Tangent line

**bluejewballs** · February 20th 2008, 02:03 PM

For what values of a and b are y = a b^x and y = 4 + x tangent at x = 0

im guessing i have to find the derivative but im not quite sure.
Thank you

**wingless** · February 20th 2008, 03:18 PM

$y = a\cdot b^x$

$y = x + 4$

There are two conditions to be made,
I) Both functions must have a common point at $x = 0$
II) Their derivatives have to be equal at that point.

I)
$y = x + 4$
When $x=0$ , $y=4$
So both functions must have a point at (0,4).
$y = a\cdot b^x$
$4 = a\cdot b^0$
$a = 4$

The function becomes,
$y = 4\cdot b^x$

II)
$y=x+4$
$y' = 1$

$y= 4\cdot b^x$
$y' = 4 b^x \ln b$
(At x = 0), $y' = 4 \ln b$

Derivatives have to be equal.. So,
$1 = 4 \ln b$
$b = e^{\frac{1}{4}}$

Finally,
$y=4e^{\frac{x}{4}}$

**bluejewballs** · February 20th 2008, 03:30 PM

thank you very much that makes perfect sense. i love how it seems so easy once someone shows me how its done

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