = 2 3 or 3 2
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