Question

Let F(x) = integral from 0 to 2 of 2tdt. Then F(x) is the area under the line y=25 from the origin to x. (a) Construct a table showing the values of F for x = 0, 1, 2, 3, 4, 5. (b) Is F increasing or decreasing when x>0? Concave up or down? Explain. (c) When t<0, the line y=2t is below the t-axis. Explain why F(-1) is positive.

I did (a) by evaluating the integral from 0-1, 1-2, etc. and came up with (0,0), (1,1), (2,3), (3,5), (4,7) and (5,9), which creates a graph where F is increasing and concave up (part b). But I'm not sure if that's correct, and I'm tripped up on (c)!

Answer 1

(3,5), (4,7) and (5,9) ... how did you get these values?

Answer 2

I took F(3)-F(2) (the antiderivative of 2t) - so (3)^2 - (2)^2 = 9-4 = 5 (to get the point 3,5). Not sure if that was the correct thing to do though?

Answer 3

"Let F(x) = integral from 0 to 2 of 2tdt."
Let F(x) = integral from 0 to x of 2tdt. ..?

"Then F(x) is the area under the line y=25 from the origin to x"
Then F(x) is the area under the line y=2t from the origin to x. ...?

Answer 4

Yes, typo - should say y=2t - whoops! Sorry for the confusion

Answer 5

and the top one?

Answer 6

Teh top one is correct - I did ask a TF about it saying t in one place and x in another, but he said x was just a dummy variable - they are essentially the same thing for the purposes of the problem - i guess writing "integral from 0 to 2 of 2t(dt)" is maybe more clear?