\title{Math 105 Test II}
\date{October 17, 1991}
\lastversion=00
 
\problem % 1
The average rate of change of the function $y=3x+ x^2 +5$ over the interval
$[1, 1.1] $ is 
\correct $5.1$
\wrong $5$
\wrong $9$
\wrong $8$
\wrong $0$

\problem %2

The second derivative of the function $f(x)= \frac15 x^{5/2} + x^2$ is $f''(x)=$

\correct $\frac34 \sqrt{x} +2$
\wrong $\frac43 x^{3/2} + 2$
\wrong $\frac43 x^{-1/2} + 6$
\wrong $2x$
\wrong  $\frac12 x^{3/2}+2x$

\problem % 3 
The value obtained by using the derivative to estimate $\sqrt{16.08}$ is

\correct $4.01$
\wrong $4.02$
\wrong $4.03$
\wrong $4.04$
\wrong $4.05$

\problem % 4
If $f(t)= 100t + 20 t^{5/2}$ then $f''(4)=$

\correct $150$
\wrong $75$
\wrong $100$
\wrong $250$
\wrong $50$

\problem % 5 
Suppose that $f(x)$ and $g(x)$ are continuously differentiable and
$f(1) = 5$, $f'(1) = 2$, $f(2)=3$, $f'(2)=4$, $g(1)=2$ and $g'(1)=8$.
Then $\left.\frac{d}{dx}\left[f(g(x))\right]\right|_{x=1}=$

\correct $32$
\wrong $8$
\wrong $10$
\wrong $16$
\wrong $15$


\problem % 6 
The graph pictured below is the graph of the function $f(x)$. 
Which of the following statements is correct?

\vskip 1in

\special{picture fig1 scaled 250}

\correct increasing and concave down in $(0, 0.5)$
\wrong increasing and concave down in $(1.5, 1)$
\wrong increasing in $(0, 2)$
\wrong concave up in $(0, 2)$
\wrong decreasing and concave down in $(0, 0.5)$ 

\problem % 7 
The graph pictured below is the graph of the function $f(x)$
near $x=5$. 
Which of the following  statements is correct?

\vskip 1in

\special{picture fig2 scaled 250}

\correct $f'(5)<0$, $f''(5)<0$
\wrong $f'(5)>0$, $f''(5)<0$
\wrong $f'(5)<0$, $f''(5)>0$
\wrong $f'(5)>0$, $f''(5)>0$
\wrong $f'(5)>0$, $f''(5)=0$

\problem % 8 
Which of the following sketches represents the graph of the function
$y= - x^3 + 12 x + 7$

\vskip .5in

\correct \special{picture fig61 scaled 100} 
\wrong  \special{picture fig62 scaled 100} 
\wrong \special{picture fig63 scaled 100} 
\wrong \special{picture fig64 scaled 100}
\wrong \special{picture fig65 scaled 100} 


\problem % 9 
The maximum of the function $f(x)= 5 + 2x - x^2$ is

\correct $6$
\wrong $5$
\wrong $2$
\wrong $7$
\wrong $0$

\problem % 10
If a function $f(x)$ satisfies $f(1) = 0$, $f'(1) = 0$, $f''(1)<0$
then $(1, 0)$ is

\correct a relative maximum
\wrong a relative minimum
\wrong a point of inflection
\wrong not a relative extremum
\wrong an absolute minimum


\problem % 11
The slope of the curve $y=(x+1)^{100}\cdot (x + 2)^2$ at $(0, 4)$ is
 
\correct $404$
\wrong $400$
\wrong $100$
\wrong $200$
\wrong $99$

\problem %12
If $y = \frac{x+1}{x^3 - 2}$, then $\frac{dy}{dx}=$

\correct $\frac{(x^3 - 2)- 3x^2(x+1)}{(x^3 - 2)^2}$
\wrong $\frac{(x^3 - 2)- 3x^2(x+1)}{3x^2}$
\wrong $\frac{1}{3x^2}$
\wrong $\frac{1}{x^3 - 2}$
\wrong $\frac{x^3 - 2}{(x^3 - 2)- 3x^2(x+1)}$


\problem %13
The graph $y= x^3 - 3x^2 - 9x +1$ has only one relative maximum.
Which point below is it?

\correct $(-1, 6)$
\wrong $(3,-26)$ 
\wrong $(0, 1)$
\wrong $(1, -10)$
\wrong $(10, 611)$.

\problem %14
Find two positive numbers, $x$ and $y$, such that $x+2y=100$ and the
product $xy$ is as large as possible

\correct $x=50$, $y=25$
\wrong $x=25$, $y=50$
\wrong $x=33\frac13$, $y=33\frac13$
\wrong $x=20$, $y=40$
\wrong $x=100$, $y=50$ 

\problem % 15
An open rectangular box with a square base is going to be constructed.
Find the dimensions of the side $x$ of the base and height $h$ of the 
box so that the volume of the box is 500 and the surface area is as 
small as possible. (The surface area is the sum of the areas of the
five faces of the box, excluding the top face.) 

\vskip 1in

\special {picture fig15 scaled 320}

\correct $x=10$, $h=5$
\wrong $x=5$, $h=20$
\wrong $x=500^{1/3}$, $h=500^{1/3}$
\wrong $x=\sqrt{50}$, $h=10$
\wrong the surface area can not be as small as possible.

\problem %16
An object moving along a straight line travels $t^2+3t$ miles after $t$ hours.
What is the instantaneous velocity after 2 hours?

\correct $7$ miles per hour
\wrong $10$ miles per hour
\wrong $10$ miles
\wrong $2$ hours
\wrong $2$ miles per hour

\problem %17
If $f(t)= 3t + 5 - \frac{50}t$, what is the instantaneous rate of change of $f(t)$
with respect to $t$ when $t=10$?

\correct$3.5$
\wrong $2$
\wrong$5$
\wrong$3$
\wrong$50$

\problem %18
Which of the following statement is true for the graph of the function
$y= x^4 - 2 x^2 + 7$?
\correct it has two relative minima
\wrong it has two relative maxima
\wrong it has one relative minimum and one relative maximum
\wrong it is always increasing and concave up
\wrong it is always decreasing and concave down

\problem %19
If $y= u^3 + 5 u$ and 
$u= x^2 -1$ then $\frac{dy}{dx}=$
\correct$ 6x(x^2-1)^2 + 10x $
\wrong$3(x^2-1)^2+5$
\wrong$2x(x^2-1)^2$
\wrong$3x^2 +5$
\wrong$2x(3x^2+5)$

\problem %20
 If $f(x)= \frac13 x^3 - \frac12 x^2$ then the maximum value of the function $f(x)$
in the interval $0\leq x\leq 3$ is 

\correct $4.5$
\wrong $0$
\wrong $\frac16$
\wrong $-\frac16$
\wrong $9$

\end
