Calculus I

Chapter $4$

Nan

$35$ points

a separate sheet of paper to me by the due date. Show all necessary work. Unsupported work will not receive full credit.

Consider the function $f(x)=x^3-2x^2-4x.$ Use calculus to determine:

a$).$ Over what interval$($s$)$ is the function increasing?

b$).$ Test for and identify any local extrema

c$).$ Test for and identify any inflection points of the function

Use calculus to determine all critical points of the function $f(x)=\frac{2x^2+2}{2x}$ and use a Test to classify each as local maximum, local minimum, or neither.

Two posts, one $12$ feet high, the other $28$ feet high, stand $30$ feet apart. They are to be stabilized by two wires attached to a single stake positioned between the posts, running from the ground to the top of each post. Use calculus to determine where the stake should be placed to use the least amount of wire.

Determine the general antiderivative:

a$).{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(x^2+3x-4x^{-2}-2x^{-4})dx$

b$).{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(1-cosx-e^{-2x})dx$

Use L'Hopital's Rule to find the limits:

a$).,\underset{x1}{lim}\frac{1-x+lnx}{x^3-2x+1}$

b$).,\underset{x0}{lim}\frac{xsinx}{1-cosx}$

Use L'Hopital's Rule and logarithmic differentiation to find the limit:

$\underset{x}{lim}(1+\frac{1}{3x}{)}^x$