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All Textbook Solutions for Calculus Volume 1
For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. 224. f(x)=x26xFor the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. 225. f(x)=x36x2For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. 226. f(x)=x46x3For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. 227. f(x)=x116x10For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. 228. f(x)=x+x2x3For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. 229. f(x)=x2+x+1For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. 230. f(x)=x3+x4For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 231. [T] f(x)=sin(x)cos(x) over x = [-1, 1]For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 232. [T] f(x)=x+sin(2x) over x=[2,2]For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 233. [T] f(x)=sinx+tanx over [2,2]For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 234. [T] f(x)=(x2)2(x4)2For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 235. [T] f(x)=11x,x1For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 236. [T] f(x)=sinxx over x=[2,2][2,0)(0,2]For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 237. f(x)=sin(x)ex over x=[,]For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 238. f(x)=Inxx,x0For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 239. f(x)=14x+1x,x0For the following exercises, determine intervals where f is increasing or decreasing, local minima and maxima of f, intervals where f is concave up and concave down, and the inflection points of f. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. 240. f(x)=exx,x0For the following exercises, interpret the sentences in terms of f, f’, and f”. 241. The population is growing more slowly. Here f is the population.For the following exercises, interpret the sentences in terms of f, f’, and f”. 242. A bike accelerates faster, but a car goes faster. Here f = Bike’s position minus Car’s position.For the following exercises, interpret the sentences in terms of f, f’ and f”. 243. The airplane lands smoothly. Here f is the plane’s altitude.For the following exercises, interpret the sentences in terms of f, f’, and f”. 244. Stock prices are at their peak. Here f is the stock price.For the following exercises, interpret the sentences in terms of f, f’, and f”. 245. The economy is picking up speed. Here f is a measure of the economy, such as GDP.For the following exercises, consider a third-degree polynomial f(x), which has the properties f(1)=0,f(3)=0 . Determine whether the following statements are true or false. Justify your answer. 246. f(x)=0 for some 1x3For the following exercises, consider a third-degree polynomial f(x), which has the properties f(1)=0,f(3)=0 . Determine whether the following statements are true or false. Justify your answer. 247. f(x)=0 for some 1x3For the following exercises, consider a third-degree polynomial f(x), which has the properties f(1)=0,f(3)=0 . Determine whether the following statements are true or false. Justify your answer. 248. There is no absolute maximum at x = 3For the following exercises, consider a third-degree polynomial f(x), which has the properties f(1)=0,f(3)=0 . Determine whether the following statements are true or false. Justify your answer. 249. If f(x) has three roots, then it has 1 inflection point.For the following exercises, consider a third-degree polynomial f(x), which has the properties f(1)=0,f(3)=0 . Determine whether the following statements are true or false. Justify your answer. 250. If f(x) has one inflection point, then it has three real roots.For the following exercises, examine the graphs. Identify where the vertical asymptotes are located. 251.For the following exercises, examine the graphs. Identify where the vertical asymptotes are located. 252.For the following exercises, examine the graphs. Identify where the vertical asymptotes are located. 253.For the following exercises, examine the graphs. Identify where the vertical asymptotes are located. 254.For the following exercises, examine the graphs. Identify where the vertical asymptotes are located. 255.For the following functions f(x), determine whether there is an asymptote at x = a. Justify your answer without graphing on a calculator. 256. f(x)=x+1x2+5x+4,a=1For the following functions f(x), determine whether there is an asymptote at x = a. Justify your answer without graphing on a calculator. 257. f(x)=xx2,a=2For the following functions f(x), determine whether there is an asymptote at x = a. Justify your answer without graphing on a calculator. 258. f(x)=(x+2)3/2,a=2For the following functions f(x), determine whether there is an asymptote at x = a. Justify your answer without graphing on a calculator. 259. f(x)=(x1)1/3,a=1For the following functions f(x), determine whether there is an asymptote at x = a. Justify your answer without graphing on a calculator. 260. f(x)=1+x2/5,a=1For the following exercises, evaluate the limit. 261. limx13x+6For the following exercises, evaluate the limit. 262. limx2x54xFor the following exercises, evaluate the limit. 263. limxx22x+5x+2For the following exercises, evaluate the limit. 264. limx3x32xx2+2x+8For the following exercises, evaluate the limit. 265. limxx44x3+122x27x4For the following exercises, evaluate the limit. 266. limx3xx2+1For the following exercises, evaluate the limit. 267. limx4x21x+2For the following exercises, evaluate the limit. 268. limx4xx21For the following exercises, evaluate the limit. 269. limx4xx21]For the following exercises, evaluate the limit. 270. limx2xxx+1For the following exercises, find the horizontal and vertical asymptotes. 271. f(x)=x9xFor the following exercises, find the horizontal and vertical asymptotes. 272. f(x)=11x2For the following exercises, find the horizontal and vertical asymptotes. 273. f(x)=x34x2For the following exercises, find the horizontal and vertical asymptotes. 274. f(x)=x2+3x2+1For the following exercises, find the horizontal and vertical asymptotes. 275. f(x)=sin(x)sin(2x)For the following exercises, find the horizontal and vertical asymptotes. 276. f(x)=cosx+cos(3x)+cos(5x)For the following exercises, find the horizontal and vertical asymptotes. 277. f(x)=xsin(x)x21For the following exercises, find the horizontal and vertical asymptotes. 278. f(x)=xsin(x)For the following exercises, find the horizontal and vertical asymptotes. 279. f(x)=1x3+x2For the following exercises, find the horizontal and vertical asymptotes. 280. f(x)=1x12xFor the following exercises, find the horizontal and vertical asymptotes. 281. f(x)=x3+1x31For the following exercises, find the horizontal and vertical asymptotes. 282. f(x)=sinx+cosxsinxcosxFor the following exercises, find the horizontal and vertical asymptotes. 283. f(x)=xsinxFor the following exercises, find the horizontal and vertical asymptotes. 284. f(x)=1xxFor the following exercises, find the horizontal and vertical asymptotes. 285. x=1andy=2For the following exercises, find the horizontal and vertical asymptotes. 286. x=1andy=0For the following exercises, find the horizontal and vertical asymptotes. 287. y=4andx=1For the following exercises, find the horizontal and vertical asymptotes. 288. x=0 .For the following exercises, graph the function on a graphing calculator on the window x=[5,5] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. 289.[T] f(x)=1x+10For the following exercises, graph the function on a graphing calculator on the window x=[5,5] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. 290. [T] f(x)=x+1x2+7x+6For the following exercises, graph the function on a graphing calculator on the window x=[5,5] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. 291. [T] limxx2+10x+25For the following exercises, graph the function on a graphing calculator on the window x=[5,5] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. 292. limxx+2x2+7x+6For the following exercises, graph the function on a graphing calculator on the window x=[5,5] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. 293. [T] limx3x+2x+5For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 294. y=3x2+2x+4For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 295. y=x33x2+4For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 296. y=2x+1x2+6x+5For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 297. y=x3+4x2+3x3x+9For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 298. y=x2+x2x23x4For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 299. y=x25x+4 .For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 300. y=2x16x2For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 301. y=cosxx,onx=[2,2]For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 302. y=exx3For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 303. y=xtanx,x[,]For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 304. y=xIn(x),x0For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior. 305. y=x2sin(x),x[2,2]For f(x)=P(x)Q(x) to have an asymptote at y = 2 then the poynomials P(x) and Q(x) must have what relation?For f(x)=P(x)Q(x) to have an asymptote at x = 0 then the poynomials P(x) and Q(x). must have what relation?If f' (x) has asymptotes at y = 3 and x = 1, then f(x) has what asymptotes?Both f(x)1x1andg(x)=1( x1)2 have asymptotes at x = 1 and y = 0. What is the most obvious difference between these two functions?True or false: Every ratio of polynomials has vertical asymptotes.For the following exercises, answer by proof, counterexample, or explanation. 311. When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points?For the following exercises, answer by proof, counterexample, or explanation. 312. Why do you need to check the endpoints for optimization problems?For the following exercises, answer by proof, counterexample, or explanation. 313. True or False. For every continuous nonlinear function, you can find the value x that maximizes the function.For the following exercises, answer by proof, counterexample, or explanation. 314. True or False. For every continuous nonconstant function on a closed, finite domain, there exists at least one x that minimizes or maximizes the function.For the following exercises, set up and evaluate each optimization problem. 315. To carry a suitcase on an airplane, the length +width + height of the box must be less than or equal to 62 in. Assuming the height is fixed, show that the maximum volume is V=h(31( 1 2 )h)2 . What height allows you to have the largest volume?For the following exercises, set up and evaluate each optimization problem. 316. You are constructing a cardboard box with the dimensions 2 m by 4 m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions of the box with the largest volume?For the following exercises, set up and evaluate each optimization problem. 317. Find the positive integer that minimizes the sum of the number and its reciprocal.For the following exercises, set up and evaluate each optimization problem. 318. Find two positive integers such that their sum is 10, and minimize and maximize the sum of their squares.For the following exercises, consider the construction of a pen to enclose an area. 319. You have 400 ft of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?For the following exercises, consider the construction of a pen to enclose an area. 320. You have 800 ft of fencing to make a pen for hogs. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area?For the following exercises, consider the construction of a pen to enclose an area. 321. You need to construct a fence around an area of 1600 ft. What are the dimensions of the rectangular pen to minimize the amount of material needed?Two poles are connected by a wire that is also connected to the ground. The first pole is 20 ft tall and the second pole is 10 ft tall. There is a distance of 30 ft between the two poles. Where should the wire be anchored to the ground to minimize the amount of wire needed?You are moving into a new apartment and notice there is a comer where the hallway narrows from 8 ft to 6 ft. What is the length of the longest item that can be earned horizontally around the corner?
A patient’s pulse measures 70 bpm, 80 bpm. then 120 bpm. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression (x70)2+(x80)2+(x120)2 ? What value minimizes it?In the previous problem, assume the patient was nervous during the third measurement, so we only weight that value half as much as the others. What is the value that minimizes (x70)2+(x80)2+12(x120)2 ?You can inn at a speed of 6 mph and swim at a speed of 3 mph and are located on the shore, 4 miles east of an island that is 1 mile north of the shoreline. How far should you run west to minimize the time needed to reach the island? For the following problems, consider a lifeguard at a circular pool with diameter 40 m. He must reach someone who is drowning on the exact opposite side of the pool, at position C. The lifeguard swims with a speed v and inns around the pool at speed w = 3v.Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, .Find at what angle the lifeguard should swim to reach the drowning person in the least amount of time.A truck uses gas as g(v)=av+bv , where v represents the speed of the truck and g represents the gallons of fuel per mile. At what speed is fuel consumption minimized? For the following exercises, consider a limousine that get m(v)=(1202v)5mi/gal at speed v, the chauffeur costs $ 15/h, and gas is $3.5/gal.Find die cost per mile at speed v.Find the cheapest driving speed. pizzas for a revenue of R(x)=ax and costs C(x)=b+cx+dx2 , where x represents the number of pizzas.Find the profit function for the number of pizzas How many pizzas gives the largest profit per pizza?Assume that R(x)=10x and C(x)=2x+x2 .How many pizzas sold maximizes the profit?Assume that R(x)=15x , and C(x)=60+3x+12x2 . How many pizzas sold maximizes the profit?For the following exercises, consider a wire 4 ft long cut into two pieces. One piece forms a circle with radius r and the other forms a square of side x. 335. Choose x to maximize the sum of their areas.For the following exercises, consider a wire 4 ft long cur into two pieces. One piece forms a circle with radius r and the other forms a square of side x. 336. Choose x to minimize the sum of their areas.For the following exercises, consider two nonnegative numbers x and y such that x+y=10 . Maximize and minimize the quantities. 337. xyFor the following exercises, consider two nonnegative numbers x and y such that x+y=10 . Maximize and minimize the quantities. 338. x2y2For the following exercises, consider two nonnegative numbers x and y such that x+y=10 . Maximize and minimize the quantities. 339. y1xFor the following exercises, consider two nonnegative numbers x and y such that x+y=10 . Maximize and minimize the quantities. 340. x2yFor the following exercises, draw the given optimization problem and solve. 341. Find the volume of the largest light circular cylinder that fits in a sphere of radius 1.For the following exercises, draw the given optimization problem and solve. 342. Find the volume of the largest right cone that fits in a sphere of radius 1.For the following exercises, draw the given optimization problem and solve. 343. Find the area of the largest rectangle that fits into the triangle with sides x=0,y=0 and x4+y6=1 .For the following exercises, draw the given optimization problem and solve. 344. Find the largest volume of a cylinder that fits into a cone that has base radius R and height h.For the following exercises, draw the given optimization problem and solve. 345. Find the dimensions of the closed cylinder volume V=16 that has the least amount of surface area.For the following exercises, draw the given optimization problem and solve. 346. Find the dimensions of a light cone with surface area S=4 that has the largest volume.For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions. 347. [T] Where is the line y=52x closest to the origin?For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions. 348. [T] Where is the line y=52x closest to point (1, 1)?For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions. 349. [T] Where is the parabola y=x2 closest to point (2, 0)?For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions. 350. [T] Where is the parabola y=x2 closest to point (0, 3)?A window is composed of a semicircle placed on top of a rectangle. If you have 20 ft of window-framing materials for die outer frame, what is the maximum size of the window you can create? Use r to represent the radius of the semicircle.You have a garden row of 20 watermelon plants that produce an average of 30 watermelons apiece. For any additional watermelon plants planted, the output per watermelon plant drops by one watermelon. How many extra watermelon plants should you plant?You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $5/ft2 and the material for the sides costs $2/ft2. You need a box with volume 4 ft2 . Find the dimensions of the box that minimize cost. Use x to represent the length of the side of the box.You are building five identical pens adjacent to each other with a total area of 1000 m2, as shown in the following figure. What dimensions should you use to minimize tine amount of fencing?You are the manager of an apartment complex with 50 units. When you set rent at $800/month, all apartments are rented. As you increase rent by $25/month, one fewer apartment is rented. Maintenance costs run $50/month for each occupied unit. What is the rent that maximizes the total amount of profit?For the following exercises, evaluate the limit. 356. Evaluate the limit limxexxFor the following exercises, evaluate the limit. 357. Evaluate the limit limxexxkFor the following exercises, evaluate the limit. 358. Evaluate the limit limxInxxkFor the following exercises, evaluate the limit. 359. Evaluate the limit limxaxax2a2 , a0For the following exercises, evaluate the limit. 360. Evaluate the limit limxaxax3a3,a0 .For the following exercises, evaluate the limit 361. Evaluate the limit limxaxaxnan,a0 .For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule. 362. limx0+x2InxFor the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule. 363. limxx1/xFor the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule. 364. limx0x2/xFor the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L'Hôpital’s rule. 365. limx0x21/xFor the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L'Hôpital’s rule. 366. limxexxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 367. limx3x29x3For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 368. limx3x29x+3For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 369. limx0( 1+x)21xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 370. limx/2cosx2xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 371. limxxsinxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 372. limx1x1sinxFor the following exercises, evaluate the limits with either L‘Hôpital’s rule or previously learned methods. 373. limx0( 1+x)n1xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 374. limx0( 1+x)n1nxx2For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 375. limx0sinxtanxx3For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 376. limx01+x1xxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 377. limx0exx1x2For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 378. limx0tanxxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 379. limx1x1InxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 380. limx0(x+1)1/xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 381. limx1xx3x1For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 382. limx0+x2xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 383. limxxsin(1x)For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 384. limx0sinxxx2For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 385. limx0+xIn(x4)For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 386. limx(xex)For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 387. limxx2exFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 388. limx03x2xxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 389. limx01+1/x11/xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 390. limx/4(1tanx)cotxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 391. limxxe1/xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 392. limx0x1/cosxFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 393. limx0x1/xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 394. limx0(11x)xFor the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods. 395. limx(11x)xFor the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 396. [T] limx0ex1xFor the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 397. [T] limx0xsin(1x)For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 398. [T] limx1x11cos(x)For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 399. [T] limx1e( x1)11xFor the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 400. [T] limx1( x1)2InxFor the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 401. [T] limx1+cosxsinxFor the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 402. [T] limx0(cscx1x)For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 403. [T] limx0+tan(xx)For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 404. [T] limx0+InxsinxFor the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s title to find the limit directly. 405. limx0exexxLet r = 0.5 and choose x0= 0.2. Either by hand or by using a computer, calculate the first 10 values in the sequence. Does the sequence appear to converge? If so, to what value? Does it result in a cycle? If so, what kind of cycle (for example, 2 - cycle, 4 - cycle.)?What happens when r =2?For r = 3.2 and r = 3.5, calculate the first 100 sequence values. Generate a cobweb diagram for each iterative process. (Several free applets are available online that generate cobweb diagrams for the logistic map.) What is the long-term behavior in each of these cases?Now let r = 4. Calculate the first 100 sequence values and generate a cobweb diagram. What is the long-term behavior in this case?Repeat the process for r = 4, but let x0=0.201 . How does this behavior compare with the behavior for x0=0.2 ?For the following exercises, write Newton’s formula as xn+1=F(xn) or solving f(x) = 0. 406. f(x)=x2+1For the following exercises, write Newton’s formula as xn+1=F(xn) or solving f(x) = 0. 407. f(x)=x3+2x+1For the following exercises, write Newton’s formula as xn+1=F(xn) or solving f(x) = 0. 408. f(x)=sinxFor the following exercises, write Newton’s formula as xn+1=F(xn) or solving f(x) = 0. 409. f(x)=exFor the following exercises, write Newton’s formula as xn+1=F(xn) or solving f(x) = 0. 410. f(x)=x3+3xexFor the following exercises, solve f(x) = 0 using the iteration xn+1=xncf(xn), which differs slightly from Newton’s method. Find a c that works and a c that fails to converge, with the exception of c = 0. 411. f(x)=x24 , with x0=0For the following exercises, solve f(x) = 0 using the iteration xn+1=xncf(xn), which differs slightly from Newton’s method. Find a c that works and a c that fails to converge, with the exception of c = 0. 412. f(x)=x24x+3 , with x0=2For the following exercises, solve f(x) = 0 using the iteration xn+1=xncf(xn), which differs slightly from Newton’s method. Find a c that works and a c that fails to converge, with the exception of c = 0. 413. What is the value of “c” for Newton’s method?For the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 414. xn+1=xn212For the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 415. xn+1=2xn(1xn)For the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 416. xn+1=xnFor the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 417. xn+1=1xnFor the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 418. xn+1=3xn(1xn)For the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 419. xn+1=xn2+xn2For the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 420.xn+1=12xn1For the following exercises, start at x0=0.6and x0=2 . Compute x1 and x2 using the specified iterative method 421. xn+1=|xn|For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 422. x210=0For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 423. x4100=0For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 424. x2x=0For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 425. x3x=0For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 426. x+5cos(x)=0For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 427. x+tan(x)=0, choose x0(2,2)For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 428. 11x=2For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 429. 1+x+x2+x3+x4=2For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 430. x3+(x+1)3=103For the following exercises, solve to four decimal places using Newton’s method and a computer or calculator. Choose any initial guess x0 that is not the exact root. 431. x=sin2(x)For the following exercises, use Newton’s method to find the fixed points of the function where f(x)=x; round to three decimals. 432. sinxFor the following exercises, use Newton’s method to find the fixed points of the function where f(x)=x; round to three decimals. 433. tan(x)onx=(2,32)For the following exercises, use Newton’s method to find the fixed points of the function where f(x)=x; round to three decimals. 434. ex2For the following exercises, use Newton’s method to find the fixed points of the function where f(x)=x; round to three decimals. 435. In(x)+2Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 436. To find candidates for maxima and minima, we need to find the critical points f(x)=0 . Show that to solve for the critical points of a function f(x) , Newton's method is given by xn+1=xnf(xn)f(xn)Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 437. What additional restrictions are necessary on the function f ? For the following exercises, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 438. Minimum of f(x)=x2+2x+4Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 439. Minimum of f(x)=3x3+2x216Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 440. Minimum of f(x)=x2exNewton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 441. Maximum of f(x)=x+1xNewton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 442. Maximum of f(x)=x3+10x2+15x2Newton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 443. Maximum of f(x)=xx3xNewton’s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton's method to the derivative function f(x) to find its roots, instead of the original function. For the following exercises, consider the formulation of the method. 444. Minimum of f(x)=x2sinx , closest non-zero minimum to x = 0Minimum of f(x)=x4+x3+3x2+12x+6For the following exercises, use the specified method to solve the equation. If it does not work, explain why it does not work. 446. Newton’s method, x2+2=0For the following exercises, use the specified method to solve the equation. If it does not work, explain why it does not work. 447. Newton’s method. 0=exFor the following exercises, use the specified method to solve the equation. If it does not work, explain why it does not work. 448. Newton’s method. 0=1+x2 starting at x0=0For the following exercises, use the specified method to solve the equation. If it does not work, explain why it does not work. 449. Solving xn+1=xn3 starting at x0=1For the following exercises, use the secant method, an alternative iterative method to Newton’s method. The formula is given by xn=xn1f(xn1)xn1xn2(x n1)f(xn2) 450. Find a root to 0=x2x3 accurate to three decimal places.For the following exercises, use the secant method, an alternative iterative method to Newton’s method. The formula is given by xn=xn1f(xn1)xn1xn2(x n1)f(xn2) 451. Find a root to 0=sinx+3x accurate to four decimal places.For the following exercises, use the secant method, an alternative iterative method to Newton’s method. The formula is given by xn=xn1f(xn1)xn1xn2(x n1)f(xn2) 452. Find a root to 0=ex2 accurate to four decimal places.For the following exercises, use the secant method, an alternative iterative method to Newton’s method. The formula is given by xn=xn1f(xn1)xn1xn2(x n1)f(xn2) 453. Find a root to In(x+2)=12 accurate to four decimal places.Why would you use the secant method over Newton's method? What are the necessary restrictions on f ?For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method. use the first guess from Newton's method. 455. f(x)=x2+2x+1,x0=1For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method. use the first guess from Newton's method. 456. f(x)=x2,x0=1For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method. use the first guess from Newton's method. 457. f(x)=sinx,x0=1For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method. use the first guess from Newton's method. 458. f(x)=ex1,x0=2For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method. use the first guess from Newton's method. 459. f(x)=x3+2x+4,x0=0In the following exercises, consider Kepler's equation regarding planetary orbits, M=Esin(E) , where M is the mean anomaly, E is eccentric anomaly, and measures eccentricity. 460. Use Newton’s method to solve for the eccentric anomaly E when the mean anomaly M=3 and the eccentricity of the orbit = 0.25; round to three decimals.In the following exercises, consider Kepler's equation regarding planetary orbits, M=Esin(E) , where M is the mean anomaly, E is eccentric anomaly, and measures eccentricity. 461. Use Newton’s method to solve for the eccentric anomaly E when the mean anomaly M=32 and the eccentricity of the orbit =0.8 ; round to three decimals.The following two exercises consider a bank investment. The initial investment is $10,000. After 25 years, the investment has tripled to $30,000. 462. Use Newton's method to determine the interest rate if the interest was compounded annually.The fallowing two exercises consider a bank investment. The initial investment is $10,000. After 25 years, the investment has tripled to $30,000. 463. Use Newton's method to determine the interest rate if the interest was compounded continuously.The cost for printing a book can be given by the equation C(x)=1000+12x+(12)x2/3 .Use Newton's method to find the break-even point if the printer sells each book for $20.For the following exercises, show that F(x) are antiderivatives of f(x). 465. F(x)=5x3+2x2+3x+1,f(x)=15x2+4x+3For the following exercises, show that F(x) are antiderivatives of f(x). 466. F(x)=x2+4x+1,f(x)=2x+4For the following exercises, show that F(x) are antiderivatives of f(x) . 467. F(x)=x2ex,f(x)=ex(x2+2x)For the following exercises, show that F(x) are antiderivatives of f(x) . 468. F(x)=cosx,f(x)=sinxFor the following exercises, show that F(x) are antiderivatives of f(x) . 469. F(x)=ex,f(x)=exFor the following exercises, find the antiderivative of the function. 470. f(x)=1x2+xFor the following exercises, find the antiderivative of the function. 471. f(x)=ex3x2+sinxFor the following exercises, find the antiderivative of the function. 472. f(x)=ex+3xx2For the following exercises, find the antiderivative of the function. 473. f(x)=x1+4sin(2x)For die following exercises, find the antiderivative F(x) of each function f(x). 474. f(x)=5x4+4x5For the following exercises, find the antiderivative F(x) of each function f(x). 475. f(x)=x+12x2For the following exercises, find the antiderivative F(x) of each function f(x). 476. f(x)=1xFor the following exercises, find the antiderivative F(x) of each function f(x). 477. f(x)=(3)3For the following exercises, find the antiderivative F(x) of each function f(x). 478. f(x)=x1/3+(2x)1/3For the following exercises, find the antiderivative F(x) of each function f(x). 479. f(x)=x1/3x2/3For the following exercises, find the antiderivative F(x) of each function f(x). 480. f(x)=2sin(x)+sin(2x)For the following exercises, find the antiderivative F(x) of each function f(x). 481. f(x)=sec2(x)+1For the following exercises, find the antiderivative F(x) of each function f(x). 482. f(x)=sinxcosxFor the following exercises, find the antiderivative F(x) of each function f(x). 483. f(x)=sin2(x)cos(x)For the following exercises, find the antiderivative F(x) of each function f(x). 484. f(x)=0For the following exercises, find the antiderivative F(x) of each function f(x). 485. f(x)=12csc2(x)+1x2For the following exercises, find the antiderivative F(x) of each function f(x). 486. f(x)=cscxcotx+3xFor the following exercises, find the antiderivative F(x) of each function f(x). 487. f(x)=4cscxcotxsecxtanxFor the following exercises, find the antiderivative F(x) of each function f(x). 488. f(x)=8secx(secx4tanx)For the following exercises, find the antiderivative F(x) of each function f(x). 489. f(x)=12e4x+sinxFor the following exercises, evaluate the integral. 490. (1)dxFor the following exercises, evaluate the integral. 491. sinxdxFor the following exercises, evaluate the integral 492. (4x+ x)dxFor the following exercises, evaluate the integral. 493. 3 x 2+2 x 2dxFor the following exercises, evaluate the integral. 494. (secxtanx+4x)dxFor the following exercises, evaluate the integral. 495. (4 x+ x 4)dxFor the following exercises, evaluate the integral. 496. ( x 1/3 x 2/3 )dxFor the following exercises, evaluate the integral. 497. 14 x 3+2x+1 x 3dxFor the following exercises, evaluate the integral. 498. ( e x+ e x)dxFor the following exercises, solve the initial value problem. 499. f(x)=x3,f(1)=1For the following exercises, solve the initial value problem 500. f(x)=x+x2,f(0)=2For the following exercises, solve the initial value problem. 501. f(x)=cosx+sec2(x),f(4)=2+22For the following exercises, solve the initial value problem. 502. f(x)=x38x2+16x+1,f(0)=0For the following exercises, solve the initial value problem. 503. f(x)=2x2x22,f(1)=0For the following exercises, find two possible functions f given the second-or third-order derivatives. 504. f(x)=x2+2For the following exercises, find two possible functions f given the second-or third-order derivatives. 505. f(x)=exFor the following exercises, find two possible functions f given the second-or third-order derivatives. 506. f(x)=1+xFor the following exercises, find two possible functions f given the second-or third-order derivatives. 507. f(x)=cosxFor the following exercises, find two possible functions f given the second-or third-order derivatives. 508. f(x)=8e2xsinxA car is being driven at a rate of 40 mph when the brakes are applied. The car decelerates at a constant rate of 10 ft/sec2. How long before the car stops?In the preceding problem, calculate how far the car travels in the time it takes to stop.You are merging onto the freeway, accelerating at a constant rate of 12 ft/sec2. How long does it take you to reach merging speed at 60 mph?Based on the previous problem, how far does the car travel to reach merging speed?A car company wants to ensure its newest model can stop in 8 sec when traveling at 75 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.A car company wants to ensure its newest model can stop in less than 450 ft when traveling at 60 mph. If we assume constant deceleration, find the value of deceleration that accomplishes this.For the following exercises, find the antiderivative of the function, assuming F(0)=0 . 515. [T] f(x)=x2+2For the following exercises, find the antiderivative of the function, assuming F(0)=0 . 516. [T] f(x)=4xxFor the following exercises, find the antiderivative of the function, assuming F(0)=0 . 517. [T] f(x)=sinx+2xFor the following exercises, find the antiderivative of the function, assuming F(0)=0 . 518.[T] f(x)=ex