286
Chapter 3 | Derivatives
203. The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by y =10+5sin x where y is the number of hamburgers sold and x represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find y ′ and determine the intervals where the number of burgers being sold is increasing. 204. [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by y ( t ) = 0.5 + 0.3cos t , where t is months since January. Find y ′ and use a calculator to determine the intervals where the amount of rain falling is decreasing. For the following exercises, use the quotient rule to derive the given equations. 205. d dx (cot x ) =−csc 2 x 206. d dx (sec x ) = sec x tan x 207. d dx (csc x ) =−csc x cot x 208. Use the definition of derivative and the identity cos( x + h ) =cos x cos h −sin x sin h to prove that d (cos x ) dx =−sin x . For the following exercises, find the requested higher-order derivative for the given functions.
d 3 y dx 3 d 2 y dx 2 d 4 y dx 4 d 2 y dx 2 d 3 y dx 3
of y =3cos x
209.
of y =3sin x + x 2 cos x
210.
of y =5cos x
211.
of y = sec x +cot x
212.
of y = x 10 −sec x
213.
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