Problem Statement (Ink Blot) Suppose a function f has two derivatives (both cont

Question

Problem Statement (Ink Blot) Suppose a function f has two derivatives (both continuous) over [0,2] and that its graph is given below. As you see, the printer was sloppy and spilled a lot of ink on the graph. Decide. if possible. whether each of the following definite integrals is positive, negative. or equal to 0. Explain your answers. Extension Questions: Suppose the ink blot covered the whole area of [0.25,1.75]. Which integrals could you still estimate? How large could the ink blot be before you could no longer estimate those integrals?

Explanation / Answer

a shows the area under the graph between 0 and 2 and hence positive as the area above x axis is much larger that the area under the x axis if there is any

b shows the area under the graph between 1 and 1.5 in the current graph and as lot of it is covered in ink , we cannot predict this

c is also positive as it shows area under 0 and 1 and it is positive in this case

d shows the area under the derivative curve . Since in this graph , the slope is increasing and coming back to the original point ,it could be 0

e is negative as the graph shown here is concave

f is also 0 as the graph is clearly straight line in the given limits

if the inkblot covers (0.25 , 1.75 ) , we can still estimate a and f

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