Determine the equation of the tangent line at x=0 and the value of f(x)at x=1.5 given $f(x)=a^{x}$ for all values of a A)y=(ln a)x+1, y=1.5(ln a)+1 B)y=(ln a)(a^1.5)x-1, y=-1 C)y=(a^1.5)[(ln a)x-1], y=-1 D)y=(ln a)x-1, y=1.5(ln a)-1 E)y=(ln a)x-1, y=1.5

Question

Quizplus · Accepted Answer

The derivative of $f(x) = a^x$ is $f'(x) = a^x \ln(a)$, which gives the slope of the tangent line at any point x. At $x = 0$, $f'(0) = a^0 \ln(a) = \ln(a)$, so the equation of the tangent line is $y = (\ln(a))x + 1$ because $f(0) = a^0 = 1$. For $f(1.5) = a^{1.5}$, the value is not directly given by the options, but the correct option must have the correct tangent line equation.

Determine the Equation of the Tangent Line at X=0 and the Value