Exam Two. Phu Vu. test Two. Take home group test April 13 ~ April 18. Your group alias: Your group members: Student name

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Millicent O’Brien’
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1 Exam Two Take home group test April 3 ~ April 8 Your group alias: Your group members: (leave it blank if you work alone on this test) Your test score Problem Score Total page /7

Problem : (chapter 9, applications of integrals) (8 points) Lets have a look at the following function: f x =x 2 x The graph of this function for x is sketched as in this

Evolution solid about x axis Evolution solid about y axis a) Compute the volume of the solid generated by evolving the curve y = f(x), where x is from to, about the x axis.

2 Problem : (chapter 9, applications of integrals) (8 points) Lets have a look at the following function: f x =x 2 x The graph of this function for x is sketched as in this picture, in case you are curious. Imagine we rotate this curve about the x axis (or y axis), we will have a cool looking solid, their pictures are also provided. Evolution solid about x axis Evolution solid about y axis a) Compute the volume of the solid generated by evolving the curve y = f(x), where x is from to, about the x axis. b) Compute the volume of the solid generated by evolving the curve y = f(x), where x is from to, about the y axis. (Hint: there were at least 2 examples in lectures which is similar to this question) page 2/7

3 Solution: a) volume = b) volume = y 2 dx= x 2 f ' dx= x 2 x 2 dx= x 4 2 x 3 x dx=[ 2 x5 x x3 x 2 2 x dx= 2 x 3 x dx=[ 2 2 x4 4 x3 3 ]x= x= 3 ]x= x= = 5 6 = 3 3 Problem 2: (chapter, Fish in a pond) (8 points) The population of fish in a lake can be modeled as in following differential equation: where y = y(t) t r = K = 25 a = 4 y' =r y y K a : is the population of fish (thousands) : time (weeks) : is the growing rate : population capacity (thousands) : (thousands) of fish was caught every week. a) Find all constant solutions to this differential equation. b) Lets call the two solutions y and y2. Discuss the meaning of y and y2. What happens when we have different number of fish to start off (initial condition). (Hint: you can also sketch roughly the solutions like in lectures) c) Use separation of variables to solve this differential equation. (Hint: first, rewrite the right hand side as (const)*(y y)(y y2), where y and y2 were computed; later use partial fraction integration; following lecture example closely, all are pretty much similar) d) Suppose that at the beginning, there are 8 thousands fish in the lake ( meaning y() = 8 ), write out the exact solution of this equation. What is the population of fish after weeks? Solution: With the given parameters, the equation is: y ' = y y 25 4 a) Constant solutions: y' =, which means y y 25 4= y 25 y2 4= y=2 y=5 b) If the population starts off from above 2 thousands, it will decrease and going page 3/7

4 down to 2 thousands. If the population starts off below 5 thousands, it will decrease and eventually there is no fish left in the lake. If the population starts off somewhere between 5 and 2, it will increase and get closer to 2 thousands. Lastly, if the population starts at one of the 2 constant solutions, it will not change. 2 5 c) Follow the hints, rewrite the equation as y' = y y y2 4 = y 4 = y 5 y Then use separation of variables to solve the equation: dy y 5 y 2 = 25 dt The right hand side is just (-/25)t + C, while the left hand side, use partial fraction to obtain: /5 y 2 /5 y 5 dy = 5 ln y 2 ln y 5 = 5 ln y 2 y 5 Therefore, bring the 5 to the right, we have ln y 2 y 5 = 5 25 t C = 3 5 t C Which yields y 2 /5 t = Ae 3 y 5 Solve for y, we get /5 t 2 5 A e 3 y = 3 /5 t Ae d) At starting time, we have y() = 8, which means 8 = 2 5 A A Solve for A we get 8 ( A) = 2 5 A, and A = -4 page 4/7

5 So now we know the formula of y(t), meaning the population after t weeks. To get the population after weeks, we just have to replace t = and get e 6 y() = ~ e 6 (thousands) Problem 3: (chapter, applications of Differential Equation) (8 points) Consider the following differential equation and initial condition: y' t = 2 y, y() = t a) Find the correct solution using separation of variables. (Hint: check homework 5) b) Use your answer in a) to compute y(3) (value of y when t = 3) c) Use Euler's method with t = to estimate y(3), how good is this result? d) Use Euler's method with t =.2 to estimate y(3), how good is this result? (You can modify and use my Excel worksheet, but understand it first. Check homework 5, question 3c) Solution a) Separation of variables: dy 2 y = dt t Solve for y to get: y = 2 A t Plug in the condition y() =, we have: = 2 (/A), meaning A = Therefore, the solution is: y = 2 t b) y(3) = 2 - (¼) = 7/4 =.75 c) t y RHS ln 2 y =ln t C 2 y = A t 3 2 estimate y ~ 2, error is.25 (absolute error), or.25/.75 ~.4 (relative error) page 5/7

6 d) t y RHS estimate y(3) =.79, error is.4 (absolute error), or.4/.75 ~.23 (relative error) Problem 4: (chapter 2, discrete random variables) (8 points) Consider the following experiment: We throw 2 dice and add up their faces. The results vary from 2 to 2. a) Create a probability table/chart for this experiment, like in the following: X p /36 2/36 /36 b) If we do this experiment many many times, what is the average of all the outcome? (Hint: expected value!!!) c) Find the variance of this random variable. d) How likely is the chance that the outcome (sum of two faces) is an even number less than? (Hint: probability of an event!!! ) Solution: a) X p /36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 /36 page 6/7

7 b) EX = x i p i = 7 c) Var[ X ] = E [ X 2 ] EX 2 = p i x 2 i EX 2 = 35/6 ~ d) This is the event that outcomes are among 2, 4, 6, 8, the total probability of that event is /36 + 3/36 + 5/36 + 5/36 = 4/36 = 7/8 ~.3889 Problem 5: (chapter 2, continuous random variables) (8 points) Some experts observed that the outcome of a experiment follows a probability density function in the form of: A f x =, x 4 x 2 where A is some coefficient we will need to find out. Question a) Find the value of A so that f can be a proper probability density function b) Find the expected value of the outcome. c) Find the probability that X is less than ( X ) d) Find the probability that X is larger than. Solution: this is a very nice problem. a) f x dx = A dx = A 4 x 2 4 We need to check that the integral Which results in A = 4 f x dx = and our pdf is f x = 4 x 4 4 x 2 b) 4 x EX = 4 x dx = 2 4 x 2 4 x [ dx= 4 ln 4 x = 4 4 x ] Note: this is normal, it means that no matter the outcome of our experiment is, we will eventually get some bigger outcome in a later experiment. c) Using our integral from a), we see that Prob[ X ] = [ = 4 x ] 4 d) Prob[ = 4 4 X ] = - Prob[ X ] = /4 page 7/7

Math 180 Written Homework Assignment #10 Due Tuesday, December 2nd at the beginning of your discussion class.

Math 180 Written Homework Assignment #10 Due Tuesday, December 2nd at the beginning of your discussion class. Math 18 Written Homework Assignment #1 Due Tuesday, December 2nd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 18 students, but