Practice Final A. Miller Spring 89 Math Find the limits: a. lim x 1. c. lim x 0 (1 + x) 1/x. f. lim x 0

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1 Practice Final A. Miller Spring 89 Math Find the limits: a. lim 1 ln() 1 b. lim 0 ln (3+ )2 9) c. lim 0 (1 + ) 1/ d. lim 2 +1 e e. lim 0 (sec()) 1 2 sec() 1 g. lim Find the integrals: a. ln( 2 ) d b. e 1+e d c. 1+ d d e 3 d e. d If f() = ln( 2 e ), then what does f (2) equal? 4. Find d d log ( + 1) f. lim 0 ln(cos()) 2 5. Two buildings, one 1 unit high, the other 2 units high are 3 units apart. A solar collector is to be placed between them on the ground, maimizing its angle of eposure. How far should it be from the shorter building? 6. A biccle is being peddled at the rate of one complete turn of the peddles per second. The peddle sprocket has radius 4 inches and the wheel sprocket has radius 1 inch. The radius of the wheel is 18 inches. The length of the chain is 6 feet and the biccle is red. How fast is the biccle moving? If the biccle is climbing a hill which is 15 degrees from the horizontal and moving 25 miles per hour, how fast is the altitude changing? 7. Derive a rule for finding d d (uv ) in terms of u, v, du dv d, and d ; where u and v are differentiable functions of. You ma use an rules for differentiation ou know. 8. Find the value of c such that the line = c bisects the region bounded b the curves = 2 and = The base of a solid is the triangle in the,-plane with vertices at (0,0), (0,1), and (1,0). The cross sections perpendicular to the -ais are squares with one side on the base. Find the volume of the solid. 10. What is the surface area of the solid obtained b revolving the curve = 3 with 0 1 about the ais? 11. What is the center of gravit of a wire joining (0, 0) and (1, 0) with densit ρ() = 1? 12. Find the suface area S(R, r) of the Diskus of Frisbius. First described b Archimedes in 275 BC, it is obtained b revolving the curve (a) consisting of the line segment joining (0, r) and (R, r), and the quarter circle of radius r centered 1

2 at (R, 0). Let V (R, r) be the volume obtained b revolving the area (b) around the ais. Show how to obtain V (R, r) b using S(R, r). Eplain ou formula. (R, r) (R, r) (0, r) (a) (0, r) (b) (R + r, 0) (R + r, 0) 13. Bzantium decas at a rate proportional to the amount present. The halflife is 500 ears. What fraction of the original amount is present after 250 ears? Final Eam from Dec Prove directl without using an differentiation rules that d d = True or false. a. A continuous function on an interval is alwas differentiable ecept at possibl finitel man points. b. If a function f() is differentiable at a point c then it is continuous at c. c. Between an two zeros of a differentiable function lies a zero of its derivitive. d. If an interval contains three distinct zeros of a twice differentiable function then it also contains a zero of the function s second derivitive. e. If the derivitive of a function is alwas positive then the function cannot have an inflection point. f. The product of two differentiable functions is continuous. g. A continuous function f on an interval [a,b] must achieve its etreme values (minimum and maimum). h. If f(c) is the maimum value of a differentiable function f, then f (c) = 0. i. Suppose f() is a continuous function on [a,b] such that for ever in [a,b], f() is in [a,b]. Then for some c in [a,b], f(c)=c. From now on assume f, g, and f k are continuous functions on the interval [a,b]. k. b a f()g()d = b a f()d b a g()d j. b a [c 1f() + c 2 g()]d = c 1 b a f()d + c 2 b a g()d where c 1, c 2 are constants. l. b a [Σn k=1 c kf k ()]d = Σ n k=1 [c k b a f k()d] where c 1,..., c n are constants. 2

3 m. b a [f()]n d = [ b a f()d]n n. b b a f()d = a f()d o. if a < c < b, then c a f()d = b a f()d b c f()d p. if a < c < b, then c a f()d b a f()d q. Suppose for ever in [a,b], f() g(), then b a f()d b a g()d r. Suppose for ever in [a,b], f() g(), then for ever in [a,b], f () g (). 3. Multiple choice: 1. Suppose 4 1 f(t)dt = 6 and 4 3 f(t)dt = 2. What is 3 (2f(t) + 1)dt =? 1 a. cannot be determined b. 13 c. 18 d What is d 2 d sin(t2 ) dt = a. sin( 4 ) sin( 2 ) b. (2 1) sin( 2 ) c. 2 2 sin( 4 ) 2 sin( 2 ) d. 2 2 cos( 4 ) 2 cos( 2 ) e. none of these 3. f() = t2 dt, g() = 2, h() = f(g()), h (1) = a. 2 b. 2 2 c. 2 d. 1 e. none of these lim = a. doesn t eist b. 1 c. -1 d. e = f() a 4. Find the following integrals: a. cos(4 2) d b. 1 2 d 1+ 1 c. tan(θ) dθ d. 3 ( 2 +1) 3 d e.( Note definite ) d 5. Let R 1, R 2, and R 3 be the regions indicated below. R 1 b = f() a R 2 b = g() = g() d c R 3 = f() Epress the following as definite integrals. (a) The volume of the solid generated when R 1 is revolved about the -ais. (b) The volume of the solid generated when R 2 is revolved about the -ais. (c) The volume of the solid generated when R 3 is revolved about the -ais. (d) The volume of the solid generated when R 3 is revolved about the -ais. 3

4 6. A compan wants to manufacture a right circular clindrical can. The material for the round ends costs 20 cents per sq cm. and the material for the side costs 10 cents per sq cm. The volume is required to be 108π cubic cm. What are the dimensions of the can which is cheapest to manufacture (Assume no waste in cutting the material). 7. The base of a solid is the triangle in the,-plane with vertices at (0,0), (0,1), and (1,0). The cross sections perpendicular to the -ais are squares with one side on the base. Find the volume of the so lid. 8. Cornucopia: Horn of Plent. The center of horn is a circle of radius 6. The cross section at the angle θ of the Horn is a circle of radius r = θ. What is the volume? r r θ (6, 0) Final Eam from Ma A search light is revolving around once ever minute, shining on a wall. When the light beam is making a 90 degree angle with the wall ou see that it is traveling at the rate of 10 feet per second. How far is the search light from the wall? 2. Use Newton s method to solve 3 = + 1 with a starting value of 0 = 1. Find 1. 4

5 3. Use the trapezoidal rule with n = 4 to estimate d. 4. Radioactive material decas at a rate proportional to the amount present. Ten megatons of radioactive waste with a half-life of one hundred centuries is buried in the Arizona desert. How much remains after one centur? 5. Find the integrals a. d 4 2. b. 6. Find the derivative of 10 log 2 (). 7. Find the limit lim 0 (1 + 2) 1/ d ln(). 8. A thin plate of constant densit is made in a shape which is bounded b the curves = and = 3 and having 0 1. Find the center of mass. 9. A solid is obtained b revolving about the ais the area bounded b the curve = 4 2 and the ais. Epress the volume and surface area as definite integrals, but do not solve them. 10. The base of a solid is the triangle in the,-plane with vertices at (0,0), (0,1), and (1,0). The cross sections perpendicular to the -ais are squares with one side on the base. Find the volume of the solid. 5

6 Answers 1. a. 1 b. ln(6) c. e d. e. e 1/2 f. 1/2 g a. (ln()) 2 + C b. ln(1 + e ) + C c. 1/2 arctan( 2 ) + C d. 3(1 e 1 ) e. 1/6 arctan( 2 3 ) + C 3. 2 ln()/(+1) ln(+1)/ (ln()) units miles per hour miles per hour. d 7. d (uv ) = u v ( dv d ln(u) + v du u d ) 8. c = ( 1 2 )2/3 9. 1/3 10. (π/27)(10 3/2 1) 11. 1/3 12. S(R, r) = π 2 Rr + 2πr 2 + πr 2. V (R, r) = r 0 π 2 Rr 2 /2 + 2πr 3 /3 + πr 2 r Answers to Final Eam from Dec 86 S(R, t) dt so V (R, r) = 2. a. F b. T c. T d. T e. F f. T g.t h. T i. T k. F j. T l. T m. F n. F o. T p. F q. T r. F c e none of these 3. b c a. 1/2 sin(4 2) + C b / + C c. ln cosθ + C d. 1/2( 1/( 2 + 1) + 1/(2( 2 + 1) 2 ) + C e. 1/2 ln(2) 5. (a) b a π(f())2 d (b) b (d) d c π(f()2 g() 2 ) d 6. radius= 3, height= /3 8. (27/4)π π a 2π(f() g()) d (c) d 2π(f() g()) d c Answers to Final Eam from Ma 89 feet megatons 5. a. ln( ln ) + C b. arcsin(/2) + C log 2 () ln(10) ln(2) 7. e 2 8. ( 8 15, 8 21 ) 9. Volume= /3 2 π(4 2 ) 2 d Surface area= 2 2 2π(4 2 ) d 6