Fall 2009 Math 151
2.
Week in Review VII
3.
ourtesy: David J. Manuel
(overing 3.8, 3.9, 3.10)
4.
1
1.
Setion 3.8
Find and simplify the rst and seond
derivatives of the following:
(a)
(b)
2.
3
2
f (x) = sin x
1
y= 2
x +1
1.
The graph of f, f ′ , and f ′′ are shown
below. Label whih is whih. Explain
your reasoning.
2.
3.
3.
4.
2
1.
Find the ftieth (50th) derivative of
f (x) = cos 2x.
4.
1
Given f (x) = , nd a formula for the
x
nth derivative (f (n) (x))
Setion 3.9
5.
Find an equation of the line tangent
to the urve parametrized by x =
sec θ, y = tan θ at the point where
π
θ= .
3
1
Find an equation of the line tangent to
the urve given by x = t2 +2t, y = t3 −t
at the point (3,0).
Find the points on the urve x = 4t −
t2 , y = 1 + t2 where the tangent line is
horizontal or vertial.
The urve x = t3 −4t, y = t2 rosses itself at the point (0, 4). Find the pointslope equations of both tangent lines.
Setion 3.10
Oil spilled from a broken tanker
spreads in a irular pattern whose radius inreases at a onstant rate of 0.6
m/se. How fast is the area of the spill
inreasing when the radius is 10m?
A man sitting on a pier 3m above water
pulls on a rope attahed at water level
to a boat at the rate of 0.5 m/s. At
what rate is the boat approahing the
pier when 5m of rope remain?
A amera is positioned 800m from a
roket launh pad. If the roket rises
vertially at 300 m/se, how fast is the
angle of elevation of the amera hanging when the roket is 1000m above
ground?
A man 6ft tall is walking at a rate of 3
ft/se toward a streetlight 18ft high.
a) How fast is the length of his shadow
hanging when he is 12ft from the
streetlight?
b) How fast is the tip of his shadow
moving at that instant?
A feed trough 4m long has a ross setion that is an isoseles triangle with a
base of 1.5m at the top and a height of
1m. If water pours into the trough at
a rate of 0.25 m3 /min, how fast is the
depth of the water hanging when the
depth is 0.4m?