Question

Find values of a and b so that y = a·b^x and the line y = x + 4 are tangent at x = 0.

I have no idea what to do here. It wasn't on the written homework due in class today and the teacher didn't go over it, an apparently this is going to be on the test tomorrow, so I need this broken down pretty well so I can memorize the steps for tomorrow. Please and thank you!

Answer 1

is this caclulus?

Answer 2

*calculus?

Answer 3

not sure how to proceed without it
on thing you know is that they have to meet at \(x=0\) so \[ab^0=0+4\] in other words \[a=4\]

Answer 4

Yes, it is calculus.

Answer 5

ok well we got \(a\) right?

Answer 6

clear or no? then we can find \(b\)

Answer 7

Yes, I understand how to get a now, I think. I'll do more practice problems later to make sure. What about b?

Answer 8

i am thinking
the answer i am getting is weird hold on a sec

Answer 9

ok i think we can to it

Answer 10

do you know what the derivative of \(b^x\) is in general?

Answer 11

The derivative of b^x is (b^x)*ln(b).

Answer 12

ok so the derivative of \[4b^x\] is \[4\ln(b)b^x\]

Answer 13

the slope of \(y=x+4\) is \(1\) that means you want the derivative to be 1 at \(x=0\)

Answer 14

set \[4\ln(b)=1\] ( that is the derivative at 0) and solve for \(b\)

Answer 15

clear how to do that, or no?

Answer 16

I think so. The answer I got was \[\sqrt[4]{e}\]

Does that sound about right? Also, sorry for the equation being on a different line, I don't know how to get it to stay in the same line as the first sentence.

Answer 17

no problem
yeah or \[b=e^{\frac{1}{4}}\]same thing

Answer 18

making your function \[f(x)=4e^{\frac{x}{4}}\]

Answer 19

looks right, check here
http://www.wolframalpha.com/input/?i=plot+4e^%28x%2F4%29,+x%2B4

Answer 20

Oh okay. Yeah, that was the correct value for b according to Webstudy. I really appreciate the help, thank you! That problem really confused me.

Answer 21

no problem, glad to help