406
Chapter 4 | Applications of Derivatives
For the following exercises, determine a. intervals where f is increasing or decreasing, b. local minima and maxima of f , c. intervals where f is concave up and concave down, and d. the inflection points of f .
239. f ( x ) = 1 4 240. f ( x ) = e x
x + 1 x , x >0
x , x ≠0
For 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. 242. A bike accelerates faster, but a car goes faster. Here f = Bike’s position minus Car’s position. 243. The airplane lands smoothly. Here f is the plane’s altitude. 244. Stock prices are at their peak. Here f is the stock price. 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 1≤ x ≤3 247. f ″( x ) =0 for some 1≤ x ≤3 248. There is no absolute maximum at x =3 249. If f ( x ) has three roots, then it has 1 inflection point. 250. If f ( x ) has one inflection point, then it has three real roots.
224. f ( x ) = x 2 −6 x 225. f ( x ) = x 3 −6 x 2 226. f ( x ) = x 4 −6 x 3 227. f ( x ) = x 11 −6 x 10 228. f ( x ) = x + x 2 − x 3 229. f ( x ) = x 2 + x +1 230. f ( x ) = x 3 + x 4
For the following exercises, determine a. intervals where f is increasing or decreasing, b. local minima and maxima of f , c. intervals where f is concave up and concave down, and d. 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( πx )−cos( πx ) over x = [−1, 1]
232. [T] f ( x ) = x +sin(2 x ) over x = ⎡ ⎣ − π 2 ,
⎤ ⎦
π 2
233. [T] f ( x ) = sin x +tan x over ⎛ ⎝ − π 2 ,
⎞ ⎠
π 2
234. [T] f ( x ) = ( x −2) 2 ( x −4) 2
235. [T] f ( x ) = 1
1− x ,
x ≠1
236. [T] f ( x ) = sin x
x over x = [2 π , 0) ∪ (0, 2 π ]
237. f ( x ) = sin( x ) e x over x = [− π , π ] 238. f ( x ) = ln x x , x >0
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