Math 131.
Chapter 1 Test V2 - Autumn 15
Name.
Instructions. Complete each of the following nine questions. Show all appropriate work and
do your best. You may use a scientific nongraphing calculator.
1. For this question, let f (x) = x2 − 3x + 5.
(a) (4 pts) Find the slope of the secant line through the points (3, f (3)) and (6, f (6)).
(b) (6 pts) Find the slope of the secant line through the points (3, f (3)) and (3+∆x, f (3+∆x))
where ∆x 6= 0. Simplify your answer.
2. Use the rectangles as sketched below to estimate the area under the curve y = 0.3x2 + 0.5
for 2 ≤ x ≤ 3; note that each rectangle has a base width of 1/2.
(a) (4 pts) Find the total area of the shaded rectangles.
y
6
4
2
x
−1
1
2
3
4
5
−2
(b) (1 pt) Does the area in (a) overestimate, underestimate or provide an exact value for the
requested area under the curve?
3. (5 pts) Fill in the blanks to complete the definition of a limit:
Let f be defined on an open interval containing c (except possibly at
). The statement
lim f (x) = L
x→c
means for each ε > 0 there exists a δ > 0 such that if
<
<
then
<
.
4. (5 pts) A manufacturer designs a ball bearing to have a volume of 332 cubic millimeters
with a possible error of ±7 cubic millimeters. That is, the ball bearing’s volume can vary
between 325 cubic millimeters and 339 cubic millimeters. How can the radius vary?
4
(Leave answers in exact form, and recall the volume of a sphere is V = πr3 ).
3
5. (5 pts) Let lim f (x) = 7, lim g(x) = 5, lim h(x) = −6. Use basic properties of limits to
x→a
x→a
x→a
find following limits if they exist. Write DNE if the limit does not exist.
1. lim f (x) + g(x)
x→a
2. lim f (x)h(x)
x→a
f (x)
x→a g(x)
3. lim
4. lim
p
3
h(x)
5. lim
p
h(x)
x→a
x→a
6. (5 pts) Use analytic techniques to evaluate the limit lim √
x→3
x2 − 9
. Show all work.
x+6−3
7. (5 pts) A function f is graphed below. Use the graph to answer the following questions;
write DNE for “does not exist.”
(a) Find lim f (x)
x→−4
(b) Find lim − f (x)
x→−1
5 y
4
3
2
1
−5 −4 −3 −2 −1
−1
(c) Find lim + f (x)
x→−1
(d) Find lim f (x)
x→−1
(e) For what value(s) c, does f have a
removable discontinuity at x = c?
−2
−3
−4
−5
x
1
2
3
4
5
8. (a) (1 pt) What is the value of the limit lim
x→0
sin x
?
x
1
.
x→0 cos(9x) + 1
(b) (2 pts) Find lim
cos(9x) − 1
. Show all work.
x→0
4x2
(c) (2 pts) Use analytic techniques to find lim
9. For this problem, let f (x) =
x2 − 4x + 3
.
x2 − 7x + 12
(a) (4 pts) Simplify f if possible, and find the domain of f .
(b) (2 pts) Write the equation for each vertical asymptote of the graph of f .
(c) (2 pts) For each vertical asymptote x = c found in part (b), determine the limits lim− f (x)
x→c
and lim+ f (x).
x→c
(d) (2 pts) Find all values of c for which there is a “hole” in the graph of f above x = c.