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Prove that if f is any cubic polynomial function f(x)=ax3+bx2+cx+dthen the coefficients of f are completely determined by the values of f(x) and its derivative at x=0 as follows

a=f"'(0)6;b=f"(0)2;c=f'(0);d=f(0)

Short Answer

Expert verified

We determine the values of a, b, c, d in terms of f and its derivatives at x=0

Step by step solution

01

Given information

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

02

Find the derivatives of the function

We have,

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

Now compute all the values at x=0

We get,

f(0)=d(1)f'(0)=c(2)f"(0)=2bb=f"(0)2(3)f"'(0)=6aa=f"'(0)6(4)

We determine the values of a, b, c, d in terms of f and its derivatives at x=0

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Most popular questions from this chapter

Every morning Linda takes a thirty-minute jog in Central Park. Suppose her distance s in feet from the oak tree on the north side of the park tminutes after she begins her jog is given by the function s(t)shown that follows at the left, and suppose she jogs on a straight path leading into the park from the oak tree.

(a) What was the average rate of change of Lindaโ€™s distance from the oak tree over the entire thirty-minute jog? What does this mean in real-world terms?

(b) On which ten-minute interval was the average rate of change of Lindaโ€™s distance from the oak tree the greatest: the first 10minutes, the second 10minutes, or the last10minutes?

(c) Use the graph of s(t)to estimate Lindaโ€™s average velocity during the 5-minute interval fromt=5tot=10. What does the sign of this average velocity tell you in real-world terms?

(d) Approximate the times at which Lindaโ€™s (instantaneous) velocity was equal to zero. What is the physical significance of these times?

(e) Approximate the time intervals during Lindaโ€™s jog that her (instantaneous) velocity was negative. What does a negative velocity mean in terms of this physical example?

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=x(4-2x);f(0)=0

For each function fgraphed in Exercises 65-68, determine the values of xat which ffails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

use the definition of derivative to directly prove the differentiation rules for constant and identity function

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f(x)=x3x+1

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