Math 105a
Final Exam
Solutions
1. (a) If f (x) =
sin x
then
x
f (x) =
(b) If g (t) = ln(sin(t2 ) then
t3 + 7 t
dt =
t4
(c)
(d)
g (t) =
1
+ 7t3
t
/4
x cos x sin x
x2
cos(t2 ) 2t
= 2t cot(t2 )
sin(t2 )
dt = ln |t|
7
+C
2t2
/4
= sin (/4) sin 0 =
cos x dx
Math 105
Quiz 1
9/14/11
SOLUTIONS
1. Consider f (x) =
3 + x2 , for x < 0
sin x,
for 0 x < 2
(a) Graph f is shown below.
(b) The domain of f is (, 2 ).
6
(c) The range of f is [1, 1] (3, ).
4
2
-2
2
4
6
2. The graph of f is shown below along with graphs fo
Math 105
Quiz 3
10/15/12
Solutions
Read directions carefully and show your work. Partial credit will be assigned based upon the correctness,
completeness, and clarity of your answers.
1. Consider y (x) = ee ln x + 3 log2 x (e2 )x 7 sin x + cos( ), nd
1
1
Math 105
Quiz 2
9/21/12
Solutions
1. Using the graph of f below we can identify features such as stationary points, local extrema, and
inection points for the graph of f .
f has stationary points at x = 2, 1, and 3 since
f (x) = 0 at x = 2, 1, and 3.
stat
Math 105
Quiz 4
10/26/12
Name
Read directions carefully and show your work. Partial credit will be assigned based upon the correctness,
completeness, and clarity of your answers.
1. (3 pts) Find f (t) if f (t) = t2 ln(sin t).
2. (3 pts) Consider y (x) =
3
Math 105: Review for Final Exam, Part II - SOLUTIONS
1. Consider the function f (x) = x3 ln x on the interval [1/e, e2] .
(a) Find the x- and y -coordinates of any and all local extrema and classify each as a local
maximum or local minimum.
f (x) = 3x2 ln
Math 105: Review for Exam II - Solutions
1. Find dy/dx for each of the following.
(a) y = x2 + 2x + e2 + e2x + ln 2 + ln (2x) + arctan 2
dy
1
= 2x + (ln 2)2x + 2e2x +
2
Note that e2 , ln 2, and arctan 2 are constants.
dx
2x
(b) y = x arctan (5x)
1
1
arcta
Math 105: Review for Exam I - Solutions
Graphical Relationships Between f , f , and f
f
f
f
positive
X
X
1. Let f (x) =
negative
X
X
increasing
positive
X
decreasing
negative
X
concave up
increasing
positive
concave down
decreasing
negative
x+1
x+1
. Note