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{\bf\underline{MATH 111. TEST III}}
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{\bf NAME:}

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Directions: To receive full credit you must show all your work. No notes, 
books, etc are to be used for the test. Calculators are only allowed to be 
used in performing simple operations (you are not allowed to use their 
graphic and programing functions, memory, etc). 

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1.(15 points) Compute the following derivatives:

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(i) $\displaystyle \frac{d}{dx}\left(\frac{2x+x^2}{x^3+1}\right)=$

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(ii) $\displaystyle \frac{d}{dx}\left[(x^3-2x+1)(2x+3)\right]=$

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(iii) $\displaystyle \frac{d}{dx}\left(4x^2+2x+1\right)^{3/2}=$



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2.(10 points) Let $y=u^2+u+1$ and $\displaystyle u=\frac{1}{x}$ . Find 
$\displaystyle \frac{dy}{dx}$ .

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3.(15 points) Find the following derivatives:

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(i) $\displaystyle \frac{d}{dx}\,e^{x^3-x+2}=$

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(ii) $\displaystyle \frac{d}{dx}\left[\ln(x^2+x+1)\right]=$

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(iii) $\displaystyle \frac{d}{dx}\,(e^x+\ln x)^2=$

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4.(15 points) Let $y$ be defined as a function of $x$ by the equation 
$$x^2y+y^3=2\;.$$

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(i) Compute $\displaystyle \frac{dy}{dx}$ using implicit differentiation.

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(ii) Write the equation of the tangent line to the above curve at the point 
$x=1$, $y=1$. 

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5.(10 points) Determine the function $y=f(x)$ which satisfies the 
differential equation $y'=-2y$ and such that $f(0)=5$. 

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6.(15 points) Simplify the following:

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(i) $\displaystyle e^{\ln3+\ln5}=$

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(ii) $\displaystyle \ln\left[(e^2)^3\right]=$

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(iii) $\displaystyle e^{2\ln3}=$

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7.(15 points) Solve the following equations for $x$:

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(i) $\displaystyle e^{x^2-3x+2}=1$ .

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(ii) $\displaystyle \left(e^{x+1}e^3\right)^2=e^8$ .

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(iii) $\displaystyle 3\ln x=6$ .

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