MA 123 – First Midterm
Instructions: You must complete 8 of these problems for full credit. Please answer them on the
paper provided and staple together before handing in. You do not need to do them in order.
1.
Consider the functions: f (x) = 2x + 1 and g(x) = 3x 2 .
a. Evaluate: f (g(1)) and f (g(x))
b. Evaluate: ( g ! f ) (1) and ( g ! f ) (x)
!#a. 7, 6x 2 +1
Solution: "
2
2
#$b. 27, 3(2x + 1) or 12x + 12x + 3
2.
State the domain of the function.
f (x) =
x+8
x!3
Solution: x > 3 or (3, !)
3.
Write the equation of the line passing through the points (2, 5) and
(!3,1) .
" y ! 5 = 45 ( x ! 2 )
$
Solution: # y ! 1 = 45 ( x + 3)
$ y = 4 x + 17
5
5
%
4.
Evaluate:
log 5 125 + log 5 5 + log 5 1 + log 5 5 3
Solution: 7
5.
Solve for the exact value of x:
Solution: x =
6.
e3x = 5
ln 5
or ln 3 5
3
If the domain of y = sin x is restricted to #$ ! "2 , "2 %& , sketch the graph of its inverse.
Solution:
7.
Evaluate the following:
2"x
lim 2
x!2 x + 4x " 12
Solution: factor to –(x-2)/((x-2)(x+6)) then cancel and substitute to get -1/8.
8.
If h(x) = sqrt(9 – x2), determine whether each of the following limits exists.
a. limx→3 h(x)
b. limx→3 h(x)
c. limx→−3 h(x)
d. limx→−3 h(x)
e. Sketch the graph of h.
+
−
+
−
Solution: a. DNE
b. 0
c. 0
d. DNE
e.
9.
Consider the nth degree polynomial function f whose leading term is axn Use limits to describe the
end behavior of f in each case:
a. a > 0 , n even
b. a < 0 , n even
Solution: a. lim f (x) = #
x!"#
lim f (x) = "
x!"
10.
b. lim f (x) = "#
x!"#
lim f (x) = #"
x!"
State the intervals of continuity for the function:
c. a > 0 , n odd
c. lim f (x) = "#
x!"#
lim f (x) = "
x!"
f (x) =
d. a < 0 , n odd
d. lim f (x) = #
x!"#
lim f (x) = #"
x!"
x
x!2
Solution: Domain is x >= 0 and x != 2, continuous over whole domain except for endpoint x = 0,
so (0, 2), (2, ∞)