91Ó°ÊÓ

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable. $$\lim _{x \rightarrow \infty}(x-\sqrt{x^{2}+4 x})$$

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=e^{2 x}$$

Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=-x^{4}-2 x^{3}+12 x^{2}$$

Absolute maxima and minima Determine the location and value of the absolute extreme values of \(f\) on the given interval, if they exist. $$f(x)=x^{1 / 3}(x+4) \text { on }[-27,27]$$

Crease-length problem A rectangular sheet of paper of width \(a\) and length \(b,\) where \(0

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{1+\sqrt{x}}{x} d x$$

Absolute maxima and minima Determine the location and value of the absolute extreme values of \(f\) on the given interval, if they exist. $$f(x)=x^{3} e^{-x} \text { on }[-1,5]$$

Suppose a continuous function \(f\) is concave up on \((-\infty, 0)\) and \((0, \infty) .\) Assume \(f\) has a local maximum at \(x=0 .\) What, if anything, do you know about \(f^{\prime}(0) ?\) Explain with an illustration.

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=2-a \cos x, a \text { constant }$$

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable. $$\lim _{x \rightarrow \infty}(\sqrt{x-2}-\sqrt{x-4})$$