Math 171
Exam II
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March 21, 2002
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Work all problems on the paper provided.
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Turn in your exam with your work.
SCHOLASTIC DISHONESTY WILL NOT BE TOLERATED.
1. (8 points each) Compute
any more.
(a) y =
(b) y =
dy
dx .
Once you solve for
dy
dx ,
you do not have to simplify your answers
√
5
3
4 − 7x + 2 + 54
x
x2 + 3x
x2 + 1
!3
(c) y 5 − 3x2 y 3 + 5x4 = 12
2
2. (8 points) Find the second derivative of y = e−x .
3. (6 points each) Compute the following.
2π x
=
x→∞
7
(b) D 50 cos(2x) =
(a) lim
sin(x − 1)
=
x→1 x2 + 5x − 6
(c) Give the exact value for lim
4. (8 points) If x = sec(θ) and y = tan(3θ 2 ) compute
dy
dx
5. (8 points) Find the equations(s) of the tangent lines where the function y = (x2 + 6)(x + 4) has
a tangent line that is parallel to 9x − y = 5.
√
6. (5 points) Use differentials to find an approximate value for the number 36.1.
7. (5 points) Solve for x.
log5 (ln x) = A
8. (8 points) Newton’s method says the formula for the n+1 approximation (i.e. next approximaf (An )
tion) is given by An+1 = An − ′
.
f (An )
(a) Show how this formula is derived.
(b) Newton’s method breaks down when f ′ (An ) = 0, since you can not divide by zero. What
does this mean, with respect to the function f (x), when this happens?
9. (8 points) Find the inverse function to y =
3x
.
5 − 2x
10. (8 points) Water is leaking out of an inverted conical tank at a rate of .01m3 /min. The tank has
height of 8m and the diameter at the top is 6m. Find the rate of change of the radius (located
at the water level of the tank) when the height of the tank is 3m. V = 31 πr 2 h