CALCULUS I, Final Exam 1

1 CALCULUS I, Final Exam1MA 125 CALCULUS IFinal Exam, December 10, 2014 Name (Print last name first):..Show all your work, justify and simplify your answer!No partial credit will be given for the answer only!PART IYou must simplify your answer when possible but you don t need to com-pute numbers: e6sin(12/5) +8 is a fine problems in Part I are 4 points Usethe definitionof the derivative to show that the derivative of the functiony=f(x) =x2isf (x) = Find the derivativef (x) iff(x) =x2sin(x).3. Find the derivativef (x) iff(x) = ln(x3+x2+ 1). CALCULUS I, Final Exam24. Find the derivativef (x) iff(x) =x3+1x3 Find the anti-derivative x2(1 + x) Find the anti-derivative sin6(x) cos(x) Find the anti-derivative x3 x4+ I, Final Exam38.

2 Solve ln(x2+ 1) = IfF(x) = x2sin(t2+ 1)dt, findF (x).10. If oil leaks from a well at the rate ofe 5t(m3/s), how much oil will leak in thefirst minute? (If you use your calculator to compute it is OK if you give anapproximate answer.) CALCULUS I, Final Exam411. Approximate 411xdxusing a Riemann sum withn= 3 terms and the midpointrule. What does this number have to do with ln(4)?12. The velocity of a particle is given byv(t) =t2+ 1 (m/s) .(a) Find the accelerationa(2) of the particle,(b) How far does the particle travel in the first 5 seconds? CALCULUS I, Final Exam513. Given the graph of the functionf(x) below answer the following (x)(a) Isf(x) one-to-one?

3 Explain!!(b) Use the graph to approximatef 1(20).(c) Use the graph to approximate (f 1) (20). CALCULUS I, Final Exam6 PART all local/absloute maxima/minima of the functionf(x) = (2x+ 1)3(1 x)5on the real line ( , ). CALCULUS I, Final (t) be the function which specifies the distance (in km) from arunner to the start line at timet(in hours) of a race. The graph ofS(t) is givenbelow:t(h)y(km)123451234S(t)Use the graph to give approximate answers to the following problems.(a) When was the runner running the fastest?(b) What happened between times 3 and 4?(c) What is the meaning ofS 1(1)? CALCULUS I, Final the functiony=f(x) =x2x2 1.

4 Findxandy-intercepts, hor-izontal and vertical asymptotes, all critical numbers, intervals of in-/de-creasing,local/absolute max/minDraw your graph on the next I, Final Exam9xyCALCULUS I, Final 10x2(5 x) I, Final the dimensions of a can ( a cylinder) of radiusr, heighthand volume 1 (m3) with minimal surface area. [Hint: the volumeV= r2handthe surface areaS= 2 rh+ 2 r2.] CALCULUS I, Final Exam12 Scratch paper