final eaxm math111
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{\bf\underline{MATH 111. FINAL EXAMINATION}}
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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. There are 25 problems worth 6 
points each, for a total of 150 points. Calculators are only allowed to be 
used in performing simple operations (you are not allowed to use their 
graphic and programming functions, memory, etc). 

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1. Find the domain of the function $\displaystyle f(x)=\frac{x^2+1}{x(x+5)}$ 
.


\vspace{25mm}
2. Find the equation of the line through the point $(2,3)$, parallel to the 
line $y+4x=10$ .

\vspace{55mm}
3. Find \hspace{5mm}$\displaystyle \lim_{x\rightarrow1}\frac{x^2-
5x+4}{x-1}$ .

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4. Find the instantaneous rate of change of $\displaystyle 
f(x)=\frac{x+2}{x^2}$ at $x=1$ .

\vspace{50mm}
5. Find $f(g(x))$, if $f(x)=e^x$ and $g(x)=1+\ln x$ . 

\vspace{50mm}
6.  Find $f(2)$ for the function $f$ given below:
\[  f(x)=\left\{ \begin{array}{ll}
            x^2+1 &  \mbox{ if $x\leq0$}\;, \\ \\
            \sqrt{x+2} & \mbox{ if $x>0$}\;. 
                \end{array}
   \right. \]

\vspace{20mm}
7. Find the $x$-coordinate of a point on the curve $\displaystyle 
y=e^{x^2+1}$ where the tangent line is horizontal.

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8. If $f(x)=(x+1)^3-(x-1)^4$, compute $f''(x)$ .

\vspace{40mm}
9. Find the minimum value of $g(x)=x^2+2x+5$ .

\vspace{40mm}
10. Find the $x$-coordinate of an inflection point of $f(x)=x^3+6x^2-
15x+20$ .

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11. If $y=e^u+u^5$ and $u=3x+1$, find $\displaystyle\frac{dy}{dx}$ in 
terms of $x$ .

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12. Compute $\displaystyle\frac{dy}{dx}\;$, if $y$ is defined implicitly in 
terms of $x$ by $y^4+x^3y=3$ .

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13. Find the $x$-coordinates of all the relative extreme points of 
$f(x)=2x^3-12x^2+7$ . Use the second derivative to decide whether they are 
maximum or minimum points.

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14. Find the dimensions of the rectangular garden of maximum area that 
can be surrounded by 300 yards of fencing.  

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15. Find $\displaystyle\frac{d}{dx}\left(x^2\ln x\right)$ .

\vspace{50mm}
16. Write the equation of the tangent line to the curve $y=x+\ln(x^2+1)$ at 
the point where $x=0$.

\vspace{50mm}
17. Simplify $\displaystyle e^{\ln5+\ln7}$ .

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18. Solve for $x$: \hspace{5mm}$\ln(x^2+3x+1)=0$ .

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19. Find the growth constant of a bacteria population which grows 
exponentially, if the size of the population 
doubles every 3 hours.

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20. Five thousand dollars is deposited in a savings account at 6 percent 
interest compounded continuously. What 
is the amount in the account after 5 years?

\vspace{45mm}
21. Compute $\displaystyle\int_1^e\left(3x^2+\frac{1}{x}\right)\,dx$ .

\vspace{45mm}
22. Find the function $f(x)$, if $\displaystyle f'(x)=e^{x}+x$ and $f(0)=2$ .

\newpage
23. Find the area of the region under the curve $y=4(x+1)^3$ from $x=0$ to 
$x=1$ .

\vspace{60mm}
24. The velocity of a rocket flying straight up in the air is given by 
$v(t)=20t$ feet per second. Find the 
distance traveled by the rocket in the first 5 seconds after launching.

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25. Find $\displaystyle\int_0^1\left(e^{4x}+x^3\right)\,dx$ .

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