Question

Suppose f (x) is a function defined on the interval [0, 6] and whose graph consists of a circular arc and two line segments as shown in the graph below.

g(x) = ∫0 x f(t)dt.


Evaluate g(3), g'(3), and g''(3).

Find the absolute minimum value for g(x) on the interval [0, 6]. Show the analysis that leads to your answer.


http://curriculum.kcdistancelearning.com/courses/CALCABx-AP-A09/b/assessments/Unit5_ExamFR/Unit5_ExamFR_1q2.jpg

Answer 1

g(3) is the net area from 0 to 3

g'(3) = f(3) (value of the function f at 3)

g''(3) = f'(3) (slope of the function f at 3, you might need to estimate)

Answer 2

in fact g''(3) is undefined, because slope at 3 is undefined

Answer 3

thank you! would there be a way to find a numerical value of g(3)?
and i still need help with part b.

Answer 4

from 0 to 2, looks like a quarter of a circle (positive area). From 2 to 3 a triangle (negative area)

Answer 5

The critical points on the given graph are at x= 2, 5, and 3.
g(2) = 02f(t)dt = pi
g(5) = 05f(t)dt =pi -2 + .5
g(3) = 03f(t)dt =pi-2
The absolute minimum would be at 3 because pi-2 is the smallest number.

will someone tell me if this is correct? (02f(t)dt is basically the definite integral on the interval of 0 to 2)