Chapter 3: Applications of the Derivative

Sketch careful, labeled graphs of each function $f$in Exercises $57-82$by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\frac{2^x}{1-2^x}$.

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=\frac{e^x}{x}$

Sketch careful, labeled graphs of each function f by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f and f' , and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=x^2{3}^x$

As you will prove in Exercises 93 and 94, exponential growth functions ${\mathrm{e}}^{\mathrm{kx}}$always dominate power functions $x^r$, and power functions $x^r$with positive powers always dominate logarithmic functions ${\log }_{b}x$. Use these facts to quickly determine each of the limits in Exercises 75-80.

$\underset{x\to \infty }{\lim }\frac{x^{101}+500}{e^x}$

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=e^{3x}-e^{2x}$

calculate the limits of$\underset{x\to \infty }{\lim }\frac{\ln \left(x^{100}\right)}{x^8}$

Sketch careful, labeled graphs of each function $f$in Exercises $57-82$by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f\text{'}$, and examine any relevant limits so that you can describe all key points and behaviors of $f$.

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Sketch careful, labeled graphs of each function $f$in Exercises $57–82$by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f\text{'}$, and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\frac{\ln x}{x}$

Calculate the limits of$\underset{x\to \infty }{\lim }{2}^x{x}^{-100}$