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 $f$.

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.

$f\left(x\right)=\cos \left(3\left(x-\frac{\mathrm{\pi }}{2}\right)\right)$

Calculate the limits$\underset{x\to \infty }{\lim }{e}^{-x}\ln \left(x\right)$

Sketch careful, labeled graphs of each function fin

determine the global extrema of function $f\left(x\right)=x\ln \left(x\right),I=\left(0,1\right),J=\left(0,\infty \right)$

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.

$f\left(x\right)={\left(\ln x\right)}^2+1$

Determine the local extrema of a function.

$f\left(x\right)=x^2\ln \left(0.2x\right),I=(0,4],J=(0,\infty )$

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.

$f\left(x\right)=\sin \left({\tan }^{-1}x\right)$

determine the global extrema of function$f\left(x\right)=x^2\ln \left(0·2x\right),I=\left(0,4\right),J=\left(0,\infty \right)$

Determine the local extrema of the function,

$f\left(x\right)=x^3{e}^{-x},I=[0,\infty ),J=(-\infty ,\infty )$.