The graph f consist of 3 line segments. the points of these line segments are (-

Question

The graph f consist of 3 line segments. the points of these line segments are (-2,0), (1,4), (2,1) and (4,-2)

let g(x) = the integral from 1 to x f(t)dt

a) Compute g(4) and g(-2) <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /?>

b) Find the instantaneous rate of change of g with respect to x at x=1.

c) Find the Absolute minimum value of g on the closed interval [-2,4]. Justify answer

d) The 2nd derivative of g is not defined at x=1 and x=2. Howmany of these valuesx-coordinates if points of inflection of the graph of g? Justify your answer.

Enter question here...

Explanation / Answer

You can work out equations for the three line segments and then do integrals on each according to the values of x for which you have to solve g. Or... Draw the graph. If we are working out the area under the curve, you can see it is composed mainly of right triangles. The area of a right triangle is half the x-dimension times the y-dimension. From -2 to 1 the area is: (3 x 4 / 2 ) = 6 From 1 to 2 the area is: (3 x 1 / 2) + 1 = 5/2 From 2 to 4 the area is: 0 (two opposite triangles) a) g(4) = area from 1 to 4 = 5/2 (from above) g(-2) = negative(area from -2 to 1) = -6 (from above) b) The instantaneous rate of change is the derivative. The derivative of an integral is the original function. Therefore, g'(1) = f(1) = 4 c) OK I see now that when we go to the left, the integral is negative. Therefore, g decreases continuously as we go to the left, and reaches a minimum where f touches the x-axis, at x=-2. g(-2) = -6. I'm not sure if this would be a formal enough answer. I guess we can note that g is positive for x > 1. d) Need to think carefully about the definition of an inflection point! One definition is that it is when the first derivative is at a maximum or minimum. We have already seen that the first derivative is the original function f. This at a maximum at x=1 but not at x=2. Therefore, there is only one point of inflection, at x=1.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.