Explanation Given information: The given function of f is, f ( x ) = − x 2 + 4 x − 3 And the lines are y = 0 , x = − 1 , x = 2 . Calculation: The given function is a polynomial function, so the domain of the function is ( − ∞ , ∞ ) . Substitute 0 for x in the given function to evaluate the y -intercept. f ( 0 ) = 0 + 0 − 3 = − 3 Since the function is polynomial, x -intercept is not used. Since the function is a polynomial, there will be no vertical or horizontal asymptotes. Differentiate the function f ( x ) = − x 2 + 4 x − 3 with respect to x and equate to 0 to evaluate the critical points. f ′ ( x ) = 0 d d x ( − x 2 + 4 x − 3 ) = 0 − 2 x + 4 = 0 x = 2 The critical point is x = 2 , so the interval is ( − ∞ , 2 ) and ( 2 , ∞ ) . Differentiate the function f ′ to check the concavity of function f ( x ) f ″ ( x ) = d d x ( − 2 x + 4 ) = − 2 Since f ″ is negative, so the graph of f is concave downward in the interval ( − ∞ , + ∞ ) . Substitute some values of x from ( − ∞ , ∞ ) and find the corresponding values of f ( x ) as shown in table below. x f ( x ) − 1 − 8 0 − 3 1 0 2 1 Use online graphing calculator to plot all the points from the above table and join them with a smooth curve to get the graph of f ( x ) = − x 2 + 4 x − 3 as shown below. Also draw the line y = 0 , x = − 1 and x = 2 to get the bounded area. Figure (1) Figure (1) shows the graph of f ( x ) = − x 2 + 4 x − 3 , x = − 1 and x = 2 . The area A consist of two areas A 1 and A 2 above and below the x − axis respectively. The area A 1 , is the region bounded above by the x − axis and below by the graph of f ( x ) = − x 2 + 4 x − 3 between ( − 1 , 1 ) . The area A 2 is the region bounded above by the graph of. f ( x ) = − x 2 + 4 x − 3 and below by g ( x ) = 0 between x = 1 to x = 2 Integrate the function f ( x ) = − x 2 + 4 x − 3 between the region [ − 1 , 1 ] for the area A 1

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Chapter 6.6, Problem 29E

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To evaluate: The graph of function f(x)=−x2+4x−3 and find the area bounded by graph of f(x) and the lines y=0, x=−1,x=2 .

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Chapter 6.6, Problem 28E

Chapter 6.6, Problem 30E

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Ch. 6.1 - What is an antiderivative? Give an example.Ch. 6.1 - Prob. 2CQ Ch. 6.1 - What is the difference between an antiderivative...Ch. 6.1 - Prob. 4CQ Ch. 6.1 - In Exercises 1-4, verify directly that F is an...Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - In Exercises 1-4, verify directly that F is an...Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E

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The graph of function f ( x ) = − x 2 + 4 x − 3 and find the area bounded by graph of f ( x ) and the lines y = 0 , x = − 1 , x = 2 .

Solution Summary: The author evaluates the graph of function f(x) and finds the area bounded by graph. Substitute 0 for x to evaluate the y -intercept.

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To evaluate: The graph of function f(x)=−x2+4x−3 and find the area bounded by graph of f(x) and the lines y=0, x=−1,x=2 .

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