Sketch the Graph Of $f ( x ) = x ^ { 4 } + x - 3$

question 128

Essay

Sketch the graph of $f ( x ) = x ^ { 4 } + x - 3$ on the interval $[ - 3,3 ]$ . Suppose that Newton's method is used to approximate the positive root of f with initial approximation $x _ { 1 } = 2$ .(a) On your sketch, draw the tangent lines that you would use to find $x _ { 2 }$ and $x _ { 3 }$ , and estimate the numerical values of $x _ { 2 }$ and $x _ { 3 }$ .(b) To approximate the negative root of f, use $x _ { 1 } = - 2$ as the starting approximation. As before, draw the tangent lines that you would use to find $x _ { 2 }$ and $x _ { 3 }$ , and estimate the numerical values of $x _ { 2 }$ and $x _ { 3 }$ .(c) Suppose that you had used $x _ { 1 } = 0$ as the starting point for approximating the negative root. Discuss what would happen.

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Definitions:

Nondisplaced Fracture

A break in a bone that has not led to a misalignment of the bone fragments; the broken pieces remain in place.

Radius

One of the two large bones in the forearm, extending from the elbow to the wrist on the side of the thumb.

Ulna

One of the two long bones in the forearm, running from the elbow to the smallest finger, and located on the side opposite the thumb.

Bone Remodeling

The continuous process by which old bone tissue is removed and new bone is created, essential for bone health and calcium homeostasis.