Et6[frm€!Please circle the name of vour section leader:Srdjan Divac Robert Kaplan (10:00)Robert Kaplan (11:00)Esther SilbersteinMathernatics 1aFinal ExaminationMay 24, 1995Show all your work.QuestionPointsScore16263 846D t68II8 II1010 101l 8T213Total100,&1. (6 points) Use the definition of derivative to find f'(x) if. !(r) = *i2. (6 points) Find the derivatives of the following functions:(") g(t) = x3 .e'(b) ft(o) = ln(sin c * cos a;): r, ,,,,.,.,. :.r ,. : ,' ,' t/ttL$}., ": : rillf j;;E;- r'I : :: :::: l:/:lr: l: : ll(8 points) Find antiderivatives of the following functions:(a) /(t) = 2tr - 13t + ;hr-3t+f(c) g(t) = tz(d)m(r)=+4. (6 points) Given the curve 3rtano - cosy,(a) nnd #(b) Find the equation of the tangent to the curve at (t,l)5. (7 points) Find the area between U = coso and U = a2 -t.4'6. ( 8 points) Each graph in the right-hand column represents the second derivative of some func-tion in the left-hand column. Match the functions and their second derivatives.Functions(a)Second Derivuives(i)FunctionFunctionFunctionFunctiou(a) has second derivative(b) has second derivative(c) has second derivative(d) has second derivative7.(9 points) A searchlight in a lighthouse 15 miles off a straight shore is revolving at the rate of 2revolutions/minute. At what speed does the beam of light pass a point 20 miles down the shorefrom the lighthouse?(9 points) The U. S. Postal Service will accept a box for domestic equipment only if the sum ofits length and girth (distance around) does not exceed 108 inches. Find the dimensions of thelargest acceptable box with a square end. (Greatest volume!)6161= 4rLcngth = r',,-6,P:n. qg points) (a) Sketch a representative family of solution curves for the differential equationh = cos P for -2r { 0 { 2r on the axis below.(b) What happens to P(t) as f --+ m if P(0) = 6?(c) What happens to P(t) as t --+ m if P(0) =3o,2'1010. (10 points) Eugene Saperstein is driving south on the Panamaniac Highway in his "Volga" sedan.Suddenly, having seen a roadblock, he steps on the brakes and decelerates at -20mf s2.If it takeshim 90m to come to a stop, how fast was he driving (in m/s) when he stepped on the brakes?11li. (8 points) (a) How many subdivisions would be needed to approxim ^t" [^'tt i.f , dc with anerror of no more than 0.11? Jo(b) Using the number of subdivisions n you found in part (a), find an upper bound [/ forft (1+0\ ,Jo too "''(c) Using the number of subdivisious n you found in part (a), find a lower bound .t for/t (1 + rz)s ,Jo -too "'t212. (13 points) Consider the function /(c) = Y.(a) Show that f,(x)= t;t,"t.(b) Show thar f,,(x)= "nf,u- t.(c) In what intervals is / increasing?(d) In what intervals is / concave up?(e) Provide the coordinates of all local max and local uriu, if any, and indicate what theyare. Do not approximate.Continued on next page.1312, continued.(f) Provide the coordinate of all inflection points. Do not approximate.(g) Does this function have any vertical asymptotes? Explain.(h) Does this function have any horizontal asymptotes?(i) Sketch the graph of the above function, labeling (without approximating) all intercepts,stationary points, inflection points, and asymptotes. Choose a scale such that all features
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