Final.2023
.pdf
School
Case Western Reserve University**We aren't endorsed by this school
Course
MATH 125
Subject
Mathematics
Date
Dec 27, 2024
Pages
2
Uploaded by CorporalArt15812
Math 125Final Exam Fall 20231. Progressive Insurance examined the driving records of 900 drivers over the past year and found the followingdata:Age16-2526-7576-92at least one accident3011040no accidents10556055(a) Find the probability that a randomly selected driver was at least 26 years old and had no accidents.(b) Find the probability that a driver was less than 76 years old or had no accidents.(c) Find the probability that a driver who had at least one accident was age 16 to 25.2. LetEandFbe events in a sample space withP(E) = 0.30,P(E∪F) = 0.58, andP(E∩F) = 0.12.(a) Find:P(F)=P(E0|F)=(b) AreEandFindependent? Justify your answer mathematically.3. There are twenty math SIs– 8 work for Terri and 12 work for Chris.(a) In how many ways can six of the twenty SIs be assigned to proctor exams in six different classrooms?(b) Eight of the 20 SIs will form a committee to plan the math gala. How many possible gala committeesinclude Nell or Nidhi (2 of the SIs)?(c) Find theprobabilitythat a committee of eight SIs includes at least one of Terri’s SIs.4. Twenty-nine percent of adults have heart disease, while 8% of adults are diagnosed with diabetes. Adultswho have diabetes are three times more likely to have heart disease than adults who do not have diabetes.Find the probability that an adult does not have heart disease and is not diabetic.5. Sketch the graph of a functionf(x) that is defined for ALL values ofxand that satisfies ALL of the following:•f(x) is continuous everywhere exceptx= 0;•f0(-2) does not exist;•f00(x)<0 only whenx >0;•f0(x)<0 only when-2< x <-1;•limx→0+f(x) =-∞;•limx→±∞=-2.6. The cost to producexbobble-heads is given byC(x) = 5√x+ 10 dollars.(a) Determine the average rate of change in the cost function (with units!) when production increases from6 bobble-heads to 15 bobble-heads.(b) Use the limit definition of the derivative (that is, the four-step process) to determine the instantaneousrate of change when 6 bobble-heads are produced. Be sure to include correct units!
7. Differentiate. Final answers should look nice– at least simplify within each term.(a)y= 2xsin5(x) cos(5x3)(b)y=3-x2x+ 54(c)y=5ex-3(5x) + 3e5-5√x3-ln(5x3)8.(a) Thefirst derivativeof a functionf(x) is given byf0(x) = 4x5e2x-9x3e2x. At what values ofxdothe relative extrema off(x) occur? Classify each as a relative min or relative max.(b) A continuous functiong(x) hasfirst derivativeg0(x) = 2xlnx-8x.Over what interval(s) isg(x)concave down?9. A pipeline is to be run from a source on the edge of a lake to a small resort community on an island 5 milesoffshore. It will cost 3 million dollars per mile to build the pipeline across land and 5 million dollars per mileto build the pipeline under water. How much of the pipeline should be constructed on land to minimize thetotal cost?source::::island:::QQQQQQQQQQQQQQQQQQ-10 mi6?5 miles10. A spotlight is on the ground 30 feet from a wall. A 5-foot tall woman walks away from the wall and towardthe light. Determine the rate at which the height of the woman’s shadow on the wall is changing when sheis 12 feet from the wall and walking 3 ft per second.11. Finddydx:5x-3x5y2= 4y+ycos(x) + 112.(a) Letf(x) = ln(3x). Write the third-degree Taylor polynomial forf(x) atx= 2.(b) A functionf(x) has a Taylor polynomial given byPn(x) = 2-6x+ 9x2-9x3+. . .+2(-3)nn!xn.Determine the value off(23)(0). (You do not need to simplify the answer.)13.(a) Solve the difference equation:an= 10-23(an-1) ,a0= 8.(b) Solve the difference equation:an=an-1-5,a0= 2.(c) Find the fixed points of the difference equation. Classify each of them as stable/unstable/semi-stable.an=13(an-1)2+ 2 (an-1)Have a great break!