Calculus I Test 2. March 24, 2003 NAME:_____________________________________________
Justify your answers. You may not use calculators, books, or notes. Do your
own work. CIRCLE answers. Failure to follow these instructions will result in
the loss of some or all points.
1.) A street light is mounted at the top of a 15-ft tall pole. A man 6ft. tall
walks away from the pole with a speed of 5ft/s along a straight path. How fast
is the tip of his shadow moving when he is 60 ft. from the pole?
2.) Let g(x) = x33x2:Find the absolute maximum and absolute minimum
values of g(x)(and where they occur) on the interval [1;3].
3.) Let g(x) = 1 9x6x2x3:Carefully sketch the graph of g, including
the coordinates of all local extrema and in‡ection points, and showing concavity.
4.) Let g(x) = x
1+x2:Carefully sketch the graph of g, including the coordi-
nates of all local extrema and in‡ection points, and showing concavity.
5.) Find the dimensions of the rectangle of largest area that has its base on
the x-axis and its other two vertices above the x-axis and lying on the parabola
y= 9 x2.
6.) A box with a square base is to be constructed. It is to have total volume
of 100 ft3. The material for both the top and bottom will cost $5=ft2while the
material for the lateral sides will cost $2=ft2. What should be the dimensions
of the base of the box so the cost will be minimal?
7.) Use L’Hospital’s rule to …nd the following limits:
(A) lim
x!1
ln x
x31
(B) lim
x!0+(1 + 3x)1=x
8.) Let g(x)=3x2=3x: Find all xand yintercepts, all local maxima
and minima, in‡ection points and any other signi…cant points and features and
carefully sketch the graph, labeling all signi…cant points and features.
EXTRA CREDIT: Two carts, Aand B, are connected by a rope 39 ft. long
that passes over a pulley P(see the …gure). The point Qis on the ‡oor 12 ft.
directly beneath Pand between the carts. Cart Ais being pulled away from Q
at a speed of 2ft/s. How fast is cart Bmoving toward Qat the instant when
cart Ais 5ft. from Q?
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