For what values of a and b are y = a b^x and y = 1 + x tangent at x = 0.
how do i start this one??
Let f(x) = y = a*b^x -------------(i)
And g(x) = y = 1 +x
If f(x) and g(x) are tangent at x=0, then then their y's there are the same.
So,
f(0) = g(0)
a*b^0 = 1 +0
a*1 = 1
a = 1 -----------------answer.
So,
f(x) = y = 1*b^x = b^x -------(ii)
If f(x) and g(x) are tangent at x=0, then their slopes are equal at x=0.
slope, m = dy/dx
For g(x) = 1 +x,
m2 = g'(x) = 1 --------**
For f(x) = b^x,
m1 = f'(x) = ln(b) *b^x
At x=0,
m1 = m2
ln(b) *b^x = 1
ln(b) *b^0 = 1
ln(b) *1 = 1
ln(b) = 1
b = e^1
b = e ------------------------answer.