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
$\displaystyle y = a\cdot b^x$
$\displaystyle y = x + 4$
There are two conditions to be made,
I) Both functions must have a common point at $\displaystyle x = 0$
II) Their derivatives have to be equal at that point.
I)
$\displaystyle y = x + 4$
When $\displaystyle x=0$, $\displaystyle y=4$
So both functions must have a point at (0,4).
$\displaystyle y = a\cdot b^x$
$\displaystyle 4 = a\cdot b^0$
$\displaystyle a = 4$
The function becomes,
$\displaystyle y = 4\cdot b^x$
II)
$\displaystyle y=x+4$
$\displaystyle y' = 1$
$\displaystyle y= 4\cdot b^x$
$\displaystyle y' = 4 b^x \ln b$
(At x = 0), $\displaystyle y' = 4 \ln b$
Derivatives have to be equal.. So,
$\displaystyle 1 = 4 \ln b$
$\displaystyle b = e^{\frac{1}{4}}$
Finally,
$\displaystyle y=4e^{\frac{x}{4}}$