Math 115 Spring 2014
Written Homework 9
Due Friday, April 18
Instructions: Write complete solutions on separate paper (not spiral bound).
If multiple pieces of paper are used, THEY MUST BE STAPLED with your
name and lecture written on each page. Please review the Course Information
document for more complete instructions.
1. Determine the exact value of the following expressions using the basic properties of logarithms:
(a) log10 (8) + 3 log10 (5)
log5 36
log5 6
1
(c) ln 3
e
(b)
2. Suppose logb x = 3, logb y = −4, logb z = 5, and logb w = 7. Use the properties of
logarithms to expand each expression and determine its value.
(a) logb (xy)
(b)
logb
x4 y 5
.
zw3
3. Solve each of the following equations:
(a) e3x = 2507
(b) 7x = 42x−1
(c) log3 (x − 4) + log3 7 = 2
(d) log2 (x − 1) − log2 (x + 1) = 3
(e) ln(x − 1) + ln 6 = ln(3x)
x2 + 1
.
x−3
(a) Determine where f is continuous. (Remember - rational functions are continuous on
their domain.)
4. Let f (x) :=
(b) Consider the limit lim+
x→3
steps to evaluate this limit.
(c) Consider the limit lim−
x→3
this limit.
x2 + 1
1
. Use the sequence 3 + and the appropriate algebraic
x−3
n
x2 + 1
. Choose an appropriate sequence and use it to evaluate
x−3
5. Use limits to determine the horizontal asymptotes of the graphs of the following rational
functions, if any. If the graph does not have any horizontal asymptotes, you must clearly
state why.
x2 − 16
(a) f (x) := 2
3x − 11x − 4
(b) g(x) :=
(c) h(x) :=
3x3 − 4x − 4
2x2 − 8
x2
x+2
+ 2x − 3
6. Determine the following limits and justify your answer. State what each limit tells you
about the graph of the function.
x+2
(a)
lim − 2
x→−3 x + 2x − 3
(b)
x2 − 16
lim
x→4 3x2 − 11x − 4
7. For each rational function, determine the domain, end behavior, and intercepts. Then
draw a rough sketch of the graph of the function. Be sure to clearly label all intercepts and
asymptotes on your graph. You may refer to work you’ve done in previous problems where
appropriate.
x2 − 16
(a) f (x) := 2
3x − 11x − 4
(b) h(x) :=
x2
x+2
+ 2x − 3
8.
(a) If lim− f (x) = 5 and lim+ f (x) = 5, what do you know about lim f (x)? What do you
x→1
x→1
x→1
know about f (1)? Explain.
(b) If lim+ f (x) = 8, but lim f (x) does not exist, what do you know about lim− f (x)?
x→2
x→2
x→2
Explain.
(c) If lim f (x) = ∞, lim f (x) = 3, and lim+ f (x) = ∞, what do you know about any
x→−∞
x→∞
x→1
horizontal and vertical asymptotes of the graph of y = f (x)? Explain.
9. Sketch the graph of a rational function with all of the following limits and values:
lim f (x) = lim f (x) = −3
x→∞
x→−∞
lim f (x) = ∞
x→3
lim f (x) = ∞
x→−2−
lim f (x) = −∞
x→−2+
Only intercepts are (−4, 0), (0, −2), (1, 0), (5, 0).