Exponen'al func'ons and exponen'al growth. UBC Math 102
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1 Exponen'al func'ons and exponen'al growth
2 Course Calendar: OSH 4 due by 12:30pm in MX 1111 You are here
3 Coming up (next week) Group version of Quiz 3 distributed by Group version of Quiz 3 due in class
4
5 Learning goals
6 The func'on 2 n
7 (1) Which of the following approxima'ons is reasonable? (A) (B) (C) (D) (E)
8 (2) Andromeda strain hwps:// This statement is (A) TRUE (B) FALSE
9 Here is the informa'on we need: Each bacterium divides once per 20 min.
10 (3) How many cells does it take to make an earth- sized colony? Each bacterium divides once per 20 min. (A) (B) (C) (D) (E) None of the above
11 (3) How many cells does it take to make an earth- sized colony? Each bacterium divides once per 20 min. Solu'on: number of cells = mass Earth/mass E coli =
12 (4) How many cells are there a_er 1 day? (Each bacterium divides once per 20 min.) (A) 72 (B) (C) (D) (E) Not sure how to compute it.
13 (4) How many cells are there a_er 1 day? (Each bacterium divides once per 20 min.) Solu'on: there are 3x 24=72 doublings, so a_er one day there are cells. (or, with a calculator, we find ) hwp://
14 (5) How long does it take to make an earth- sized colony? (That is, how long un'l we have cells?)
15 (5) How long does it take to make an earth- sized colony? (That is, how long un'l we have cells?) (A) 30 hrs (B) 40 hrs (C) 50 hrs (D) 100 hrs (E) 500 hrs
16 (5) How long does it take to make an earth- sized colony? (That is, how long un'l we have cells?) So it takes (roughly) 122 doublings = 122 x (1/3)hr A total of 40.6 hours!
17 Alternate solu'on using logarithms (we ll study this next 'me): Solve for n: Using logarithms: 122 doublings = x (1/3)hr =40.72 hr See also Example same idea but with natural logarithms
18 Exponen'al growth applica'on POLYMERASE CHAIN REACTION - A way of amplifying DNA for research, gene'cs, crime- scene forensics - hwps:// - See also the PCR Song: - hwps://
19 A smooth func'on 2 x
20 (6) Why do we want a smooth func'on for 2 x? (A) So that the limit 2 x will exist as x à 0. (B) So that we can find 2 n for nega've n. (C) So that y=2 x will actually be a func'on. (D) So we can use tools of calculus (such as the deriva've). (E) None of the above.
21 Other bases: y=a x
22 What is the deriva've of y=a x? Use the defini'on of the deriva've to calculate dy/dx.
23 (7) What is the deriva've of y=a x? (A) (B) (C) where C is some constant (D) (E) where C is some constant
24 (7) What is the deriva've of y=a x? Use the defini'on of the deriva've to calculate dy/dx. This thing is a constant
25 (8) What is special about the base e? (A) It is the only base such that d(e x )/dx=e x. (B) It is the only base for which e 0 =1. (C) It is the only base that has an inverse (e x ). (D) It is the only base that has an inverse (ln(x)). (E) More than one of the above.
26 The value of e For the base e, this constant = 1, so that Write the value of e in terms of a limit
27 (9) The value of e x The value of e can be wriwen in terms of a limit: (A) (B) (C) (D) (E)
28 (9) The value of e x The value of e can be wriwen in terms of a limit: For small h: In the limit as h à 0
29 (10) The graph of e x looks like which of these?
30 Find the equa'on of this tangent line Tangent line to the curve y=e x at x=0.
31 (11) Find the equa'on of this tangent line Tangent line to the curve y=e x at x=0. (A) y=x + 1 (B) y=e x (C) y=x (D) y=cx (E) y=e x + 1
32 Find the equa'on of this tangent line Tangent line to the curve y=e x at x=0. Solu'on: The slope is At x=0 slope is m= TL goes through b=1, and equa'on so has y- intercept
33 Inverse func'ons and logarithms
34 The natural logarithm is an inverse func'on for
35 Restate the rela'onship in the form Now use implicit differen'a'on to find dy/dx
36 Restate the rela'onship in the form Now use implicit differen'a'on to find dy/dx Result:
37 Answers 1 C 2 B 3 D 4 D 5 B 6 D 7 E 8 A 9 A 10 C 11 A
38 Prac'ce: Exam ques'on Note: Part (a) uses logarithm in final step.. See next lecture
39 Prac'ce: conceptual midterm ques'on Here is the graph of y = C e kt for some constants C, k, and a tangent line. Use data from the graph to determine C and k.
40 Prac'ce: conceptual midterm ques'on Here is the graph of y = C e kt for some constants C, k, and a tangent line. Use data from the graph to determine C and k.
41 Prac'ce midterm ques'on Sketch the graph of the func'on
42 Prac'ce Exam ques'on
43 Prac'ce Exam Ques'on:
44 Prac'ce Exam Ques'on
45 Prac'ce Exam ques'on for Implicit differen'a'on
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