Chapter 4 | Applications of Derivatives
437
2 x −5 4 x
281. f ( x ) = x 3 +1 x 3 −1 282. f ( x ) = sin x +cos x sin x −cos x 283. f ( x ) = x −sin x
lim x →∞
262.
x 2 −2 x +5 x +2
lim x
263.
→∞
3 x 3 −2 x x 2 +2 x +8
lim x
264.
→−∞
284. f ( x ) = 1 x − x For the following exercises, construct a function f ( x ) that has the given asymptotes. 285. x =1 and y =2 286. x =1 and y =0 287. y =4, x =−1 288. 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 ) = 1 x +10
x 4 −4 x 3 +1 2−2 x 2 −7 x 4
lim x
265.
→−∞
3 x x 2 +1
lim x
266.
→∞
4 x 2 −1 x +2
lim x
267.
→−∞
4 x x 2 −1
lim x
268.
→∞
4 x x 2 −1
lim x
269.
→−∞
270. 2 x x − x +1 For the following exercises, find the horizontal and vertical asymptotes. 271. f ( x ) = x − 9 x lim x →∞
x +1 x 2 +7 x +6
290. [T] f ( x ) =
x 2 +10 x +25
291. [T] lim x
272. f ( x ) = 1
→−∞
1− x 2
x +2 x 2 +7 x +6
292. [T] lim x
273. f ( x ) = x 3
→−∞
4− x 2
3 x +2 x +5
293. [T] lim x →∞
274. f ( x ) = x 2 +3 x 2 +1 275. f ( x ) = sin( x )sin(2 x ) 276. f ( x ) =cos x +cos(3 x )+cos(5 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.
294. y =3 x 2 +2 x +4 295. y = x 3 −3 x 2 +4 296. y = 2 x +1 x 2 +6 x +5 297. y = x 3 +4 x 2 +3 x 3 x +9
x sin( x ) x 2 −1
277. f ( x ) =
278. f ( x ) = x
sin( x )
279. f ( x ) = 1
x 3 + x 2
280. f ( x ) = 1
x
x −1 −2
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