This is again exploiting the fact that a definite integral is computing the area between a curve and the x-axis (but regions below the x-axis have negative areas). Note, since g(x) is the definite integral between 0 and x, that means g(1) is the definite integral between 0 and
1, while g(2) is the definite integral between 0 and 2, etc.
Part (a)
Visualize the area each value is describing and solve it using the fact the shapes are just rectangles and triangles. Remember! Areas below the x-axis are negative.
Part (b)
Almost everyone is going to get this question wrong, because they won't read the graph properly. Don't be guilty of jumping to
conclusions.
One way to answer this question is to observe your answers to part (a). That should give you a pretty good idea where g is decreasing.
A more sophisticated way is to use the Fundamental Theorem of Calculus (see my Lesson 11, question 10). Note, you have been given that g(x) is a definite integral. You can compute g'(x) using the Fundamental Theorem. Now that you have computed g'(x),
you need to establish where g'(x) is negative (since that is where g(x) is decreasing). The key is to establish how does g'(x) relate to the graph of f(x) you have been given. Therefore, how can you establish where g'(x) is negative.
Part (c)
There are several ways you can solve this. The easiest is probably by looking at your answers to part (a). You can also make a sign diagram for g'(x) using the Fundamental
Theorem of Calculus again.