Explanation Provide the steps to graph a polynomial function as follows: Identify where the extrema of the function occur. Calculate the intervals at which the function is increasing and decreasing. Calculate the point at which the graph of the function is concave up and concave down. Sketch the general shape of the graph of the function. Plot some specific points, such as local maximum and minimum points, points of inflection, and intercepts. Then sketch the curve. Example: Consider the function y = x 4 − 4 x 3 + 10 . The domain of the function is ( - ∞ , ∞ ) . Provide the function as shown below. y = x 4 − 4 x 3 + 10 (1) Differentiate equation (1) with respect to x . y ′ = 4 x 3 − 4 × 3 x 2 = 4 x 3 − 12 x 2 = 4 x 2 ( x − 3 ) (2) Differentiate equation (2) with respect to x . y ″ = 4 × 3 x 2 − 12 × 2 x = 12 x 2 − 24 x = 12 x ( x − 2 ) The critical points occurs at a point where y ′ = 0 . Calculate the critical points as shown below. 4 x 2 ( x − 3 ) = 0 4 x 2 = 0 or x − 3 = 0 x = 0 or x = 3 The curve increases on ( 3 , ∞ ) and decreases on ( − ∞ , 0 ) and ( 0 , 3 )

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

4th Edition

ISBN: 9780136880912

Author: Joel Hass, Christopher Heil

Publisher: PEARSON+

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 4, Problem 13GYR

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Provide the steps to graph a polynomial function with an example.

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Chapter 4, Problem 12GYR

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Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

Ch. 4.1 - In Exercises 16, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 3E Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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