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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

If f ( x ) = x 46 + x 45 + 2 1 + x , calculate f ( 46 ) ( 3 ) . Express your answer using factorial notation: n ! = 1.2.3..... ( n 1 ) . n .

To determine

To express: f463using factorial notation.

Explanation

1) Concept:

After simplifying the given function, we can find the derivative of f(x) after differentiating fx,46  times and using particular value of x.

2) Formula:

i.

ddxfxn=nfxn-1

ii.

dndxnfx+gx=dndxnfx+dndxngx

iii.

nn-1n-2n-33.2.1=n!

3) Given:

fx=x46+x45+21+x

4) Calculations:

f(x) can be rewritten as,

fx= x451+x+21+x

fx=x45+21+x

f(46)x=d46dx46x45+21+x

f46x=d46dx46x45+d46dx4621+x

x45 is a polynomial with degree 45. Since degree is less than the number of derivative, it ends with zero. So first term of in the above expression is zero.Thus

 f46x=0+2d46dx461+x-1

Differentiating 1+x-1 with respect to x,

ddx1+x-1= -1+x-2

d2d

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