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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.

(b) Use a graph of f″ to give better estimates.

f(x) = sin 2x + sin 4x, 0 ≤ xπ

(a)

To determine

To estimate: The intervals of concavity and the inflection points of the given function.

The function is concave upward on (0.8,1.6),(2.3,π), concave downward on (0,0.8),(1.6,2.3) and the inflection points are (0.8,0.7),(1.6,0) and (2.3,0.7).

Explanation

Given:

The function f(x)=sin2x+sin4x on the interval [0,π].

Estimation:

Use the online graphing calculator to draw the graph for the function f(x) as shown in below Figure1.

From Figure 1, it is observed that the tangent lines for all the points before 0.8 in the x-axis lies above the curve and hence, the function f is concave downward on (0.8,1.6). Similarly, the tangent lines for all the points from 2.3 in the x-axis to π lies above the curve and hence, the function f is concave downward on (2.3,π).

Also from Figure 1, it is observed that the tangent lines for all the points from 0.8 in the x-axis to 1.6 lies below the curve and hence, the function f is concave upward on (0,0.8) and the tangent lines for all the points from 1

(b)

To determine

To estimate: The intervals of concavity and the inflection points from the graph of f.

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