Free Math Essay Sample

. Differentiate with respect to x

(a)      

F(x) =   ¹/₃ x⁷

F(x) = ¹/₃x ^-7

F^I(x) = -⁷/₃x^-8

f^I(x)= -⁷/₃x⁸

(b)     

If f(x) =  ^g(x)/_h(x)

f^I(x)= (h(x) g^I(x) - g(x)h^I(x))/h(x)²

=(x-5) (12x²-4x³) / (x-5)²

= (12x³-60x²-4x³)/ (x²-10x+25)

= (  8x³-60x²) /(x²-10x+25)

(c) 

F(x) = (2x²-6)³(3x³-3)

F(x) = g(x)h(x)

f^I(x)= g(x) h^I(x) + h(x)g^I(x)

f^I(x) = (2x²-6)²(3x³-3)^I+((2x²-6)³)^I(3x³-3)

= (2x²-6)²(2x²-6)27x²+((2x²-6)³)^I(3x³-3)

=((2x²-6)³)^I=U³=3U²=3(2x²-6)²(4x)

= (2x²-6)²(2x²-6)27x²+12x (2x²-6)(3x³-3)

=(4x⁴-24x²+36)(2x²-6)27x²+12x (4x⁴-24x²+36)(3x²-3)

=(4x⁴-24x²+36)(54x⁴-162x²)+ (48x⁵-288x³+432x)(3x²-3)

=216x⁸-1296x⁸+1944x⁴-648x⁸+3888-5832x² +144x⁷+864x⁵+1296x³-144x⁷+864x⁵-1296x

f^I(x)=-1728x⁸+1720x⁵+1944x⁴+1296x³-5832x²-1296x+3888

    (d)

F(x) = xln(2x)

We know ^dlnU/_dx=U^I/U

Also F(x)=g(x)h(x)

f^I(x)=g(x)h^I(x)+h(x)g^I(x)

= ^2x/_2x+ln (2x)

Answers to question 2

TR=P x Q,                           TR- Total Revenue

TR = 400Q - Q²/50

Expression for marginal revenue is,

MR = 400 - Q/25

MR = 400 - 10000 / 25

MR=0

Price elasticity demand when price=100

P = 400 - Q/50

Q/50 = 400 - P

Q=400 x 50 - 50P,

Therefore Gradient =-50

Price elasticity=gradient x P/Q

=-50 x 100/10000

=-¹/₃ hence inelastic

c) Henry can maximize daily revenue by either increasing the quantity of the commodities. Increase in price may not affect demand for the commodity due to the inelastic demand for the commodity. Increasing the quantity is the only way to maximize profit hence revenue.

Question three answers

a)     A=P(1+R)ⁿ

A=P(1+²⁴/₁₂₀₀)¹²

A=P(1+0.02)¹²

=P(1.02)¹²

=1.2682P

I=PRT

R=I/PT = 0.2682P / P=0.2682

=26.82%

b) A=P(1+R)ⁿ

A=P(1+^R/₄)⁴ⁿ

A= P(1+⁶/₄₀₀)²⁴

A=12000(1.015)²⁴

A=1.4295 x 12000

=$17154.03

C) ⁵/₁₀₀x 1000x 40 = £ 2000

4. Express the first and the second order derivatives of;

(a)

f^I(x,y) = 12x + 5 ^dy/_dx

f^II(x,y) = 12 + d²y/dx²

(b) f(x,y) = 3x⁵y⁴

f^I(x,y)= 15x⁴y⁴+3x⁵4y³dy/dx

f^II(x,y)= 60x³y⁴+15x⁴4y^3dy/_dx+15x⁴y^3dy/_dx+3x⁵12y²d²y/dx²

c) f(x,y)=(10+x²y)³

f^I(x,y)=U³=3U².U^I

=3(10+x²y)² (2xy+ x^2dy/_dx)

Using f(x) =h(x)g(x)

f^I(x,y)= h(x)g^I(x)+h^I(x)g(x),

f^I(x,y) = (10+x²y)²(2y+ 2xdy/dx+2xd²y/dx²)+(2xy+x²dy/dx)

f^I(x,y)=(300+60x²y+3x⁴y²)(2xy+x²dy/dx)

=(600xy+120x³y²+6x⁵y³+300x²dy/dx+60x⁴ydy/dx+3x⁶y²dy/dx)

f^II(x,y)=600y+600x^dy/_dx+360x²y²+120x³2y^dy/_dx+30x⁴y³+6x⁵3y²dy/dx+600x^dy/_dx+240x³…y^dy/_dx+60x⁴d²y/dx²+18x⁵y²dy/dx+3x⁶2yd²y/dx².

 

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