Lecture 26

Faculty of Science; Department of Mathematics
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Summary

Group 7: This is a review class. Feel free to create examples and discussions on any topic in the course.

Here are some general areas that I would like to review.

assignment #2 question 2 but with bounds of ax and bx, where a and b are some real number.

Review of Riemann sum applications.

Review of Recursive shapes (snowflakes)

How to find median

1. A bag contains 5 red balls, 3 green balls, and 2 yellow balls. If the balls are always replaced into the bag after being drawn, what is the probability of drawing the same color out of the bag on two successive attempts?

P(RR)+P(GG)+P(YY)

=(5/10)*(5/10)+(3/10)*(3/10)+(2/10)*(2/10) =(38/100) or 38%

2. At the beginning of a poker game, each player receives two cards randomly selected from a deck of cards. Suppose you are one of the players. What is the probability that the two cards you receive are two aces?

sample space C(52,2)

     number of outcomes C(4,2)

Hence the probability is (6/(26*51)=0.0045

3. A drawer contains 4 black,6 brown, and 2 blue socks. 2 are drawn at random from the drawer, one after the other. What is the probability that both socks will be the same colour?

P(2 black)=C(4,2)/c(12,2)=(6/66)

P(2 brown)=C(6,2)/C(12,2)=(15/66)

P(2 blue)=C(2,2)/C(12,2)=(1/66)

Add them up and you get (1/3)

4. A recent graduate plans to spend a month looking for a job in each of Vancouver, Calgary, and Toronto in that order. The graduate will accept the first job offered and stop looking at that time. The probabilities that a person with similar qualifications as the grad student will find a job in Vancouver, Calgary, or Toronto during a one-month period are 0.3, 0.6, and 0.5 repectively. Find the expected number of cities visited by the graduate.

Let x=number of cities visted by grad

P(x=1)=0.3

P(x=2)=0.7(0.3)=0.42

P(x=3)=0.7(0.4)1=0.28

Use the expected value formula of sigma x(p(x))dx = 1(1)+2(0.42)+3(0.28) =1.98

5. Write the Taylor Series for sin(2x) upto and including x^5

Use the taylor series for sin(x) with the substitution of (2x)

2x-((2^3)*x^3)/3!)+((2^5)*x^5)/5!)....

6. Consider the differential equation dy/dt=ry(y-a)(1-(y/k)). Find the steady state solutions

ry(y-a)(1-y/k)=0 therefore y=a, y=K and y=0

7. A man has 1 nickel, 2 dimes, and 1 quarter in his pocket. Suppose he reaches in his pocket and draws 2 coins at random. What is the expected amount of money that he draws?

let x=amount of money drawn--possible amounts are 15, 20, 30, and 35 cents with probabilities, (2/6), (1/6), (1/6), (2/6) respectively.

Using the discrete probability mean formula we get the sum of x(p(x))

=15*(2/6)+20*(1/6)+30*(1/6)+35*(2/6)

=25 cents

8. Calculate the probability of getting 3 heads by tossing a fair coin 6 times.

P(6H)=C(6,3)(1/2)^3*(1/2)^3

=5/16

9. What is the probability of getting atleast 5 heads?

P(atleast 5 H)=C(6,5)(1/2)^5(1/2) +C(6,6)(1/2)^6(1/2)^0

=7/64

10. Suppose that X measures the time (in hours) it takes a student to complete an exam. Assume that all students are done within 2 hours and the density function for X is given by p(x)=x^3/4 from 0 to 2. What is the proportion of students that take between 1.5 to 2.0 hours to finish the exam?

integral from 1.5 to 2 of (1/4)(x^3)dx

=(1/4)(x^4/4) evaluated at 1.5 and 2

=175/256

11. What is the mean time for students to complete the exam?

integral from 0 to 2 of x(1/4*x^3)dx =(1/4)(x^5)/5 evaluated at 0 and 2 =32/20 or 1.6 hours

12. What is the mean?

integeral from 0 to m of f(x)dx

=(1/4)*(x^4/4) evaluated at 0 and m

=1/16[x^4]evaluated at 0 and m

=m^4=8 m=1.68 hours

13. Suppose f(x) is a function with f(1)=2, f`(1)=0 and f``(1)=3. Estimate f(1.1)

f(x)=f(1)+f'(1)(x-1)+f(1)/2)*(x-1)^2 =2+(3/2)*(x-1)^2

sub in x=1.1 and you get 2.105

14. Determine the Taylor Series of the function f(x)=sinx^2

Using the taylor series of sin(x)=x-(x^3)/3!)... just substitute in x^2 and you get

x^2-(x^6/3!)+(x^10/5!)...

15. Using the first 3 terms of taylor series find a numerical estimate for e

1+1+(1^2/2!)....=2.5

16. An antiderivative of (x/(4-x^2) is :

let u =4-X^2 du =-2xdx

=-1/2int of du/u -1/2ln(4-X^2)+C

17. Find the average value of f(x)=cos^2(x) over [0,2pi]

f(x)=1/2pi*intgergal from 0 to 2pi of cos^2(x)dx = (1/2pi)*integral from 0 to 2pi of ((1+cos(2x)/2)dx =(1/2pi)*(1/2)integral from 0 to 2pi of (1+cos^2(x))dx

=(1/4pi)[x+0.5sin(2x)]evaluated at 0 and 2pi

=1/2

18. What determines if a steady state is stable?

If the solution moves farther away from the steady state it is unstable. However, if it comes back to the steady state, it is considered stable.

19. Solve the differential equation by separation of variables

(dy/dx)=Y^2*cos(x)

=integral of(1/Y^2)dy= integral of cosxdx

=-1/y=sinx +C

y=(-1/(sinx+C))

20. Solve the differential equation (dy/dx)=(1+x)/(xy)) y(1)=-4

int of ydy=int of (1+x)/x=(1/x)+(x)+C

Y^2/2= ln(x)+x+C

sub in initial conditions to find C

y^2=2(lnx+x+7)

y=-(2(lnx+x+C)^0.5

21. Find the solution of dP/dt=2P-2Pt that satisfies P=5 when t=0

dP/dt=dP(1-t)

int(dP/P)= int(2(1-t))dt

=ln(P)=dt-t^2 +C

=(P)=e^2t-t^2*e^c --let e^c =D

P=+/-D*e^2t-t^2

=Ee^2t-t^2

using intial conditions P(t)=e^2t-t^2

22. I received four bug bites during a trip to Mexico where 10% of the bugs carried a deadly virus. The chance to get a bite from each bug was equal. What is the probability that atleast one of the bugs that bit me was a carrier of the virus?

Let E denote that atleast one bug was the carrier Let D denote that no bug was the carrier

P(D)=(0.9)^4 Therefore P(E)=1-(0.9)^4= 34.4%

23. An absent-minded person picks 2 socks at random from a drawer that contains 4 black and 6 white socks. Find the probability that the socks match.

P(B)=C(4,2)=6 P(W)=C(6,2)=15 Total Sample Space=C(10,2)=45

Therefore (6+15)/45=(7/15)

24. A club has 10 members, 4 men and 6 women. A delegation of 3 of the members is chosen at random. What is the probability that atleast 2 women are chosen?

Let E=Event of atleast 2 women chosen and S=sample space

S=C(10,3)=120

P(E)=C(6,2)+C(6,3)=35

Therefore 35/120=7/24

25. It is claimed that the waiting time x(in years) between getting a scheduled appointment for surgery and th actual operation has a probability density function of the form f(x)=k(1-x^3). Find k.

integral from inf to 0 of 0dx +int from 0 to 1 (k(1-x^3)dx +int from 1 to inf of 0dx=1

k(1-x^3)evaluated at 0 and 1 =1

k=4/3

26. Using the sum from k=0 to N=infinity of (x^k) find the taylor series of 1/1-x

use values of k starting at 0 and you get 1+x+x^2+x^3+x^4...

27. Using the sum from k=1 to N=infnity of (-1)^k-1/k*x^k find the taylor series of ln(1+x)

=x-X^2/2+ x^3/3-x^4/4 ....

28. Using the sum of k=1 to N=infinity of (-1)^(k-1)/2k-1 of x^2k-1 find the taylor series of arctan x

x-x^3/3+x^5/5-x^7/7....

29. Seeds are dispersed from a tree in all directions uniformly. The density of the seeds that land on the ground at a distance r away is approximately described by the funtion s(r)=50sin(pir/10) [0,10]. At was distance from the tree are the seeds most densely distributed?

sin is mas at 1=pi/2

make the argument =to pi/2 allowing r=5

30. what is sin(pi/2-x)?

This is equal to cos(x)

31.What is the pythagorean identity?

sin^x(x)+cos^2(x)=1

32. sin^(x)=?

(1-cos(2x)/2)

33. The antiderivitive of ln(x) is (1/x). True or false?

False. The derivative is (1/x) not the antiderivative.

34. WHen is the techniqe of partial fractions used?

When there are polynomials or rational expressions in the integral that can't be solved with substitution.

35. The growth rate of a plant during the first week after germination is given by g(t)=ae^t/2, where the time t is measured in days and a is a constant. Determine the height of the plant h(t) as a function of the time t, assuming that it was zero at t=0.

h(t)=int of g(t)dt

=2ae^t/2+C

at h(0)=0 so C=-2a

Therefore h(t)=2a(e^t/2-1)

36. On the first day, the plant grew 1 mm. Determine the constant a.

1=2a(e^t/2-1) 0.5=a(e^t/2-1)

a= 1/(2(e^t/2-1))

37. 1-2x^2 is the second order Taylor Polynomial for which function?

cos(2x) beause if you use the taylor series of cos(x) and sub in a 2x you get the above function.

Derivative and Trig Review-----------

38. What is the derivative of a^x

a^xln(x)

39. what is the derivative of csc(x)?

-csc(x)cot(x)

40. Derivative of arcsin(X)?

1/sqrt(1-x^2)

41. Derivative of arccos(x)?

-1(sqrt(1-x^2))

42. An integral from a to a =?

This is taking the integral from point a to a which is 0.

43. What is the antiderivative of sec(x)dx?

ln(sec(x)+tan(x))+C

44. What is a major difference between definite and indefinite integrals?

Definite integrals give you a numerical answer and have numerical bounds of integration while indefinite have no bounds of integration and give the antiderivative + a constant.

45. what is the antiderivative of cot(x)dx?

ln(sinx)+c

46. How to tell if an integral converges or diverges?

Take the limit and see if it goes to infinity or comes to a number.

47. integral of dx/(x(lnx)^4)?

set u=lnx and take the integral to get (-/3(ln(x)^3)+C

48. Integral of xsec^xdx

u=x du=dx dv=sec^x v=tanx

plug into the formula to get

xtanx +ln(cosx)+C

49. The velocity of a car is given by 10t+C. How do you find the position?

Take the integral with respect t to get 5t^2+C

50. The acceleration is given by the function e^-x dx. What is the velocity?

Take the antiderivative to get -e^-x +C

51. What is the derivative of sec(x)?

sec(x)tan(x)

52. What is the antiderivative of sin(x)?

-cos(x)

53. What is the best function to put at x in integration by parts?

the one that is easiest to take the derivative of.

54. Which should be used at dv in IBP?

usually the one that is easy to take the antiderivative of.

55. What is the usual for of a ODE?

P(t)=Ce^kt

56.