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Chapter 2: Q. 4 (page 200)

Suppose f is ant cubic polynomial function f(x)=ax3+bx2+cx+dprove that coefficients of f a, b, c, d can be expressed in terms of values of f(x) and its derivatives at the point x=2

Short Answer

Expert verified

We give the values of a, ,b, c, d in terms of function and its derivative at x=2

Step by step solution

01

Given information

We are given a cubic functionf(x)=ax3+bx2+cx+d

02

We find all the necessary derivatives

We get,

f(x)=ax3+bx2+cx+df'(x)=3ax2+2bx+cf"(x)=6ax+2bf"'(x)=6a

Compute the derivative at x=2

We get,

f(2)=8a+4b+2c+d(1)f'(2)=12a+4b+c(2)f"(2)=12a+2b(3)f"'(2)=6a(4)

03

Write in terms of the function and its derivative at x=2

We get,

Substitute 4 in 3

f"(2)=2f"'(2)+2bb=f"(2)-2f"'(2)2(5)a=f"'(2)2(6)

Substituting 5 and 6 in 2 we get

role="math" localid="1648269750376" c=2f"''(2)-2f"(2)+f'(2)

Substituting all values in 1 we get\

a=f(2)-2f'(2)+2f'"(2)-86f"'(2)

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Most popular questions from this chapter

Suppose h(t) represents the average height, in feet, of a person who is t years old.

(a) In real-world terms, what does h(12) represent and what are its units? What does h' (12) represent, and what are its units?

(b) Is h(12) positive or negative, and why? Is h'(12) positive or negative, and why?

(c) At approximately what value of t would h(t) have a maximum, and why? At approximately what value of t would h' (t) have a maximum, and why?

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=3x2+18-7x-23

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=2-3x+1

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c)for each function f and value x=c in Exercises 23鈥�38.

23.f(x)=x2,x=-3

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=1x2-1
See all solutions

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