Chapter 7.4, Problem 71E

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Investment The value of an investment of $100 earning 10% interest compounded annually is V ( I , R ) = [ 1 + 0.01 ( 1 − R ) 1 + I ] 10 where I is the annual rate of inflation (in decimal form) and R is the tax rate (in decimal form) for the person making the investment. Calculate V I ( 0.03 , 0.28 ) and V R ( 0.03 , 0.28 ) . Determine whether the tax rate or the rate of inflation is the greater “negative” influence on the growth of the investment. To determine To calculate: The value of VI(0.03,0.28) and VR(0.03,0.28) if the value of an investment is given by function V(I,R)=1000[1+0.10(1R)1+I]10. Also, determine whether the tax rate is the greater negative influence on the growth of the investment. Explanation Given Information: The value of an investment of$1000 earning 10% interest compounded annually is given by function V[I,R]=1000[1+0.10(1R)1+I]10.

Formula used:

The power rule of differentiation,

ddx(xn)=nxn1

Where, n is any real number.

The constant rule for differentiation,

ddx(cf(x))=cf(x)

Where, c is a constant.

Calculation:

Consider the function,

V(I,R)=1000[1+0.10(1R)1+I]10

Partially differentiate the above function with respect to I,

VI(I,R)=I(1000[1+0.10(1R)1+I]10)=1000(1+0.10(1R))10I(1+I)10=1000(1+0.10(1R))10×(10)(1+I)11=10000×(1+0.10(1R))10(1+I)11

Now, substitute I=0.03 and R=0.28 in the above equation,

VI(0.03,0.28)=10,000×(1+0.10(10

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