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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

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