Explanation Given: The function is f ( θ ) = θ + sin θ , [ 0 , 2 π ] . Formula used: Test for concavity: Let f ″ ( x ) exist for all x ∈ ( a , b ) , then · y = f ″ ( x ) > 0 for all x ∈ ( a , b ) , here the function f is concave upward on ( a , b ) . · y = f ″ ( x ) < 0 for all x ∈ ( a , b ) , here the function f is concave downward on ( a , b ) . y = f ( x ) has a point of inflection if f ″ ( x ) = 0 . To find the critical points substitute f ′ ( x ) = 0 . Second derivative test: If f ′ ( c ) = 0 and f ″ ( c ) exists, then f ( c ) is a local maximum if f ″ ( c ) < 0 . f ( c ) is a local minimum if f ″ ( c ) > 0 . The test fails if f ″ ( c ) = 0 . When test fails then use the first derivative test. Calculation: The function is f ( θ ) = θ + sin θ , [ 0 , 2 π ] . First, find the derivative. f ′ ( θ ) = 1 + cos θ Now, find the second derivative. f ″ ( θ ) = − sin θ Find the value of θ by substitute f ″ ( θ ) = 0 . − sin θ = 0 sin θ = 0 sin θ = sin π θ = π Thus, the zeroes of the f ″ ( θ ) are θ = π . This divides the x − axis into four intervals: ( 0 , π ) , and ( π , 2 π ) . Interval Point Function Sign of f ″ Behavior of f ( x ) ( 0 , π ) π 2 − sin θ = − sin π 2 = − 1 − v e Concave down ( π , 2 π ) 3 π 2 − sin θ = − sin 3 π 2 = − ( − 1 ) = 1 + v e Concave up Thus, f is concave up for interval ( π , 2 π ) and f is concave down for interval ( 0 , π )

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 4.4, Problem 49E

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Find the intervals on which f is concave up or down, the points of inflection, the critical points and the local minima and maxima.

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Chapter 4.4, Problem 48E

Chapter 4.4, Problem 50E

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Let the region R be the area enclosed by the function f(x) = ex — 1, the horizontal line y = -4 and the vertical lines x = 0 and x = 3. Find the volume of the solid generated when the region R is revolved about the line y = -4. You may use a calculator and round to the nearest thousandth. 20 15 10 5 y I I I | I + -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 -5 I -10 -15 I + I I T I I + -20 I + -25 I I I -30 I 3.5 4 x

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Find the intervals on which f is concave up or down, the points of inflection, the critical points and the local minima and maxima .

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Find the intervals on which f is concave up or down, the points of inflection, the critical points and the local minima and maxima.

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