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Advanced Mathematics II Homework 1 & 2 (2011)
Homework 1
Sect11-1 Q22 A family paid $40,000 cash for a house. Fifteen years later, they sold the house for $100,000. If interest is compounded continuously, what annual nominal rate of interest did the original $40,000 investment earn?
Solution. Using the continuous compound interest formula we have
100000 = 40000e15r ln(2.5) = 15r r = 0.061 where we have put t = 15, P = 40000 and A = 100000. Thus, the annual nominal rate should be 6.1% of interest for the investment.
Sec11-2 Q14 Find the derivative of y = 5e−x − 6ex .
Solution.
dy
= −5e−x − 6ex . dx Sec11-2 Q32 Find the derivative of f (x) =
x+1 ex and simplify.
Solution.
d d dy ex dx (x + 1) − (x + 1) dx ex*…show more content…*

dt t The rate of learning after 10 and 100 hours of instruction and practice are N (t ) = 6 = 0.6, and 10 6 N (100) = = 0.06 100 words per minute typed per hour, respectively. N (10) = Sec11-4 Q12 Find dy/dw, dw/du, du/dx, and dy/dx of y = ew ; w = terms of x. Solution. Since dy dew = = ew , dw d√ w 1 dw d u = = √, du du 2u du d ln x 1 = =. dx dx x 3 √ u; u = ln x. Express dy/dx in By using chain rule, we have dy dw du dy 11 = = ew √ . dx dw du dx 2 ux √ u Since ew = e √ ln x , =e thus √ √ dy 11 e ln x =e u √ =√. dx 2 ln x x 2x ln x Sec11-4 Q20 Find the relative rate of change of f (x) = 15x + 2x ln x. Solution. The rate of change of f (x) with respect to x is f (x) = 15 + 2 ln x + 2 = 17 + 2 ln x. The relative rate of change of the function f (x) is simply the fraction f (x) 17 + 2 ln x = . f (x) 15x + 2x ln x Sec11-4 Q26 Given the price-demand equation p + 0.01x = 50. (A) Express the demand x as a function of price p. (B) Find the elasticity of demand, E ( p). (C) What is the elasticity of demand when p = $10? If this price is decreased by 5%, what is the approximate change in demand? (D) What is the elasticity of demand when p = $45? If this price is decreased by 5%, what is the approximate change in demand? (E) What is the elasticity of demand when p = $25? If this price is decreased by 5%, what is the approximate

dt t The rate of learning after 10 and 100 hours of instruction and practice are N (t ) = 6 = 0.6, and 10 6 N (100) = = 0.06 100 words per minute typed per hour, respectively. N (10) = Sec11-4 Q12 Find dy/dw, dw/du, du/dx, and dy/dx of y = ew ; w = terms of x. Solution. Since dy dew = = ew , dw d√ w 1 dw d u = = √, du du 2u du d ln x 1 = =. dx dx x 3 √ u; u = ln x. Express dy/dx in By using chain rule, we have dy dw du dy 11 = = ew √ . dx dw du dx 2 ux √ u Since ew = e √ ln x , =e thus √ √ dy 11 e ln x =e u √ =√. dx 2 ln x x 2x ln x Sec11-4 Q20 Find the relative rate of change of f (x) = 15x + 2x ln x. Solution. The rate of change of f (x) with respect to x is f (x) = 15 + 2 ln x + 2 = 17 + 2 ln x. The relative rate of change of the function f (x) is simply the fraction f (x) 17 + 2 ln x = . f (x) 15x + 2x ln x Sec11-4 Q26 Given the price-demand equation p + 0.01x = 50. (A) Express the demand x as a function of price p. (B) Find the elasticity of demand, E ( p). (C) What is the elasticity of demand when p = $10? If this price is decreased by 5%, what is the approximate change in demand? (D) What is the elasticity of demand when p = $45? If this price is decreased by 5%, what is the approximate change in demand? (E) What is the elasticity of demand when p = $25? If this price is decreased by 5%, what is the approximate

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