The deriva*ve. Analy*c, geometric, computa*onal. UBC Math 102
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1 The deriva*ve Analy*c, geometric, computa*onal UBC Math 102
2 Deriva*ve: Analy*c Use the defini*on of the deriva*ve to compute the deriva*ve of the func*on y = x 3 MT testtype ques*on Done fast? Try to extend your method to y = x n Where n is any posi*ve integer. Sols: see last slides More advanced
3 Math Learning Center The MLC is a space for undergraduate students to study math together, with friendly support from tutors (grad students in math). Located at LSK310 and LSK302 Hours: open 5 days a week. Note: MLC is not a place to check HW answers or receive solu*ons. MLC aim is to aid students in becoming bever learners and to develop cri*cal thinking skills. hvp://
4 Diagnos*c test results Final Grade in M102 DT score + 10% If your DT < 40% à high risk of failing Possible ac*on: take M180 instead of M102 OR : prepare to work extra- hard
5 UBC Math 102 Help with homework
6 Assignment2: Problem 16 Evaluate the limit, if it exists. If a limit does not exist, enter DNE. Hint: consider factoring top and bovom UBC Math 102
7 Assignment2: Problem 18 Evaluate the limit by guessing We have not learned enough about this func*on to actually compute the limit, but we can use a value close to x=0 to guess the limit UBC Math 102
8 Guessing To guess a value for this limit: I would input which of the following into WebWork? (A) (B) (C) (D) (E) None of the above UBC Math 102
9 The deriva*ve Geometric UBC Math 102
10 Tangent line It is the line we see when we zoom into the graph of a func*on at some point o UBC Math 102
11 Simple cases: What is the deriva*ve of these func*ons? y = C y = a x
12 Simple cases: The deriva*ve is y = C y = 0 y = a x y = a
13 Remembering slopes Posi*ve zero nega*ve infinite (undefined)
14 Zooming into the graph of At the point x=1.5 UBC Math 102 See course notes
15 Zooming into the graph of At x=0 UBC Math 102 See course notes
16 A func*on with a cusp At the cusp, the deriva*ve does not exist. UBC Math 102 See course notes
17 When does the deriva*ve exist? If a func*on is discon*nuous at x=a then its deriva*ve is not defined at that point. If a func*on has a cusp at x=a, then its deriva*ve does not exist (is not defined) at that point. If a func*on blows up (goes to inky) at x=a, then its deriva*ve does not exist at that point.
18 1. What is this func*on? A x+1 B x-1 C - x+1 D - x-1 E - (x-1) UBC Math 102
19 2. This func*on is A discon*nuous B not differen*able anywhere C has no limit as x->0 D differen*able except at x=1 E I am confused UBC Math 102
20 Sketch the deriva*ve of this func*on UBC Math 102
21 3. The deriva*ve of this func*on looks A like: B UBC Math 102
22 Daily temprature (Forecast for Wedn Sept 20) Sketch the rate of change of temperature.
23 Solu*on in class The local max and min are places where the deriva*ve is zero
24 Solar power Rookop solar collectors Sustainability and clean energy
25 Solar power (KW) What happened here?
26 Note the date
27 Sketch the rate of change of this func*on
28 Solu*on (improved from the one in class)
29 Predator-prey popula*on cycles We have a great record of cycles of predators and prey (Lynx and Hare) in Canadian arc*c why??
30 Overall trend Predator and Prey (in thousands)
31 Just ther prey Sketch the rate of change dp/dx of the prey popula*on
32 Just ther prey Sketch the rate of change dp/dx of the prey popula*on
33 Just ther prey Sketch the rate of change dp/dx of the prey popula*on
34 Tangent lines Now we consider the slopes of the tangent lines
35 Tangent lines Put them all along a single line
36 Tangent lines Figure out where slope is zero, posi*ve or nega*ve
37 Tangent lines Figure out where slope is zero, posi*ve or nega*ve
38 Tangent lines Now sketch the slopes on a new graph, careful to align the places where it is zero
39 Tangent lines Add the rest of the sketch
40 Tangent lines Now we consider the slopes of the tangent lines
41 Rate of change ot the prey popula*on We can use desmos to graph the dervia*ve
42 Deriva*ve on desmos Prey popula*on Rate of change of prey popula*on hvps://
43 Desmos: tangent line hvps:// Experiment with slider Experiment with zoom! UBC Math 102
44 Answers 1. D 2. D 3. A
45 Solu*ons
46 Prac*ce calcula*on we did in class UBC Math 102
47 Challenge: Deriva*ve of y = x n from the defini*on of the deriva*ve: See Appendix E.1 By the binomial theorem, c 1 =n, so f (x) = n x n-1
48 Problems to test your skill A selec*on of problems from tests and exams to help you study concepts in this lecture
49 Related test problem Shown in the graph is the velocity of a par*cle. Use this to sketch the accelera*on of the par*cle. (Hint: the accelera*on is the deriva*ve of the velocity) UBC Math 102
50 See P 67 prob 2.20 Examtype ques*on
51 See P 85 prob 3.7 a, b Graph the deriva*ve of f(x)
52 Given the deriva*ve f (x), graph the original func*on f(x) See P 85 prob 3.7 c, d
53 Related test problem (MT1, 2008) Microtubules (MT) are biological polymers important in cell structure, cell division, and transport of material inside cells. The length of microtubules ( MT length ) grows and shrinks as shown in the following figure from Janulevicius et al (2006) Biophys J 90: Use this figure to draw a sketch of MT growth rate (i.e. rate of change of microtubule length per unit *me) over the same *me interval. Which has a greater magnitude: the rate of shrinkage (per unit *me) or the rate of growth (per unit *me)? Examtype ques*on
54 Solu*ons to previous problems First try out the problems at the end of the last lecture slides. Only then should you peek at the answers.
55 Secant and tangent lines
56 A secant line is (A) A line whose slope is instantaneous velocity (B) A line connec*ng two points on a graph (C) The same as an average velocity (D) The same all along the curve (E) Not sure. UBC Math 102
57 Which of the below is an average velocity? (A) Av veloc= (v 1 +v 2 +v 3 +v 4 )/4 (B) Av veloc = (y 1 +y 2 +y 3 +y 4 )/4 (C) Av veloc = (y 1 +y 2 +y 3 +y 4 )/(t 1 +t 2 +t 3 +t 4 ) (D) Av veloc = (y 1 +y 2 ) /(t 1 +t 2 ) (E) Av veloc = (y 2 -y 1 )/(t 2 t 1 ) UBC Math 102
The deriva*ve. Geometric view. UBC Math 102
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