Math 19A
MIDTERM EXAM 2
Math 19A
11/19/2010 , Dr. Frank Bäuerle, UCSC
Note: Show your work. In other words, just writing the answer,
even if correct, may not be sufficient for full credit. Scientific calculators are allowed, but no programmable and/or graphing calculators.
And please put away your cell phones and other electronic devices,
turned off or in airplane mode.
Your Name:
Your TA’s Name:
Your Section Time:
Problem 1:
out of 20
Problem 2:
out of 10
Problem 3:
out of 10
Problem 4:
out of 10
Problem 5:
out of 15
Problem 6:
out of 20
Problem 7:
out of 15
Total:
out of 100
Good luck and have a relaxing weekend!
1
1. (20 points) Compute the derivatives
(a) y = ln(1 +
√
dy
of the following functions:
dx
x)
(b) y = earctan(πx)
(c) y = ln(x2 + 2x ) + tan(tan x) + etan(2λ) , where λ is a constant
(d) y = xcosh(x) , where x > 0 . Hint: Use logarithmic differentiation.
2
2. (10 points) Find the equation of the tangent line to the curve
ey = xy
2
at the point ( e2 , 2).
(Extra Credit, 3 points) For what values of x does the equation ey = xy
have real solutions ? Hint: The following is a sketch of the curve.
3
3. (10 points)
Use the linear approximation
for a = 32 to the function
√
√
5
5
f (x) = x to approximate the number 32.16.
4. (10 points) Show that sinh x ≈ x for x VERY close to 0. In other words,
show that the linear approximation to the function f (x) = sinh x for
a = 0 is given by L(x) = x.
4
5. (15 points) Find the equations of both the tangent lines to the ellipse
(see picture) given by
x2 y 2
+
=1
81
9
that pass through the point (27, 3).
5
6. (20 points)
(a) Carefully state the Extreme Value Theorem.
2
2
(b) Let f (x) = x 3 − x + 4. Find all critical numbers of f (x).
3
(c) Let g(x) = 3x5 − 5x3 + 1. Find the following:
i. Find the critical numbers of g(x).
ii. Give the absolute maximal and absolute minimal values of g on
the interval [−1, 2]
6
7. (15 points) After heavy rainfall it is observed that the depth of water in
a conical reservoir of radius 10 meters and height 30 meters is increasing
at 5 meters/hour when the depth is 5 meters. How fast is the water
filling the reservoir at this instant? Recall that the volume of a cone is
1
given by V = πr2 h. Draw a picture and show all your work.
3
7