Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers. x 1 x b 1 sn(x) (b) (4 ponts) Please evaluate lm x π 1 cos(x). (c) (4 ponts) Please evaluate lm x x2 e x. (d) (4 ponts) Please wrte the sum 7 + 1 + 7 + 2 + 7 + 3 + 7 + 4 + 7 + 5 + + 7 + (n 1) + 7 + n n sgma notaton and fnd a closed form expresson for t. What s the value of the sum when n = 10? Soluton to Queston 1. (a) usng l Hoptal s rule, we get (b) lm x π (c) Usng l Hoptal s rule twce we get (d) x a 1 lm x 1 x b 1 = lm ax a 1 h 1 bx = a b 1 b. sn(x) 1 cos(x) = sn π 1 cos π = 0 1 ( 1) = 0. lm x x2 e x = lm x x 2 = lm e x x 3x = lm e x x 2e x 7 + 1 + 7 + 2 + 7 + 3 + 7 + 4 + 7 + 5 +... + 7 + (n 1) + 7 + n = (7 + ) When n = 10, then the sum equals 7 10 + = 7 + = 7n + (n (n 1))/2. 10 (10 1) 2 = 70 + 45 = 115.
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 2 Queston 2. (20 ponts) True/False. Determne whether the followng statements are true or false, and crcle your response. Please gve a bref explanaton (n a complete sentence!). (a) (4 ponts) If a dfferentable functon f s defned at c, and c s a crtcal pont of f, then f has ether a local mnmum or a local maxmum at c. TRUE FALSE (b) (4 ponts) The functon f(x) = sn(cos(x)) + x3 + 7 e x s ntegrable on the nterval [a, b] for any real numbers a and b. TRUE FALSE (c) (4 ponts) Let f(x) be a functon, and P n (x) ts order n Maclaurn polynomal. Suppose that P n (0.1) = 3.728 and E n (0.1) 0.005. Then we know for certan that f(0.1), rounded to two decmal places, s 3.73. TRUE FALSE (d) (4 ponts) The area of the regon bounded by sn x and the x-axs, between x = π and x = π s computed by π π sn x dx. TRUE FALSE (e) (4 ponts) To estmate the area below the functon y = x 2 + 4 between x = 0 and x = 2, we may wrte a Remann sum ( ( ) 2 2 S n = + 4) 2 n n. Ths sum S n s less than the actual area for all n. TRUE FALSE Soluton to Queston 2. (a) FALSE. The functon y = x 3 has a crtcal pont at x = 0, but nether a local max nor a local mn there. (b) TRUE. Ths functon s contnuous at all real numbers x : each of the functons sn(x), cos(x), x 3 + 7, e x s contnuous, and composton, sums, and quotents are contnuous (snce e x 0, there are no ssues when we have t n the denomnator). A theorem from 5.3 says that f f(x) s contnuous on [a, b], t s ntegrable on [a, b]. (c) FALSE. If E n (0.1) = 0.005, then f(0.1) = P n (0.1) + E n (0.1) = 3.728 0.005 = 3.723 whch s 3.72 when rounded to two decmal places. (d) FALSE. Because from π to 0, n the above ntegral, the area between π to 0 has a mnus sgn (below the x-axs. (The ntegral becomes actually 0 because of the symmetry of sn x around the orgn). (e) TRUE. The functon s decreasng on [0, 2] but stays postve.
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 3 Queston 3. (20 ponts) Consder the functon f(x) = 1 x2 x 2 4. We wll get you started by computng the frst two dervatves: f (x) = (a) Fnd all crtcal and sngular ponts of f. (b) Fnd all asymptotes of f. (c) Where s f ncreasng and decreasng? (d) Where s f concave up and concave down? 6x (x 2 4) 2 and f (x) = 18x2 + 24 (x 2 4) 3. (e) Usng what you have calculated n the frst four parts of ths problem, please graph the functon f(x) = 1 x2 x 2 4 on the axes gven below. There s a practce sheet of graph paper on the last sheet of ths exam that you may wsh to use to make a frst attempt at the graph. Soluton to Queston 3. (a) Set f (x) = 0 to get 6x = 0, so we have crtcal ponts when x = 0. We have sngular ponts when f (x) s undefned but f(x) = 0 s defned. Note that f (x) s undefned f (x 2 4) 2 = 0,.e x = ±2. But f(2), f( 2) are undefned, so there are no sngular ponts. (b) To fnd vertcal asymptotes, we look for 0 n the denomnator, so x = ±2. To fnd horzontal asymptotes, observe that the lm x ± f(x) = 1, so we have a horzontal asymptote y = 1. We only get oblque asymptotes f the degree of the numerator s equal to the degree of the denomnator plus 1, so there are no oblque asymptotes. (c) f s decreasng n (, 2) and ( 2, 0), and ncreasng n (0, 2) and (2, ). (d) f s concave up when 2 < x < 2, and concave down when x > 0 or x < 2.
Math LL10 (Sprng 2009) Prelm 3 (A4/21/2009) (e) Usng what you have calculated n the frst four parts of ths problem, please graph the functon 1-x2 f(x) : x2-4 on the axes gven below. There s a practce sheet of graph paper on the last sheet of ths exam that you may wsh to use to make a frst attempt at the graph. Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 4 FINAL ANSWER:... ' "...,... l II y=l J- -r- --. t -..:... -..-:- --.. :: -.-:- ".. (e) Queston 4. (9 ponts) Suppose that a functon f satsfes f(1) = 3 and ts DERIVATIVE s d ( (Keep ) gong!!) f(x) = e x2. dx Please show your work when you answer the questons below. (a) Fnd the lnear approxmaton to f about x = 1. (b) Use the lnear approxmaton to estmate the value of f(1.5). (c) Is you answer n (b) an under- or over-estmate? Please gve an argument supportng your answer. (Hnt: you do not need to compute the exact value of the error.) Soluton to Queston 4. (a) L(x) = f(1) + f (1)(x 1), so L(x) = 3 + (x 1)/e. (b) f(1.5) L(1.5) = 3 + 0.5/e = 6e 1 2e. (c) The formula for the error s E 1 (x) = f (s) 2 (x 1)2, wth 1 s x. We have x = 1.5, and an easy calculaton shows that f (x) = 2xe x2. Snce s s postve, we have f (s) < 0, whch means that the error s negatve. Ths means that the answer n part b s an overestmate.
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 5 Queston 5. (15 ponts) A clown s fllng a sphercal balloon wth helum from a tank that emts helum at a rate of 250 cm 3 /s. Please answer the followng questons, showng your work and specfyng the unts of your answers. You may wsh to refer to the formulæ at the end of ths exam. (a) How fast s the radus of the balloon changng when the radus s 5 cm? (Hnt: your answer should have a π n t.) (b) How fast s the surface area of the balloon changng when the radus of the balloon s 5 cm? (c) After 20 seconds, the balloon escapes the clown s grp. Helum blows out of the balloon at a rate of 350 cm 3 /s. How fast s the radus changng when the radus of the balloon s 10 cm?
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 6 Soluton to Queston 5. (a) The formula for the volume fo a sphere s V = 4 3 πr3, so dv dt and substtutng for r and dv dr, we get dt dt = 1 dv 4πr 2 dt = 250 100π cm/sec. (b) The formula for the surface area of a sphere s S = 4πr 2, so ds dt = 8πrdr dt 100cm 2 /sec. (c) Usng the formula dv dr = 4πr2 dt dt 1 350 ( 350) = 400π 400π cm/sec. = 4πr2 dr dt. Rearrangng = 40π 250 100π cm/sec = we derved before, we get dr dt = 1 4πr 2 ( 350) = Queston 6. (20 ponts) You are n the woods 4 km due north of the nearest pont P on a straght east-west road. You need to get to the bus stop on the road that s 10 km west of the pont P. You can walk 3 km/h n the wood and 5 km/h on the road. Please answer the followng questons. (a) Draw a fgure ndcatng your poston, the pont P, and the bus stop. Suppose that you walk to a pont on the road, x km west of P. Wrte the travel tme T (hours) as a functon of x (km). (b) Suppose that the doman of T(x) s all real numbers. Lst the sngular ponts and crtcal ponts of T(x), and determne all local maxmum and local mnmum of T(x). (c) To what pont on the road between P and the bus stop should you head n order to mnmze the travel tme to the bus stop? Justfy your answer. (d) The last bus of the day leaves the stop n three hours. Can you make t, or wll you be sleepng under the stars? Please gve evdence to support your answer. Soluton to Queston 6. Let x be the dstance on the road beween P and the pont A you head. (a) The travel tme s wrtten as T(x) = 10 x 5 x2 + 4 + 2. 3 (b) To fnd sngular and crtcal ponts on the doman, compute T T (x) = 1 5 + x 3 x 2 + 4 2.
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 7 We don t have sngular ponts. The crtcal ponts are computed by settng T (x) = 0. 1 5 + x 3 x 2 + 4 2 = 0 5x = 3 x 2 + 4 2 x > 0, x 2 = 9 x = 3. Here note that the rght hand sde of 5x = 3 x 2 + 4 2 s postve, so x must be postve. We should compute T(3) = 7/5 + 5/3 = (21 + 25)/15 = 46/15 = 3 + 1/15. T (x) < 0 on x < 3. (you can plug n 0 nto T (x)) T (x) > 0 on x > 3. (you can take lmt x or plug n some number > 3, lke 10, and see that t s greater than 3 + 1/15.) Therefore x = 3 gves a local mnmum and there s no local maxmum. (c) x = 3 gves a mnmum of T(x) snce the functon s ncreasng on x > 3 and decreasng on x < 3. Therefore we should head to the pont 3 km west of the pont P. (d) Snce 3 + 1/15 s greater than 3, we can not make t to the bus.