Essential Tips for Success in Math 21-120 Midterm Exam
School
Vms Institute Of Management Batala**We aren't endorsed by this school
Course
BUS 23902
Subject
Mathematics
Date
Dec 11, 2024
Pages
10
Uploaded by JudgeChimpanzeeMaster1177
Math 21-120 Midterm Exam 2. Oct. 25, 2024Your NameAndrew ID•Note: Your exam will be scanned and uploaded to Gradescope for grading.Only the FRONT of each page is going to be scanned. Do NOT write anythingto be graded on the back side of any pages on this exam (including on scratchpaper), as the back sides will not be scanned or graded.• There are eight problems worth a total of 100 points.•Show all your work(except where explicitly stated otherwise); insufficientlyjustified answers might not receive full credit.Please circle or clearly indicateyour final answer for each problem.• You must not communicate with other students during this exam.• No books, notes, calculators, or electronic devices allowed.•Do not turn this page until instructed to.• If you run out of room on the page for a particular problem, you may use a pieceof scratch paper to write the rest of your solution. Please ensure that you indicatethat the problem is continued on scratch paper.• Please raise your hand if you need any additional paper for scratch work oradditional room to complete a problem.• There is one sheet of scratch paper attached at the end of the exam. You maydetach it from your exam, but you need to ensure that it is stapled to your examwhen you turn it in if it contains work to be graded.•Note: If you would like to leave the exam room before turning in your exam,bring your exam and your phone to the front desk and leave them there whileyou leave the room.Violations of academic integrity will be taken extremely seriously, and will be handledunder the procedures of the Carnegie Mellon University policy on academic integrity.
1.(18 points) Findd ydxin each of the cases below. (You do not need to simplify your answer, but youranswer should start withd ydx=....)a.y=x2earcsin(x2+1)b.yesinx=xcosyc.y=(cosx)px
2.(12 points) Find the following limits if they exist. You may NOT use L’Hôpital’s Rule. If some limitdoes not exist, explain why.a. limx→0sinxtan(7x)b. limx→0cosx·cos(2x)-cosx4x
3.(10 points) Gravel is being dumped from a conveyor belt at a rate of 20 ft3/min, and its coarsenessis such that it forms a pile in the shape of a cone whose base diameter and height are always equal.How fast is the height of the pile increasing when the pile is 12 ft high? Recall that the volume of a rightcircular cone isV=13πr2h. (You may leave your answer in ready-to-calculate form.)
4.(8 points) Using an appropriate linear approximation, estimate ln(1.04)+3p1.04. (You may leaveyour answer in ready-to-calculate form.)5.(6 points) Letf(x)=2x+ex. Find an equation of the tangent line tof-1at the point (1,0).
6.(12 points) Letf(x)=ln(1+x2).a. Find the intervals on whichfis increasing and decreasing.b. Find the local minimum and maximum values off(if any).c. Find the inflection points off(if any), and the intervals of concavity.
7.(18 points) In this problem, you are asked to provide examples of certain functions. You can eithergive an example by drawing the function’s graph or by providing a formula that defines the function.(The former may be simpler.)a. Give an example of a function which is continuous on (-5,5) but has neither a global minimumnor a global maximum on [-5,5].b. Give an example of an everywhere-continuous function which has a global minimum atx=5,butf0(5)6=0.c. Give an example of a functionfdefined on [-5,5] such thatf(-5)=f(5),fis differentiable on(-5,5), but there is no pointcin (-5,5) such thatf0(c)=0.
8.(16 points) Determine whether each statement is true or false. Give a short justification for eachanswer – an answer without any explanation will receive no credit.(a) For any functionf, iffis differentiable atx=0 andf0(0)=0, thenfmust have a local maximumor minimum atx=0.TrueFalse(b) For any functionf, iffis differentiable on (-∞,+∞) and the global maximum offover theinterval [2,3] is attained atx=2, then it must be the case thatf0(2)=0.TrueFalse(c) Letfbe a function which is differentiable on (-∞,+∞), such thatf(1)= -2,f(3)=0.It ispossible forfto satisfyf0(x)≥1 for allx.TrueFalse(d) Iffis differentiable, thenddx(f(5px))=5f0(x)2px.TrueFalse
ADDITIONAL WORK AREA: You may use this page to finish any problems that you did not have roomfor on the previous pages. Do NOT detach this sheet from your exam.
SCRATCH PAPER: You may detach this page from the exam, but be sure to reattach it if it containswork to be graded.