Question

more calculus problems... see attached

Answer 1

\[\int\limits_{3}^{5}[f(x)+g(x)]dx = \int\limits_{3}^{5}f(x)dx+\int\limits_{3}^{5}g(x)dx\]

Answer 2

f(x) = g(x)+7, so now we have:
\[\int\limits_{3}^{5}(g(x)+7)dx+\int\limits_{3}^{5}g(x)dx\]
splitting apart the sum we obtain:
\[\int\limits_{3}^{5}7dx+2\int\limits_{3}^{5}g(x)dx\]

Answer 3

I got that far but was stuck when converting the answer to one of the choices

Answer 4

the first integral is evaluated as 7(5-3) = 14, the second you just leave alone.

Answer 5

oh! ok that makes sense

Answer 6

so would it be A?

Answer 7

Also, do you think you could help me with the second problem?

Answer 8

i think the correct answer would be E. because the 14 is there.

Answer 9

and for the second problem, i believe the answer is C.

Answer 10

oh. but wouldn't the 2 have to be included because it's 2g(x)? and then the two is distributed in A to both g(x) and 7?

Answer 11

whoops, forgot all about the 2. I dont think that 2 in A is with the 7 behind the integral as well. I think B is the right one. it has the 2 and the 14 in the back.

I dont know why they wrote it that way =/ it should be:
\[14+\int\limits_{3}^{5}g(x)dx\]

Answer 12

oh I see, because the dx is only in front of the g(x). got it, thanks.

Answer 13

also, how did you figure out the second one?

Answer 14

what im thinking is, because the graph of g(x) starts to decrease at some point, there must be "negative area" somewhere in f(x).

Negative area is when f(x) < 0, or below the x axis. You want the point where g(x) starts to decrease to match the part where f(x) goes below the x-axis. So that eliminates B, D, and E as possible answer choices.

Answer 15

Then, you look at how the graph of g(x) starts. It starts at 0, and increases rapidly till it starts to decrease. The graph that best matches that is C.

Answer 16

Thank you so much!