Math 121 Test 2
EF:
October 12, 2021
Name
1
2
3
4
5
Total
Directions:
1. No books, notes or running out of Macaroni and Cheese. You may
use a calculator to do routine arithmetic computations. You may not
use your calculator to store notes or formulas. You may not share a
calculator with anyone.
2. You should show your work, and explain how you arrived at your answers. A correct answer with no work shown (except on problems
which are completely trivial) will receive no credit. If you are not sure
whether you have written enough, please ask.
3. You may not make more than one attempt at a problem. If you make
several attempts, you must indicate which one you want counted, or
you will be penalized.
4. You may leave as soon as you are finished, but once you leave the exam,
you may not make any changes to your exam.
5. This test has 5 problems.
1. (20 points) If f (1) = 2, g(1) = 1, f 0 (1) = 3, g 0 (1) = 2,
(a) and s(x) = f (x) + g(x), find s0 (1).
(b) and p(x) = f (x) · g(x), find p0 (1).
(c) and q(x) =
f (x)
, find q 0 (1).
g(x)
(d) and h(x) = (f ◦ g)(x), find h0 (1).
2. (20 points) Find the derivatives of the following functions:
√
√
7
(a) f (x) = x3 + x sin x
x3 + 2
(b) f (x) = tan
x−x
(c) f (x) = arcsin(x) −
√
1 − x2 .
(d) f (x) = 2 sinh(3x) + 3 cosh(2x)
3. (20 points)
dy for y = x21 + 21x + 2121
(a) Find dx
dy for y = x(x21 )
(b) Find dx
(c) If f (x) = x2 cos(2x) find f 00 (0).
(d) If f (x) = e21x find f 2021 (x) (the 2021th derivative of f (x)).
4. (20 points)
(a) Find the slope of the tangent line to
x2 y + y 4 = 4 + 2x
at the point (−1, 1)
(b) At what point(s) on the ellipse
3x2 + y 2 = 21
is the tangent line at that point parallel to the straight line
y = −2x + 6?
5. (20 points)
(a) If f (x + h) − f (x) = h cos h + h2 x2 + 3hx, find f 0 (−1).
(b) A lovely bowl of petunias is dropped from a point 12 feet above
the ground, so that it will land 4 feet from the base of a 12 foot
lamppost. At what rate is the shadow of the bowl moving along
the ground when
the bowl is 8 feet above the ground and falling
4. (15 points) A bowl of petunias is dropped from a point 12 feet above the
so that it will land 4 feet from the base of a 12 foot lamppost.
at 16 ft/sec. ground,
At what rate is the shadow of the bowl moving along the ground when
the bowl is 8 feet above the ground and falling at 16 ft/sec.
0
