Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the polynomial f(t) of degree 3 such that $f (1) = (1), f (2) = 5, f' (1) = 2,$ and $f' (2) = 9$ , where $f' (t)$ is the derivative of $f (t)$ . Graph this polynomial.

The graphical representation of the cubic polynomial $f' (t) = - 5 + 13 t - 10 t^{2} + 3 t^{2}$ is,

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the points and form the equations

The equation is, $f (t) = a + b t + c t^{2} + d t^{3}$

The derivative of the equation is, $f' (t) = b + 2 c t + 3 d t^{2}$

Substitute the points in the above equation to form the equations.

localid="1659349415086" $(1, 1) : f (1) = a + b (1) + c {(1)}^{2} + d {(1)}^{3} = 1 a + b + c + d = 1 . . . . (1) (2, 5) : f (2) = a + b (2) + c {(2)}^{2} + d {(2)}^{3} = - 1 a + 2 b + 4 c + 8 d = 5 . . . . . . (2) (1, 2) : f' (1) = b (3) + 2 (c) + 3 d {(3)}^{2} = 2 b + 2 c + 3 d = 2 . . . . . . . (3) (2, 9) : f' (2) = b (3) + 2 c {(3)}^{2} + 3 d {(3)}^{2} = 9 b + 4 c + 12 d = 9 . . . . . . . . . . . . . (4)$

Step 2: Consider the matrix

Re-write the equations in terms of a matrix.

$|\begin{matrix} a + b + c + d = 1 \\ a + 2 b + 4 c + 8 d = 5 \\ b + 2 c + 3 d = 2 \\ b + 4 c + 12 d = 9 \end{matrix}|$

The matrix form is,

$|\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 & 5 \\ 0 & 1 & 2 & 3 & 2 \\ 0 & 1 & 4 & 12 & 9 \end{matrix}|$

Step 3: Solve the matrix

Consider the matrix.

$|\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 & 5 \\ 0 & 1 & 2 & 3 & 2 \\ 0 & 1 & 4 & 12 & 9 \end{matrix}|$

Using row Echelon form to reduce the matrix.

$[\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 & 5 \\ 0 & 1 & 2 & 3 & 2 \\ 0 & 1 & 4 & 12 & 9 \end{matrix}] = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 3 & 7 & 4 \\ 0 & 0 & - 1 & - 4 & - 2 \\ 0 & 0 & 0 & 1 & 3 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 & - 5 \\ 0 & 1 & 0 & 0 & 13 \\ 0 & 0 & 1 & 0 & - 10 \\ 0 & 0 & 0 & 1 & 3 \end{matrix}]$

The values are, $a = - 5, b = 13, c = - 10, d = 3$

Therefore, the polynomial is,

$f (t) = a + b t + c t^{2} + d t^{3} = - 5 + 13 t - 10 t^{2} + 3 t^{3}$

Step 4: Sketch the graph

The graphical representation of $f (t) = - 5 + 13 t - 10 t^{2} + 3 t^{3}$ is, ss

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the polynomial f(t) of degree 3 such that $f (1) = (1), f (2) = 5, f' (1) = 2,$ and $f' (2) = 9$ , where $f' (t)$ is the derivative of $f (t)$ . Graph this polynomial.

Step by Step Solution

Step 1: Consider the points and form the equations

Step 2: Consider the matrix

Step 3: Solve the matrix

Step 4: Sketch the graph

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Find the polynomial f(t) of degree 3 such that f(1)=(1),f(2)=5,f'(1)=2,and f'(2)=9 , where f'(t) is the derivative of f(t) . Graph this polynomial.

Step by Step Solution

Step 1: Consider the points and form the equations

Step 2: Consider the matrix

Step 3: Solve the matrix

Step 4: Sketch the graph

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Find the polynomial f(t) of degree 3 such that $f (1) = (1), f (2) = 5, f' (1) = 2,$ and $f' (2) = 9$ , where $f' (t)$ is the derivative of $f (t)$ . Graph this polynomial.