\begin{homework}\
\label{homework:5}
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\exercise %1.
Compute the following values using the tangent line approximation with an
appropriate $f(x)$, and in each case see how close you are by computing the
true value by calculator or computer:

\noindent
(a) $(2.1)^2$, (b) $1/1.1$, (c) $1/0.99$, (d) $1/101$, (e) $\sqrt{4.01}$,
(f) $\sqrt{26}$, (g) $\sqrt{24}$, (h) $\root 3 \of{8.012}$.

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\exercise %2.
What is the $y$-coordinate of the point on the circle $x^2+y^2=25$ whose
$x$-coordinate is 3.08?  Get an approximate answer using the tangent line
approximation.

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\exercise %3.
One liter is 1.0567 quarts.  Use the tangent line approximation to answer
the question: One quart is how many liters?

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\exercise %4.
You want to \index{ tile} tile a square floor, which you guess to be 15 feet by 15 feet.
Based on your guess, you buy 225 square feet of tile.  But after you've
used up all of the tile, you still have 5 square feet not covered with
tile.  What are the actual dimensions of the square floor?  Use the tangent
line approximation.

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\exercise %5.
Roughly speaking, an \index{ acre} acre is about 200 feet by 200 feet.  More precisely,
an acre is 43560 square feet.  What are the dimensions of an acre of land
in the shape of a square?  Use the tangent line approximation.

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\exercise %6.
If a right triangle has legs 6 and 8, then the smallest circle in which one
can put this right triangle has area $25\pi$ (its diameter is the
hypotenuse of the triangle, which is 10).  Suppose that one leg of the
right triangle is known to be exactly 6, but the other leg is known to be 8
with an error of $\pm h$.  What is the area of the smallest circle
containing the triangle?  Give the error estimate for this area in terms of
$h$.

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\exercise %7
You measure a cubic container, and find it to be about 10 cm on a side.
From this you conclude that it holds 1000 cc (cubic centimeters).  However,
your measurement is accurate only to within $\pm 0.1$ cm.  In other words,
you can only be sure that a side of the cube is between 9.9 and 10.1 cm.
Use the tangent line approximation to estimate the error in your value of
1000 for the volume.  Then give the {\it percent error} in your measurement
of the length of a side and the {\it percent error} in your value for the
volume.

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\exercise %8.
The distance in feet that an object falls in $t$ seconds is given by the
formula $y=16t^2$.  (a) If an object is dropped from a height of 400 ft,
how long will it take to fall to the ground?  (b) If your value of 400 ft
for the initial height of the object is accurate only to $\pm 1\%$, how
accurate is your answer to part (a)?  Determine both the \index{ absolute error} absolute error and
the \index{ percent error} percent error.  (c) If your value of 400 ft is accurate to $\pm p\%$,
then how accurate (as a percent error, expressed in terms of $p$) is your
answer to part (a).

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\exercise %9
Suppose that you know that a circle has area $A$ with an error of $\pm
1\%$.  (a) Find a formula for the radius $r$ in terms of the area $A$.  (b)
If your value for $A$ is off by $\pm 1\%$, how far off (as a percent error)
is your value for $r$ obtained using the formula in part (a)?  (c) What if
your value for $A$ is off by $\pm p\%$?

\smallskip

\exercise %10.
If you know that the hypotenuse of a right triangle is exactly 10, and one
of the legs is $6\pm 0.20$, then what is the other leg?  Include an error
estimate in your answer.

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\exercise %11.
The diameter of a sphere is measured with an error of $\pm p\%$, and this
measurement is used to calculate the volume of the sphere.  What is the
percent error in the value computed for the volume?

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\exercise %12.
What is the permissible percent error in measuring the side of a cube if
you want at most a $\pm p\%$ error in your value for (a) the surface area
of the cube, (b) the volume of the cube.

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\exercise %13
The distance around a sphere's \index{equator} equator is measured with an accuracy of $\pm
p\%$.  This value is used to compute the sphere's (a) radius, (b) volume,
(c) surface area.  In each case, determine the percent error.

\smallskip

\exercise %14.
A spherical container is filled with water, and the volume of water the
container holds is measured to within $\pm p\%$.  This information is then
used to compute the surface area of the spherical inside surface of the
container.  What is the percent error in the value obtained for this
surface area?

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\exercise %15.
The height of a \index{ cylinder} cylinder is known exactly, but its diameter is measured
only to within $\pm p\%$.  This information is then used to compute (a) the
volume of the cylinder, (b) the circumference of the top of the cylinder,
(c) the lateral surface area of the cylinder (top and bottom not included).
In each case determine the percent error.

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\exercise %16.
Use the tangent line approximation to find an $x$ near 2 such that
$x^5+x^2=35$.  (Hint: Replace the left hand side by its tangent line
approximation at $x=2$ and solve for $h$.)

\smallskip

\exercise %17.
A company is making solid cube \index{ paperweights} paperweights.  The material used to make
the cubes costs 2\cnt~per cubic cm, while the paint used to paint the
6 square faces costs 0.5\cnt~per square cm.  If the paperweight is
$3\,{\rm cm}\times 3\,{\rm cm}\times 3\,{\rm cm}$, then we calculate
that the total cost of the material is $54\cnt+27\cnt=81\cnt$.  If you
want the total cost to be 75\cnt~rather than 81\cnt, by how much do
you have to decrease the side of the paperweight?  Use the tangent
line approximation. 

\end{homework}


