Show all work on test paper. Work on scratch paper will not be ...

MATH 150A
F<strong>all</strong>, 2008
Test 1
Form A
Name _____________________________
(print)
SECTION ___________________
<strong>Show</strong> <strong>all</strong> <strong>work</strong> on test paper. Work on scratch paper will
not be graded.
Pledged: ___________________________
I have neither given nor received aid on this test, nor will I discuss
it with anyone until <strong>all</strong> students have taken the test.
Students: DO NOT WRITE BELOW THIS LINE.
INSTRUCTORS: RECORD THE NUMBER OF POINTS MISSED ON
EACH QUESTION IN THE APPROPRIATE LOCATION BELOW.
Problem 1 Problem 2 Problem 3 Problem 4
Problem 5 Problem 6 Problem 7 Problem 8
Problem 9 A, B, C Problem 9 D, E Problem 10 Problem 11
Problem 12
SCORE: _________________

[8 points] 1. True/False: Circle T for each statement that is always true, otherwise circle F.
T F a) If
sin x =
1
, then
2
3
x .
2
cos =
T F b) If f ( x)
≤ g(
x)
for <strong>all</strong> x, then lim f ( x)
≤ lim g(
x)
provided the limits exist.
x→a
x→a
T F c) The function f (x) = sin x is an odd function.
2
T F d) The function f ( x)
= cos( x − 3x)
is an algebraic function.
T F e) If f ( −1)
= −1
and f ( 1) = 1, then f ( c)
= 0 for some number c.
T F f) The rational function f (x) = x 2 −1
x −1 = x +1
T F g) The equation sin x = cos x has a solution in the second quadrant.
T F h) If lim f ( x)
= lim f ( x)
, then f is continuous at x = a .
−
x→a
+
x→a
[9 points] 2. Fill in each blank to make a true statement.
a) In slope intercept form an equation of the line through (2,3) and (0,7) is _______________________.
b) The solution set in interval notation of 2 x + 3 < 5 is __________________________________.
c) The domain of the tangent function is ___________________________________________.
d) The range of the secant function in interval notation is ________________________________.
e) The function
x + 3
f ( x)
=
has vertical asymptote(s) when x = _____________.
2
x − x − 6
f) The slope of a line with y-intercept 3 that is par<strong>all</strong>el to the line 2 x + 3y
+ 5 = 0 is _____________.
[4 points] 3. For the two functions f ( x)
= 3x 2 + 6 and g ( x)
= 7x
− 2 , find the following:
a) f ( g(
x))
= (Simplify)
b) In interval notation the domain of f ( g(
x))
is .

[6 points] 4. Find the domain of the function
notation.
f ( x)
=
2
x − 6x
. Express the answer in interval
3x
+ 2
Domain:
[7 points] 5. Find <strong>all</strong> values of x in [ 0,2π ] satisfying 2 + cos 2x
= 3cos
x .
⎛
[8 points] 6. Use the ε−δ definition of limit to prove that lim 1− x ⎞
⎜ ⎟ = 2.
x→−2 ⎝ 2⎠
[6 points] 7. Use the Squeeze Theorem to show that lim x 2 ⎛
2 + cos π ⎞
⎜ ⎟ = 0 .
x→0 ⎝ x ⎠

[8 points] 8. Determine if lim f (x) exists.
x→5
⎧ 2x − 4 if x ≤ 5
⎪
f (x) = ⎨ x − 5
if x > 5
⎩
⎪
2x −1 − 3
[16 points] 9. Evaluate each of the following limits:
(a)
lim
x→ 5π 3
tan x
x
sin 2 x
(b) lim
x→0 1− cos x
(c)
⎛
⎜ 1
lim −
x→0⎜
⎝
x
1−
x
x
2
⎞
⎟
⎟
⎠
(d) lim
x→7
1
7
− 1 x
7 − x
(e)
⎛
lim⎜
−
x→0
⎝
1
x
1 ⎞
− ⎟
x
⎠

[12 points] 10. Sketch the graph and give the domain and range for each of the following functions.
For the trig functions show the curve on −π,π ( ).
(a) y = 1+
x − 2
(b) y = cos x
(c) y = cot x
Domain ___________ _____________ ____________
Range ____________ _____________ ____________
[8 points] 11. (a) A function f (x)
is continuous at x = a if ____________________________.
⎧ b + cx, x > 2
⎪
(b) Let f (x) = ⎨ 3, x = 2. Mathematic<strong>all</strong>y determine values of b and c that will make f
⎪
⎩ c + bx, x < 2
continuous at x = 2 .
3
[8 points] 12. <strong>Show</strong> that the equation x − x −1
= 0 has a solution in the interval (1,2). What
theorem did you use