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Math 131 WIR, copyright Angie Allen
Math 131 Week-in-Review #3 (Sections 2.1, 2.2, and 2.3)
1. For the function f whose graph is shown below, find each of the following.
a) State the domain of f .
y
f(x)
8
7
6
5
4
3
2
b) Find f (1).
1
8
7 6 5
4 3
2 1
1
−1
c) Find lim f (x).
x→1
−2
−3
−4
−5
−6
−7
−8
d) Find lim f (x).
x→−3−
e) Find lim f (x).
x→6+
f) Find lim f (x).
x→6
g) Find f (6).
h) Find lim f (x).
x→2
2 3 4
5
6
7 8
x
2
Math 131 WIR, copyright Angie Allen
2. Sketch the graph of a function that satisfies the following conditions:
lim f (x) = −4, lim f (x) → ∞, and f (2) is undefined.
x→2+
x→2−
x2 + 8x + 15
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
x→−5 x2 + x − 20
3. Evaluate lim
4. Evaluate lim
x→1
x−1
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
(x − 1)2
3
Math 131 WIR, copyright Angie Allen
5. Given lim f (x) = 4, lim g(x) = −2, and lim h(x) = 0, find the following (if they exist).
x→6
x→6
x→6
a) lim [ f (x) + 4g(x)]
x→6
b) lim
x→6
g(x)
h(x)
c) lim x2 f (x)
x→6
6. If an object is launched into the air with a velocity of 50 feet per second, its height t seconds later is given by
s(t) = −16t 2 + 50t feet.
a) Find the height of the object after 2 seconds.
b) Find the average velocity between t = 2 and t = 3.
c) Estimate the instantaneous velocity when t = 2 by calculating the average velocity over the following time intervals:
[2, 2.1], [2, 2.01], [2, 2.001], and [2, 2.0001].
Math 131 WIR, copyright Angie Allen
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7. Evaluate the following limits numerically, if they exist. If a limit does not exist, use limits to describe the way in which
it does not exist.
a) lim (x2 − 5)
x→−4
b) lim
x→3
1
x−3
8. If a ball is thrown in the air with a velocity 34 ft/s, its height in feet t seconds later is given by y = 34t − 16t 2.
(a) Find the average velocity for the time period beginning when t = 1.5 and lasting (i) 0.5 second and (ii) 0.1 second.
(b) Find the average velocity during the time periods (i) [1.5, 1.55] and (ii) [1.5, 1.51].
(c) Estimate the instantaneous velocity when t = 1.5.
Math 131 WIR, copyright Angie Allen
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x2 − 7x + 1
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
x→3 (x − 3)2
9. Evaluate lim
10. After h hours, the number of bacteria in a petri dish during a certain experiment is given by N(h) = 1000(.6)h bacteria.
a) How many bacteria are present in the dish after 2.5 hours? (Round to the nearest integer.)
b) Estimate the slope of the tangent line to N at h = 2.5 by calculating slopes of nearby secant lines. (Round to the
nearest integer.)
c) What does your answer in part b represent?
Math 131 WIR, copyright Angie Allen
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√
√
11. The point P(8, 1) lies on the curve y = x − 7. If Q is the point (x, x − 7), find the equation of the line tangent to the
curve at the point P (i.e., x = 8) by calculating slopes of secant lines between P and Q.
12. Find lim
x→−3
1
3
+ 1x
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
3+x
7
Math 131 WIR, copyright Angie Allen
13. Consider the function f (x) below and answer the following questions:
a) Find f (3), if it exists.
b) Find f (−2), if it exists.
c) Find lim f (x), if it exists.
x→0
d) Find lim f (x), if it exists.
x→−2−
e) Find lim f (x), if it exists.
x→3
f) Find lim f (x), if it exists.
x→−1
g) Find lim f (x), if it exists.
x→2

10x − 4x2








x+1
2
f (x) =
x −x−2





2


 x −8
23−x
x < −2
−2 < x ≤ 3
x>3
Math 131 WIR, copyright Angie Allen
√
x2 + 11 − 6
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
14. Find lim
x→−5
x+5
15. Find lim
x→1+
1
1
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
−
x − 1 |x − 1|
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