Question

Find k so that the following function is continuous on any interval.

**function attached inside**

k=_________

:)

Answer 1

:)

Answer 2

k doesn't matter

Answer 3

it doesn't?

Answer 4

The functions must be equal at the point where x=5pi

ASSUMING there's a typo and it's meant to be

x > 5pi

Answer 5

This is called a piecewise defined function.

To the left of x=5pi, actually, between 0 and 5pi, the function takes on the values y=k*cos x. to the right of x=5pi, the function takes on the values y=13=x.

For this function to be continuous at x=5pi, the half function k*cos x MUST equal the half function 13=x.

Substitute 5pi for x: k*cos 5pi = 13-5pi. then k must equal (13-5pi) / (cos 5pi). I agree with agent0smith's second comment.

Note that cos 5pi = cos pi = -1, so k must equal 5pi-13.

Answer 6

for the bottom part?

Answer 7

ohh so what would happen from here on out? :/

Answer 8

I think we did a similar one earlier, with simpler functions

The two functions must be equal when x=5pi (the boundary between the two functions)

So plug that in for x, set them equal, find k.

Answer 9

so 13-5pi ?

Answer 10

Substitute 5pi for x: k*cos 5pi = 13-5pi. then k must equal (13-5pi) / (cos 5pi).

This is equal to -(5pi-13) / (cos 5pi), or

-(5pi-13) / (-1), or 5pi-13. k=5pi-13.

Answer 11

ohh okay so you set the equal to each other :)

and after those steps, that's as simplified as it gets? and then you isolate the k? just wanna make sure i'm understanding correctly haha

Answer 12

We did "isolate the k" here: k=5pi-13.

Answer 13

okay i see :) awesome!! thank you:)

Answer 14

:)