Fall 11

Math 124, Section E, Fall 2011, Midterm II
November 22, 2011
Name
TA/Section
Instructions.
• There are 4 questions. The exam is out of 40 points.
• You are allowed to use one page of notes written only on one side of the sheet in your own handwriting.
Hand in your notes with your exam paper.
• You may use a calculator which does not graph and which is not programmable. Even if you have a
3
is exact, 0.7 is an approximation for the same number.)
calculator, give me exact answers. ( 2 ln
π
• Show your work. If I cannot read or follow your work, I cannot grade it. You may not get full credit
for a right answer if your answer is not justified by your work. If you continue at the back of a page,
make a note for me. Please BOX your final answer.
Question points
1
2
3
4
Total
1
dy
for the following:
dx
q
√
(a) (3 points) y = 1 + 1 + sin x
1. Compute
(b) (3 points) x = 4t + 1, y = e−5t
(c) (4 points) y = (1 + sin2 x)x
2
2. For the function
f (x) = 2x3 + 3x2 − 12x + 5
(a) (3 points) Find the critical points.
(b) (4 points) Determine if the critical points are local minimum, local maximum or neither.
(c) (3 points) Find the absolute minimum and absolute maximum for f (x) on the interval [−5, 5].
3
3. A curve is given by the equation
y 3 + 3xy 2 − 4x4 = 16.
(a) (5 points) Find the equation of the tangent line to this curve at the point (1, 2).
(b) (2 points) Approximate the value of a if (a, 1.95) is on the curve.
(c) (3 points) Is your approximation above more or less than the actual value. Explain your answer.
4
4. Two tanks, one shaped like a cylinder, the other like an inverted cone are connected by a pipe. The
cylindrical tank has radius 0.5 meters and a height of 3 meters. Initially it is full. The conical tank
has radius 0.4 meters and a height of 4 meters. Water is pumped from the cylindrical tank into the
conical tank. If the level of the water y in the cylindrical tank is falling at a rate of 0.1 meters per
minute, at what rate is the water level h rising in the conical tank when the water level is 1.5 meters
in the conical tank? Hint: The total volume of water in the system remains constant and the volume
of a cone is a third of base area times its height.
5

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