Help with the FTC I using a graph

[spiesma12]

A continuous function f is defined on [0, 10] and its graph is given above. If g(x)= integral of f(t) dt from [0, X] for x in [0, 10], answer the following.

a) is g(0) negative, positive, or zero? -- I've answered as zero but I do not really know why or how to accurately articulate it.

b) is g(10) negative positive or zero? ---I know that this is basically asking what the area under the curve is but I am having a hard time thinking of a way to do that without a function given and without a graph that is easy to calculate the area from (blocks of triangles, squares etc..) .

c) Find all x's where g'(x)=0 --> since g'(x)=f(x)=0 these numbers would be x=0,2,4,6,8,10 correct?

d) Determine the intervals where g is increasing/decreasing. g would be increasing where the graph of f(t) is positive (above the x axis) and decreasing where f(t) is negative or below the x axis, correct? Again, if someone could articulate why exactly this is the case that would be great.

Thanks for the help. I really appreciate it!

 

[pka]

View attachment 2776
View attachment 2778
A continuous function f is defined on [0, 10] and its graph is given above. If g(x)= integral of f(t) dt from [0, X] for x in [0, 10], answer the following.

a) is g(0) negative, positive, or zero? -- I've answered as zero but I do not really know why or how to accurately articulate it.

b) is g(10) negative positive or zero? ---I know that this is basically asking what the area under the curve is but I am having a hard time thinking of a way to do that without a function given and without a graph that is easy to calculate the area from (blocks of triangles, squares etc..) .

c) Find all x's where g'(x)=0 --> since g'(x)=f(x)=0 these numbers would be x=0,2,4,6,8,10 correct?

d) Determine the intervals where g is increasing/decreasing. g would be increasing where the graph of f(t) is positive (above the x axis) and decreasing where f(t) is negative or below the x axis, correct? Again, if someone could articulate why exactly this is the case that would be great.

a) is correct.

b) is there more area above the x-axis than below?

c) correct


d) it is increasing for values of f(x) above the x-axis.

 

[spiesma12]

Thanks for your input. I have another question based off of the given information. If h(x)=g(sqrt(x)) then what is h'(9)? Im pretty lost on this one and do not know where to start. Thanks again!

 

[pka]

Thanks for your input. I have another question based off of the given information. If h(x)=g(sqrt(x)) then what is h'(9)? Im pretty lost on this one and do not know where to start. Thanks again!

\(\displaystyle h'(x) = g'\left( {\sqrt x } \right)\dfrac{1}{{2\sqrt x }} = \dfrac{{f\left( {\sqrt x } \right)}}{{2\sqrt x }}\)