91Ó°ÊÓ

At the beginning of this section we saw that the derivative of \(f(x)=e^{x}\) is \(f^{\prime}(x)=e^{x} .\) Use this information to find all tangent lines to the graph of \(f(x)=e^{x}\) that pass through the origin.

Either use factoring or the quadratic formula to solve the given equation. $$ 2^{2 x}-12\left(2^{x}\right)+35=0 $$

In Problems 45 and 46 , graph the given equations on the same rectangular coordinate system. $$ y=3^{-x}, x=3^{-y} $$

Determine the hydrogen-ion concentration \(\left[\mathrm{H}^{+}\right]\) of a solution with the given \(\mathrm{pH}\). $$ 7.3 $$

For \(f(x)=b^{x},\) show that: (a) \(\frac{f(x+h)-f(x)}{h}=b^{x}\left(\frac{b^{h}-1}{h}\right)\) (b) \(f\left(x_{1}+x_{2}\right)=f\left(x_{1}\right) f\left(x_{2}\right)\)

Show that the \(x\) -intercept of the tangent line to the graph of \(f(x)=e^{x}\) at \(x=x_{\mathrm{o}}\) is one unit to the left of \(\left(x_{\mathrm{o}}, 0\right)\).

In Problems \(47-50\), the given function \(f\) is one-to-one. Find \(f^{-1}\) and give its domain and range. $$ f(x)=2+4^{x} $$

Either use factoring or the quadratic formula to solve the given equation. $$ (\ln x)^{2}+\ln x=2 $$

Determine the hydrogen-ion concentration \(\left[\mathrm{H}^{+}\right]\) of a solution with the given \(\mathrm{pH}\). $$ 6.6 $$

In Problems \(47-50\), the given function \(f\) is one-to-one. Find \(f^{-1}\) and give its domain and range. $$ f(x)=10^{x+3}-10 $$