Chapter 5.5, Problem 59E

To determine

To calculate the linearization of $f\left(x\right)=e^x$

Answer to Problem 59E

The linearization of $f\left(x\right)=e^x$ is $y=ex$ . The correct answer is option B

Explanation of Solution

Given information:

  $x=1$

The option is given as,

A. $y=e$

B. $y=ex$

C. $y=e^x$

D. $y=x-e$

E. $y=e\left(x-1\right)$

Linearization is the approximation of the derivative of the function by a tangent line at a particular point.

  $\begin{array}{l}f\left(x\right)=e^x\\ f\text{'}\left(x\right)=e^x\end{array}$

At $x=1$ , $f\text{'}\left(1\right)=e$

So the tangent of slope “e” passes through the point (1,e).

The equation of the tangent line is,

  $\begin{array}{l}\left(y-e\right)=e\left(x-1\right)\\ y=ex-e+e\\ y=ex\end{array}$

Therefore, the linearization of $f\left(x\right)=e^x$ is $y=ex$