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MATH 151 (17-19), Dr. Z. ,  {\bf Answers for the Practice For Midterm II}, 

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These posted answers refer to the Nov. 20, 7:30pm version. If you have an earlier
version, please download the latest version of the Practice for Midterm II.
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Disclaimer: Even I make mistakes. If you find any errors, please E-mail me:
zeilberg at math dot rutgers dot edu  . (Subject: MathIsFun).
I will pay 1 dollar to the first person to detect any error (untlil Sunday
Nov. 21, 8:00pm). Of course, if your answer diagrees, it is more likely
that you made a mistake, so please check it three times before you report
an error.
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Nov. 21, 2004, 8:11am: Prashan found an error in 5(moderate-iii),
it should be $2+.03/e^2$. Here is the corrected version.
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{\bf 1}. (warm-up)  
(a) $f(x)=12x^2+6 \quad$, (b) $f(x)={{20} \over {x^6}}\quad$,
(c) $f(x)={{-4} \over {x^2}} \quad $.
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{\bf 1}. (moderate)
(a) $f''(t)=-2 \sin t-t \cos t \quad$,
(b) $f''(t)=e^t- 4t \cos t  +(t^2-2)\sin t \quad $,
(c) $f''(s)=(s^2+4s+2) e^s \quad$,
(d) $ f''(x)= { {-4} \over {(x+3)^3}} +  2e^x$.
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{\bf 1}. (hard) 
(a) 
$$
f''(x)= { {6x(2+x^6+ (1-x^6)^{3/2}) }  \over {(\sqrt{1-x^6})^3} }
\quad 
$$

(b) 
$$
f''(x)= -{ {1+2x \tan^{-1} x }  \over { ((1+x^2)\tan^{-1} x)^2 }  }
\quad 
$$


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{\bf 2}. (warm-up) :
Horiz. Asymp.: none. Vertical Asympt.: none.
decreasing: $-\infty < x < 2$ ; increasing: $2<x<\infty$;
local max: none; local min: $(2,-3)$; concave up: $-\infty<x<\infty$;
concave down: never; inflection point: none.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).
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{\bf 2}. (medium) 
Horiz. Asymp.: none. Vertical Asymp.: none.
decreasing: $1<x<2$; increasing: $-\infty <x<1$ AND $ 2<x<\infty$;
local max: $(1,6)$; local min: $(2,5)$; concave up: $3/2<x<\infty$;
concave down: $-\infty<x<3/2$;
inflection point: $(3/2,11/2)$.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).

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{\bf 2}. (hard) 
Horiz. Asymp.: none. Vertical Asympt.: none.
decreasing: $-\infty<x<1$; increasing: $1<x<\infty$ ;
local max: none; local min: $(1,4)$; concave up: $-\infty <x<0$ AND $2/3<x<\infty$;
concave down: $0<x<2/3$;
inflection points: $(0,5)$ (horizontal inflection point!) and $(2/3,119/27)$.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).
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{\bf 3}. (warm-up-i) 
Horiz. Asymp.: $y=0$ (on both ends!). Vertical Asymp. $x=1$.
decreasing: $-\infty<x<1$ AND $1<x<\infty$ ; increasing: never.
local max: none; local min: none; concave up: $1<x<\infty$ ;
concave down: $-\infty<x<1$;
inflection points: none.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).

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{\bf 3}. (warm-up-ii) 
Horiz. Asymp.: $y=0$ (on both ends!). Vertical Asymp. $x=2$.
decreasing: $2<x<\infty$ ; increasing: $-\infty<x<2$ ; 
local max: none; local min: none; 
concave up: $-\infty<x<2$ AND $2<x<\infty$ ;
concave down: never ;
inflection points: none.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).
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{\bf 3}. (warm-up-iii) 
Horiz. Asymp.: $y=0$ (on both ends!). Vertical Asymp. $x=-1$.
decreasing: never; increasing: $-\infty<x<-1$ AND $-1<x<\infty$; 
local max: none; local min: none; 
concave up: $-\infty<x<-1$ ;
concave down: $-1<x<\infty$ ;
inflection points: none.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).


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{\bf 3}. (moderate-i) 
Horiz. Asymp.: $y=1$ (on both ends!). Vertical Asymp. $x=1$.
decreasing: $-\infty < x < 1$ AND $1<x<\infty$ ; increasing: never.
local max: none; local min: none; 
concave up: $1<x<\infty$
concave down: $-\infty<x<1$
inflection points: none.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).



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{\bf 3}. (moderate-ii) 
Horiz. Asymp.: $y=2$ (on both ends!). Vertical Asymp. $x=-1$.
increading: $-\infty < x < -1$ AND $-1<x<\infty$ ; decreasing: never.
local max: none; local min: none; 
concave up: $-\infty<x<-1$
concave down: $-1<x<\infty$
inflection points: none.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).

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{\bf 3}. (hard-i) 
Horiz. Asymp.: $y=0$ (on both ends!); Vertical Asymp. $x=-2$ AND $x=2$;
increasing: never; decreasing: $-\infty < x < -2$ AND $-2<x<2$ AND $2<x<\infty$ ; 
local max: none; local min: none; 
concave up: $-2<x<0$ AND $2<x<\infty$
concave down: $-\infty<x<-2$ AND $0<x<2$.
inflection points: $(0,0)$.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).

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{\bf 3}. (hard-ii) 
Horiz. Asymp.: $y=0$ (only from the right!); Vertical Asymp.: none;
increasing:$-\infty < x< 1/2$ ; decreasing: $1/2<x<\infty$;
local max: $(1/2,1/(2e))$; local min: none; 
concave up: $1<x<\infty$
concave down: $-\infty<x<1$
inflection points: $(1,1/e^2)$.
(Sorry, I am too lazy to prepare a diagram on the computer or to scan).

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{\bf 4}. (war-up) $R$ is decreasing at a rate of $5$ Ohms per second.
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{\bf 4}. (moderate-i) 
$R$ is decreasing at a rate of $59/36$ Ohms per second.
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{\bf 4}. (moderate-ii) The radius is increasing at a rate of $1$ centimeter per second.

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{\bf 4}. (moderate-iii) 
The bottom of the ladder is moving away from the wall
at a speed of $4/3$ feet per second.

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{\bf 4}. (hard)
The water level is rising at a speed of ${{5} \over {24}}$
meters per second.
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{\bf 5}. (warm-up) $1.02$.
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{\bf 5}. (moderate-i) $(\sqrt{2}/2) \cdot (1.01)$
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{\bf 5}. (moderate-ii) $3{{1} \over {108}}$
or ${{325} \over {108}}$.
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{\bf 5}. (moderate-iii) $2+.03/e^2$.
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{\bf 6}. (warm-up)
$x_2=2$.
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{\bf 6}. (moderate-i)
$x_2=4/3$, $x_3=20/63$.
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{\bf 6}. (moderate-ii)
$x_2=95/48$, $x_3=1152220351/532329100$.
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{\bf 7}. (warm-up) It is continuous in $[0,1]$ and differentiable in $(0,1)$
(every polynomial is!, everywehere!). $c=0.5$.
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{\bf 7}. (moderate) It is continuous in $[0,1]$ and differentiable in $(0,1)$;
$c=(3.2)^{1/4}$.
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{\bf 7}. (hard) $\ln x$ is continuous and differentiable whenever $x>0$, in 
particular in the interval $[e^2,e^3]$; $c=e^2(e-1)$.
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{\bf 8}. (warm-up) min value is $1$; max value is $2$.
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{\bf 8}. (moderate)
min value is $0$; max value is $49$.
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{\bf 9}. (warm-up-i) $10$ and $10$.
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{\bf 9}. (warm-up-ii) The maximal possible area is $1250$ square feet,
the dimensions are $25$ by $50$ feet.
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{\bf 9}. (medium-i) 
$(500)^{1/3} \times (500)^{1/3} \times 2 \cdot (500)^{1/3}$.
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{\bf 9}. (medium-ii) $(2,\sqrt{2})$.
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{\bf 9}. (hard) radius is $(150)^{1/3}$ inches and height is
$2 \cdot (150)^{1/3}$ inches.
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{\bf 9}. (hard) 
$(550)^{1/3} \times (550)^{1/3} \times (1000)/(550)^{2/3}$ cm.
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{\bf 10}. (warm-up) $x^2+3x+C$ 
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{\bf 10}. (moderate-i) 
$$
f(x)= {{x^2} \over {2}}- {{1} \over {x}} +C
$$ 
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{\bf 10}. (moderate-ii) 
$$
f(x)=-3\cos x + 4 \sin x + e^x +C
$$ 
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{\bf 10}. (hard-i) 
$$
f(x)={{3x^2} \over {2}} +5\sin^{-1} x +C
$$ 
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{\bf 10}. (hard-ii)
$$
f(x)=3{{x^2} \over {2}} + 5 \tan^{-1} x +C
$$ 
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{\bf 11}. (warm-up)  $f(x)=x^2$.
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{\bf 11}. (moderate-i)  $f(x)=2x^2+2/x^3+2$.
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{\bf 11}. (moderate-ii) $f(x)=-\cos x + 3$.
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{\bf 11}. (hard-i) $f(x)=x^4/12+x^2/2+2x/3-1/4$.
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{\bf 11}. (hard-ii) $f(x)=x^4/24-x^3/6+5x^2/4-x/6+1/24$.





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