Math 121 Test 3 - A 2019
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Math 121 Test 3 - A
November 19, 2019
1. (15 points)
(a) lim
(b) Find the value of a so that
x(l)x ln ╱ 1 + 、 = e2
(c) Find the number c guaranteed by the Mean Value Theorem for the function f (x) =x(1) on the interval [1, 3].
(a) 2^3 (b) ← (c) ^2 (d)
(e) ^3 (f) 2 (g) 3^2 (h) none of these
2. (15 points)
(a) Sketch the graph of a continuous function f (x) that satisfies all of the following:
● f\ (x) > 0 only when 1 < x < 3 or x > 8
● f\ (1) does not exist
● f\ (x) = 0 only when x = 3 or x = 8
● f\ (x) is increasing on the interval (5, &)
● f\ (x) is decreasing on the intervals (一&, 1) U (1, 5)
(b) If a continuous function g(x) has
g\ (3) = 0 and g\\ (x) = x 一
20
^x2 + 16
Then, by the Second Derivative Test, g(x) must have must have
a(n) at x = 3.
3. (15 points)
e…
(a) Let g(x) = ln(ln t) dt, which of the following is g\ (x):
2
(a)
(e) x
(b)
(f) ex ln(x)
(c) ex
(g) ln(x)
(d) ln(ln x)
(h) none of these
(b) Newton’s method is applied to f (x) = x4 + x3 一 20 = 0 starting at x1 = 2, what is the value of x2 ?
(a) 1.909 (b) 2.012 (c) 1.903 (d) 2.231 (e) 1.877 (f) 1.986 (g) 2.101 (h) 1.887 (i) none of these
(c) Below is the graph of a particle velocity (in meters/second). Find the displacement and distance traveled for 0 ≤ t ≤ 4.
1
1 2 3 4
-1
4. (15 points) Olivia has just baked a batch of oatmeal raisin cookies, which are Nina’s favorite. Unfortunately, Nina is on a sailboat 12 miles away from shore and Olivia is 17 miles up the coast. If Nina can sail at a rate of 5 miles/hour and can run at a rate of 13 miles/hour, to which point P on the shore should she sail?
12 miles
Nina
17 miles
5. (15 points)
c c b
(a) If f (x) dx = 2 and f (x) dx = 5 , what is 2f (x) dx?
a b a
(a) 一14 (b) 一7 (c) 一6 (d) 一3
(e) 3 (f) 6 (g) 7 (h) none of these
1
(b) Compute _1 ╱ 1 一 ^1 一 x2 、 dx
2
(c) Compute 1 ╱8x3 + 3x2 、 dx
6. (15 points)
(a) Compute 6x5 (x6 + 17)22 dx
(b) Compute dx
e4x
(c) Compute e8x + 1 dx
7. (10 points) True or False.
a) If f\\ (1) < 0 then f (x) has a local maximum at T F
x = 1.
b)
If f (x) has an inflection point at x = 2, then f\ (x) must have a relative extreme point at x = 2.
T F
c) If f\\ (3) = 0 then the concavity of f changes at T F
x = 3.
d)
e)
If f (x) is decreasing when x < 4 and f (x) is in-
creasing when x > 4, then f (x) has a relative
minimum at x = 4.
If f (x) and f\ (x) are both continuous and dif-
ferentiable on [1, 3] and f (1) = 1, f (2) = 2 and
f (3) = 3 then there exists a c between 1 and 3
such that f\\ (c) = 0
2022-11-19