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\preface
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\title
\noindent
{\bf Math 105: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl October 16, 1997}\hfill
Time of MWF class:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are not allowed. There are 19 multiple choice questions, worth 5 points each, and you get an 
additional 5 points for writing your name.  
Record your answers  to the multiple choice problems by  placing an $\times$ 
through one letter for each problem on this answer sheet.  

\vskip .25cm \noindent {\bf
You are taking this exam under the honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{5in}

\note %1
\problem  
If $\displaystyle f(x) = {{1} \over {x^2}}+2x+5$ compute $\displaystyle f''(x)$.

\correct $\displaystyle {{6} \over {x^4}}$
\wrong $\displaystyle -{{6} \over {x^4}}$
\wrong $\displaystyle -{{2} \over {x^3}}+2$
\wrong 2
\wrong $6x^4$

\note %2
\problem
Find  $\displaystyle {{d^2}\over{dx^2}}\bigg(x^4-3x^3+x^2-x+24\bigg)\bigg\vert_{x=0}$.

\correct 2
\wrong 12
\wrong $-3$
\wrong 24
\wrong 0

\note %3
\problem
The average rate of change of the function $f(x)=x^3-x^2+2x+1$ over the interval from 1 to 3 is  

\correct 11
\wrong 22
\wrong 25 
\wrong 3
\wrong 23

\note %4
\problem
Find the instantaneous rate of change of the function $g(t) = \sqrt{3t-2}$ at $t=2$.

\correct $\displaystyle {{3}\over{4}}$
\wrong 2
\wrong $\displaystyle {{3}\over{2}}$
\wrong $\displaystyle {{3}\over{8}}$
\wrong it does not exist

\note %5
\problem
The profit obtained by a manufacturer when producing $x$ units of some product is given in dollars by the function
$$P(x)=0.004x^4+2x\;.$$
Find the marginal profit at production level $x=10$ units of product.

\correct 18
\wrong 16
\wrong 21
\wrong 420
\wrong 250 

\note %6
\problem
The cost of producing $x$ units of a product is given in dollars by the function
$$C(x)=0.05x^2-3x+100\;.$$
Find the production level $x$ at which the marginal cost is 10 dollars/unit. 

\correct 130 units
\wrong 135 units
\wrong 126 units
\wrong 50 units
\wrong 200 units

\note %7
\problem
The position function of an object moving on a straight line is given by 
$$s(t)=5t^2+4\sqrt{t}\;,$$
where $s$ is expressed in miles and $t$ in hours. What is the velocity of the object 
at time $t=4$ hours?

\correct 41 mph
\wrong 65 mph
\wrong 35 miles
\wrong 39 mph
\wrong 88 miles


\note %8
\problem
Suppose $T(t)$ is the temperature on a hot summer day at time $t$ hours. If $T'(10)=4$, by approximately 
how much will the temperature rise from 10:00 to 10:30 ? 

\correct 2
\wrong 4
\wrong $-4$
\wrong 1
\wrong 3

\note %9
\problem
Which of the following is true for the function $f$ whose graph is pictured below? (Only one answer is true.)

\vskip 4cm

\correct $f$ is decreasing and concave up on the interval $(0,1)$
\wrong $f$ is increasing in the interval $(-1,3)$
\wrong $f$ is concave up in the interval $(-1,3)$
\wrong $f$ is decreasing and concave down in the interval $(-1,0)$
\wrong $f$ is increasing and concave up on the interval $(2,3)$

\note %10
\problem
The graph shown below is the graph of a function $f(x)$ near $x=2$. Which of the following statements is true?

\vskip 4cm

\correct $f'(2)>0$ and $f''(2)<0$
\wrong $f'(2)<0$ and $f''(2)<0$
\wrong $f'(2)=0$ and $f''(2)<0$
\wrong $f'(2)<0$ and $f''(2)=0$
\wrong none of the above statements is true

\note %11
\problem
For what value(s) of $x$ does the function 
$$f(x)=x^3+6x^2-36x+7$$
have a relative minimum point? 

\correct $x=2$
\wrong $x=2$ and $x=-6$
\wrong $x=-6$
\wrong $x=-20$
\wrong $f$ has no relative minimum point

\note %12
\problem
What is the absolute maximum value of the function $f(x)=-5x^2+10x+3$?

\correct 8
\wrong $-10$
\wrong 3
\wrong 12
\wrong $f$ has no absolute maximum value

\note %13
\problem
For what value(s) of $x$ does the function $f(x)=2x^4-12x^2-5$ have an inflection point?

\correct $x=-1$ and $x=1$
\wrong $x=-1$ and $x=0$
\wrong $x=0$
\wrong $x=2$ and $x=-2$
\wrong $f$ has no inflection points

\note %14
\problem
Consider the function
$$f(x)=2x+3+{{8}\over{x}}\;,$$
defined for $x>0$. Find the absolute minimum value of $f(x)$.

\correct 11
\wrong 0
\wrong 15
\wrong 7
\wrong $f$ has no absolute minimum value

\note %15
\problem
On which of the following intervals is the function $f(x)=(x-1)^3$ decreasing?

\correct $f$ is never decreasing
\wrong $(-\infty,0)$
\wrong $(-\infty,1)$
\wrong $(0,+\infty)$
\wrong $(1,+\infty)$

\note %16
\problem
Which of the graphs sketched below best represents the graph of the function
$$f(x)=x^3+{{3}\over{2}}x^2-6x\;?$$

\vskip 9cm

\correct I
\wrong II
\wrong III
\wrong IV
\wrong V

\note %17
\problem
A ball is thrown straight up in the air and its height in feet (above the ground) after $t$ seconds is given by
$$h(t)=6+64t-16t^2\;.$$
What is the maximum height reached by the ball?

\correct 70 
\wrong 6 
\wrong 65 
\wrong 75 
\wrong 2


\note %18
\problem
A rectangular garden with two sides of length $x$ and two of length $y$ (in feet) is to be surrounded by a wooden fence along 
the sides of length $x$ and by a wire fence along the sides of length $y$. The cost of the wooden fence is 10 dollars per foot 
and the cost of the wire fence is 5 dollars per foot. Find the maximum area that the garden can have (in sq feet) if the total 
cost of fencing is 400 dollars.

\correct 200
\wrong 192
\wrong 220
\wrong 198
\wrong 150

\note %20
\problem
An open rectangular box (i.e. no top) has a square base and has a volume of 4 cubic feet. What is the smallest surface area (in sq feet) that the box can
have?  (By surface area we mean the sum of the areas of the 5 faces of the box.)

\correct 12
\wrong 16
\wrong 10
\wrong 14
\wrong 8



\end
