Section 1.1 Limits
1) Below is a graph of the function f(x).
Find the following
a) f(1)
b) f(2)
c) f(3)
d) lim− 𝑓(𝑥)
e) lim+ 𝑓(𝑥)
f) lim 𝑓(𝑥)
g) lim− 𝑓(𝑥)
h) lim+ 𝑓(𝑥)
i) lim 𝑓(𝑥)
j) lim− 𝑓(𝑥)
k) lim+ 𝑓(𝑥)
l) lim 𝑓(𝑥)
𝑥→1
𝑥→2
𝑥→3
𝑥→1
𝑥→2
𝑥→3
𝑥→1
𝑥→2
𝑥→3
2) Below is a graph of the function f(x).
Find the following:
a) f(1)
b) f(3)
c) f(2)
d) lim− 𝑓(𝑥)
e) lim+ 𝑓(𝑥)
f) lim 𝑓(𝑥)
g) lim 𝑓(𝑥)
h) lim+ 𝑓(𝑥)
i) lim 𝑓(𝑥)
𝑥→ 1
𝑥→3
𝑥→ 1
𝑥→3
𝑥→ 1
𝑥→3
Section 1.1 Limits
3) Below is a graph of the function f(x).
Find the following
a) f(-4)
d)
lim 𝑓(𝑥)
𝑥→ −4−
b) f(2)
e)
lim 𝑓(𝑥)
𝑥→ −4 +
c) f(4)
f) lim 𝑓(𝑥)
𝑥→ −4
g) lim− 𝑓(𝑥)
h) lim+ 𝑓(𝑥)
i) lim 𝑓(𝑥)
j) lim− 𝑓(𝑥)
k) lim+ 𝑓(𝑥)
l) lim 𝑓(𝑥)
𝑥→2
𝑥→4
𝑥→2
𝑥→4
𝑥→2
𝑥→4
4) Below is the graph of a function y = f(x).
Find the following
a) f(-1)
d)
lim 𝑓(𝑥)
𝑥→ −1−
b) f(0)
e)
lim 𝑓(𝑥)
𝑥→ −1+
c) f(1)
f) lim 𝑓(𝑥)
𝑥→ −1
g) lim− 𝑓(𝑥)
h) lim+ 𝑓(𝑥)
i) lim 𝑓(𝑥)
j) lim− 𝑓(𝑥)
k) lim+ 𝑓(𝑥)
l) lim 𝑓(𝑥)
𝑥→0
𝑥→1
𝑥→0
𝑥→1
𝑥→0
𝑥→1
Section 1.1 Limits
5) Below is a graph of the function f(x).
Find the following:
a) f(2)
b) f(0)
c) f(4)
d) lim− 𝑓(𝑥)
e) lim+ 𝑓(𝑥)
f) lim 𝑓(𝑥)
g) lim− 𝑓(𝑥)
h) lim+ 𝑓(𝑥)
i) lim 𝑓(𝑥)
𝑥→ 2
𝑥→0
𝑥→ 2
𝑥→0
𝑥→ 2
𝑥→0
6) Below is a graph of the function f(x).
Find the following:
a) f(1)
d)
lim 𝑓(𝑥)
𝑥→ −2−
b) f(0)
e)
lim 𝑓(𝑥)
𝑥→ −2+
c) f(-2)
f) lim 𝑓(𝑥)
𝑥→ −2
g) lim− 𝑓(𝑥)
h) lim+ 𝑓(𝑥)
i) lim 𝑓(𝑥)
j) lim− 𝑓(𝑥)
k) lim+ 𝑓(𝑥)
l) lim 𝑓(𝑥)
𝑥→0
𝑥→ 2
𝑥→0
𝑥→ 2
𝑥→0
𝑥→ 2
Section 1.1 Limits
7) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
8) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
Section 1.1 Limits
9) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
10) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
Section 1.1 Limits
11) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
12) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
Section 1.1 Limits
13) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
14) Below is a graph of the function f(x). Find the value of each limit (if it exists)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
Section 1.1 Limits
#15-21: Complete the table(s) and find the requested limits.
15) f(x) = 3x + 5, find
a) lim ( 3x + 5)
𝑥→2−
x
f(x)
1.5
1.9
1.99
1.999
2.5
2.1
2.01
2.001
b) lim (3x + 5)
𝑥→2+
x
f(x)
c) Use the results from part a and b to find: lim (3x + 5)
𝑥→2
d) Use algebra to find lim (3𝑥 + 5)
𝑥→2
16) f(x) = 2x - 3
a) lim (2x − 3)
𝑥→4−
x
f(x)
3.5
3.9
3.99
3.999
4.5
4.1
4.01
4.001
b) lim+ 𝑓(𝑥)
𝑥→4
x
f(x)
c) Use the results from part a and b to find: lim 𝑓(𝑥)
𝑥→4
d) Use algebra to find lim (2𝑥 − 3)
𝑥→4
Section 1.1 Limits
𝑥+2
17) f(x) = 𝑥−1 find
a) lim
𝑥+2
𝑥→2− 𝑥−1
x
f(x)
b) lim
1.5
1.9
1.99
1.999
2.5
2.1
2.01
2.001
𝑥+2
𝑥→2+ 𝑥−1
x
f(x)
𝑥+2
c) Use the results from part a and b to find: lim 𝑥−1
𝑥→2
𝑥+2
d) Use algebra to find lim 𝑥−1
𝑥→2
18) f(x) =
𝑥+5
𝑥+2
𝑥+5
𝑥→1− 𝑥+2
a) lim
x
f(x)
.5
.9
.99
.999
1.5
1.1
1.01
1.001
𝑥+5
b) lim+ 𝑥+2
𝑥→1
x
f(x)
𝑥+5
c) Use the results from part a and b to find: lim 𝑥+2
𝑥→1
𝑥+5
d) Use algebra to find lim 𝑥+2
𝑥→1
Section 1.1 Limits
19) f(x) =
√𝑥−3
,
𝑥−9
find
√𝑥−3
𝑥→9− 𝑥−9
a) lim
x
f(x)
8.5
8.9
8.99
8.999
9.5
9.1
9.01
9.001
√𝑥−3
𝑥→9+ 𝑥−9
b) lim
x
f(x)
√𝑥−3
𝑥→9 𝑥−9
c) Use the results from part a and b to find: lim
√𝑥−3
𝑥→9 𝑥−9
d) Use algebra to find lim
20) f(x) =
√𝑥−2
𝑥−4
√𝑥−2
𝑥→4− 𝑥−4
a) lim
x
f(x)
b) lim+
𝑥→4
x
f(x)
3.5
3.9
3.99
3.999
4.5
4.1
4.01
4.001
√𝑥−2
𝑥−4
c) Use the results from part a and b to find:
√𝑥−2
𝑥→4 𝑥−4
d) Use algebra to find lim
√𝑥−2
lim
𝑥→4 𝑥−4
Section 1.2 Limits part 2
#1-20: Find the following limits using Algebra.
1) lim (2𝑥 + 6)
2) lim (5𝑥 − 7)
3) lim (𝑥 2 + 5𝑥 − 4)
4) lim (𝑥 2 − 3𝑥 − 7)
5) lim √𝑥 + 5
6) lim √2𝑥 + 10
𝑥→5
𝑥→ −3
𝑥→3
𝑥→2
𝑥→4
𝑥→ −3
3𝑥+6
𝑥→2 𝑥−5
8) lim
2𝑥+1
𝑥→4 𝑥+7
𝑥 2 +5𝑥+6
𝑥→ −2 𝑥 2 +3𝑥+2
10) lim
7) lim
𝑥 2 +4𝑥−5
𝑥→1 𝑥 2 −1
9) lim
𝑥 2 −9
11) lim 𝑥 2 −4𝑥+3
12) lim
𝑥→3
𝑥→−2
2𝑥 2 −3𝑥−2
2𝑥 2 −7𝑥+3
13) lim 3𝑥 2 −2𝑥−8
14) lim 3𝑥 2 −7𝑥−6
𝑥→2
15) lim
𝑥→3
√𝑥−4
16) lim
𝑥→16 𝑥−16
17) lim
√𝑥−6
18) lim
√𝑥−8
𝑥→64 𝑥−64
√𝑥−11
20) lim
𝑥→121 𝑥−121
21) f(x) =
√𝑥−7
𝑥→49 𝑥−49
𝑥→36 𝑥−36
19) lim
𝑥 2 −4
𝑥 2 −3𝑥+2
√𝑥−5
𝑥→25 𝑥−25
2𝑥 2 +3𝑥+5
𝑥 2 +4𝑥−5
2𝑥 2 +3𝑥+5
𝑥→∞ 𝑥 2 +4𝑥−5
a) Complete the table to determine lim
x
f(x)
100
1000
100,000
b) Use your calculator to sketch a graph of f(x) to confirm your answer to part a.
2𝑥 2 +3𝑥+5
𝑥→∞ 𝑥 2 +4𝑥−5
c) Use algebra to find lim
1,000,000
Section 1.2 Limits part 2
6𝑥 2 +2𝑥+5
22) f(x) = 3𝑥2 +4𝑥−4, find
6𝑥 2 +2𝑥+5
,
𝑥→∞ 3𝑥 2 +4𝑥−4
a) Complete the table to determine lim
x
f(x)
100
1000
100,000
1,000,000
b) Use your calculator to sketch a graph of f(x) to confirm your answer to part a.
6𝑥 2 +2𝑥+5
𝑥→∞ 3𝑥 2 +4𝑥−4
c) Use algebra to find lim
23) f(x) =
6𝑥 3 −𝑥 2 +2𝑥+5
3𝑥 4 +4𝑥 2 −5𝑥
6𝑥 3 −𝑥 2 +2𝑥+5
𝑥→∞ 3𝑥 4 +4𝑥 2 −5𝑥
a) Complete the table to determine lim
x
f(x)
100
1000
100,000
1,000,000
b) Use your calculator to sketch a graph of f(x) to confirm your answer to part a.
6𝑥 3 −𝑥 2 +2𝑥+5
𝑥→∞ 3𝑥 4 +4𝑥 2 −5𝑥
c) Use algebra to find lim
24) f(x) =
2𝑥 2 +2𝑥−5
3𝑥 4 −5𝑥+2
6𝑥 3 −𝑥 2 +2𝑥+5
𝑥→∞ 3𝑥 4 +4𝑥 2 −5𝑥
a) Complete the table to determine lim
x
f(x)
100
1000
100,000
b) Use your calculator to sketch a graph of f(x) to confirm your answer to part a.
2𝑥 2 +2𝑥−5
𝑥→∞ 3𝑥 4 −5𝑥+2
c) Use algebra to find lim
1,000,000
Section 1.2 Limits part 2
25) f(x) =
6𝑥 5 −𝑥 2 +2𝑥+5
3𝑥 4 +4𝑥 2 −5𝑥
6𝑥 5 −𝑥 2 +2𝑥+5
𝑥→∞ 3𝑥 4 +4𝑥 2 −5𝑥
a) Complete the table to determine lim
x
f(x)
10
20
30
40
b) Use your calculator to sketch a graph of f(x) to confirm your answer to part a.
26) f(x) =
2𝑥 7 +2𝑥 3 −5
3𝑥 4 −5𝑥
2𝑥 7 +2𝑥 3 −5
𝑥→∞ 3𝑥 4 −5𝑥
a) Complete the table to determine lim
x
f(x)
10
20
30
b) Use your calculator to sketch a graph of f(x) to confirm your answer to part a.
#27-38: Find the following limits using Algebra.
27) lim
3𝑥+6
𝑥→∞ 2𝑥−4
4𝑥 2 −3𝑥+6
𝑥→∞ 5𝑥 2 +2𝑥−4
29) lim
31) lim
3𝑥+6
𝑥→∞ 2𝑥 2 −4
28) lim
3𝑥 2 +2𝑥−5
𝑥→∞ 7𝑥 2 +4𝑥−2
30) lim
32) lim
34) lim
3𝑥 2 +6
𝑥→∞ 2𝑥−4
36) lim
4𝑥 3 −3𝑥+6
𝑥→∞ 5𝑥 2 +2𝑥−4
38) lim
35) lim
37) lim
3𝑥+1
𝑥→∞ 5𝑥 2 −4
4𝑥 2 −3𝑥+6
𝑥→∞ 5𝑥 3 +2𝑥−4
33) lim
3𝑥+1
𝑥→∞ 5𝑥−4
3𝑥 2 +2𝑥−5
𝑥→∞ 7𝑥 3 +4𝑥−2
3𝑥 2 +1
𝑥→∞ 5𝑥−4
3𝑥 3 +2𝑥−5
𝑥→∞ 7𝑥 2 +4𝑥−2
40
Section 1.2 Limits part 2
#39-44: Find the following limits using Algebra.
3𝑥+6
𝑥→5 𝑥−5
40) lim
𝑥 2 +5𝑥+6
𝑥→ −1 𝑥 2 +3𝑥+2
42) lim
39) lim
41) lim
𝑥 2 −9
43) lim 𝑥 2 −4𝑥+3
𝑥→1
2𝑥+1
𝑥→−7 𝑥+7
𝑥 2 +4𝑥−5
𝑥→−1 𝑥 2 −1
𝑥 2 −4
44) lim 𝑥 2 −3𝑥+2
𝑥→1
Section 1.3 Continuity
I will use the graphs below to discuss the concept of being continuous at a point x = a. These 4 graphs
are not part of any homework questions.
A function is continuous at a point (a) provided the limit exists at (a) and the value of the function equals
the value of the limit.
Section 1.3 Continuity
#1-10: Find all values of x = a where the function is discontinuous. For each point of discontinuity state
which of the following three reasons for the point of discontinuity (state all reasons that are applicable).
I) The function is not defined at the point of discontinuity (far left picture)
II) The limit does not exist at the point of discontinuity (bottom picture)
III) The limit does not equal the function value at the point of discontinuity (middle graph)
1)
3)
2)
4)
Section 1.3 Continuity
5)
7)
9)
6)
8)
10)
Section 1.3 Continuity
#11-28: Find all values of x=a where the function is discontinuous. There are a few problems that do
not have any values of x = a where the function is discontinuous, for these problems simply state that
the function is continuous everywhere and that there are no values of x = a where the function is
discontinuous. State the reason for each point of discontinuity.
𝑥−3
11) 𝑓(𝑥) = 𝑥+4
−4
13) 𝑓(𝑥) = 𝑥 2 +6𝑥+5
15) 𝑓(𝑥) =
𝑥 2 −4
𝑥+2
𝑥 2 +6𝑥+7
17) 𝑓(𝑥) = 𝑥 2 +3𝑥+2
𝑥+2
19) 𝑓(𝑥) = 𝑥 2 +5𝑥+6
2
21) 𝑓(𝑥) = 𝑥 2 +9
23) 𝑓(𝑥) =
𝑥−4
𝑥 2 +12𝑥+38
𝑥+1
12) 𝑓(𝑥) = 𝑥−5
2
14) 𝑓(𝑥) = 𝑥 2 −6𝑥−7
16) 𝑓(𝑥) =
𝑥 2 −9
𝑥+3
𝑥 2 +4𝑥+3
18) 𝑓(𝑥) = 𝑥 2 −2𝑥−3
𝑥−4
20) 𝑓(𝑥) = 𝑥 2 −3𝑥−4
3
22) 𝑓(𝑥) = 𝑥 2 +25
24) 𝑓(𝑥) =
𝑥+1
𝑥 2 +4𝑥+7
25) f(x) = 2x – 6
26) f(x) = 3x – 2
27) f(x) = x2 + 6x – 7
28) f(x) = x2 – 4x – 5
1.4 Rates of Change
#1-8: Find the average rate of change for each function over the given interval. Sketch a graph to model
your answer. (You may use your calculator obtain the graph, be sure to label the necessary points.)
1) f(x) = x2 - 5x between x = 3 and x = 4
2) f(x) = x2 - 5x +2 between x = 3 and x = 4
3) f(x) = √𝑥 − 5 between x = 9 and x = 14
4) f(x) = √𝑥 + 2 between x = 2 and x = 7
5) s(t) = 2t between t = 0 and t = 2
6) s(t) = 3t between t = 0 and t = 1
7) c(t) = ln(t) between t = 1 and t = e
8) c(t) = ln(t) between t = 1 and t = e2
9) A climber is on a hike. After 2 hours he is at an altitude of 400 feet. After 6 hours, he is at an altitude
of 700 feet. What is the average rate of change?
10) A scuba diver is 30 feet below the surface of the water 10 seconds after he entered the water and
100 feet below the surface after 40 seconds. What is the scuba divers average rate of change?
11) A rocket is 1 mile above the earth in 30 seconds and 5 miles above the earth in 2.5 minutes. What is
the rockets average rate of change in miles per second?
12) A teacher weighed 160 lbs in 1996 and weighs 210 lbs in 2013. What was the average rate of change
in weight?
13) This problem has been deleted.
14) Michael started a savings account with $300. After 4 weeks, he had $350 dollars, and after 9 weeks,
he had $400. What is the average rate of change of money in his savings account per week?
15) A plane left Chicago at 8:00 A.M. At 1: P.M., the plane landed in Los Angeles, which is 1500 miles
away. What was the average speed of the plane for the trip?
16) After 30 baseball games, A-Rod had 25 hits. If after 100 games he had 80 hits, what are his average
hits per baseball game?
1.4 Rates of Change
#17 – 24: Find the instantaneous rate of change at the given value. Sketch a graph to model your
answer. (You may use your calculator obtain the graph, be sure to label the necessary points.)
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
ℎ→0
Use the instantaneous rate of change formula: lim
17) f(x) = x2 – 3 at x = 2
18) f(x) = x2 – 5 at x = 4
19) f(x) = 3 – x2 at x = 1
20) f(x) = 5 – x2 at x = -1
21) g(t) = t2 + 2t – 3 at t = -2
22) g(t) = t2 – 5t + 1 at t = -3
23) h(t) = 5t2 – 2t + 3 at t = 0
24) h(t) = 6t2 – 3t + 1 at t = 4
25) A toy rocket is launched straight up so that its height s, in meters, at time t, in seconds, is given by
s(t)=−2t2+30t+5. Calculate the instantaneous rate of change (velocity) of the rocket at t=3.
26) If a baseball is projected upward from ground level with an initial velocity of 64 feet per second,
then its height is a function of time, given by s(t) = -16t2 + 64t. Calculate the instantaneous rate of
change (velocity) of the ball at t = 2 seconds.
27) A pebble is dropped from a cliff, 50 m high. After t sec, the pebble is s meters above the ground,
where s(t)=50−2t2. Calculate the instantaneous rate of change (velocity) of the pebble at t = 2 seconds.
28) A cannon ball is dropped from a building. Suppose that the height of the cannon ball (in meters)
after t seconds is given by the quadratic function: f(t) = -4.4t2 + 50. Calculate the instantaneous rate of
change (velocity) of the ball at t = 1 seconds.
29) The profit from sale of x car seats is given by the formula: P(x)= 45x - 0.0025x2 - 5000
a) Find the profit from selling 800 car seats.
b) Find the instantaneous rate of change at a production level of 800 car seats. (The instantaneous rate
of change of a profit function is often called the marginal profit.)
1.4 Rates of Change
30) The profit from sale of x cell phones is given by the formula: P(x)= 450x - 0.055x2 - 300000
a) Find the profit from selling 1000 cell phones.
b) Find the instantaneous rate of change at a production level of 1000 cell phones. (The instantaneous
rate of change of a profit function is often called the marginal profit.)
31) The cost of manufacturing x chairs is given by the function: C(x) = x2 + 40x + 800
a) Find the cost of producing 30 chairs.
b) Find the instantaneous rate of change when 30 chairs are produced. (The instantaneous rate of
change of a cost function is often called the marginal cost.)
32) The cost of manufacturing x books is given by the function: C(x) = x2 + 30x + 50
a) Find the cost of producing 50 books.
b) Find the instantaneous rate of change when 50 books are produced. (The instantaneous rate of
change of a cost function is often called the marginal cost.)
Section 1.5 Definition of Derivatives
#1-4: Find the slope of the tangent line at the given point (x,y).
1)
2)
Section 1.5 Definition of Derivatives
3)
4)
Section 1.5 Definition of Derivatives
#5-14: For each problem complete the following.
a) Use the definition of the derivative to find f’(x)
b) Find f’(4)
5) f(x) = x2 + 3x – 4
7) f(x) = 6x2 + 12
9) f(x) = 3x2 – 4x + 2
11) f(x) =
13) f(x) =
2
𝑥
5
𝑥
6) f(x) = x2 – 5x + 7
8) f(x) = 3x2 - 4
10) f(x) = 5x2 – 6x + 1
12) f(x) =
14) f(x) =
3
𝑥
7
𝑥
#15-24: For each problem complete the following:
a) Find a formula to find the slope of a tangent line.
b) Find the equation of the tangent line through the given value of x.
15) f(x) = x2 + x – 4, x = 3
17) f(x) = 3x2 + 7, x = 5
19) f(x) = 3x2 – 2x + 3, x = 1
21) f(x) =
23) f(x) =
−8
,
𝑥
−3
,
𝑥
16) f(x) = x2 – 2x + 3, x = 4
18) f(x) = 2x2 - 1, x = -2
20) f(x) = 5x2 – 2x + 8, x = 0
6
𝑥
−4
,
𝑥
x = -3
22) f(x) = − , x = -5
x=2
24) f(x) =
x=2
25) A toy rocket is launched straight up so that its height s, in meters, at time t, in seconds, is given by
s(t)=−2t2+30t+5.
a) Find s’(t)
b) Find s’(2) and interpret your answer
26) If a baseball is projected upward from ground level with an initial velocity of 64 feet per second,
then its height is a function of time, given by s(t) = -16t2 + 64t
a) Find s’(t)
b) Find s’(2) and interpret your answer
Section 1.5 Definition of Derivatives
27) A pebble is dropped from a cliff, 50 m high. After t sec, the pebble is s meters above the ground,
where s(t)=50−2t2.
a) Find s’(t)
b) Find s’(1) and interpret your answer
28) A cannon ball is dropped from a building. Suppose that the height of the cannon ball (in meters)
after t seconds is given by the quadratic function: f(t) = -4.4t2 + 50.
a) Find f’(t)
b) Find f’(1) and interpret your answer
29) The profit from sale of x car seats for is given by the formula: P(x)= 45x - 0.0025x2 - 5000
a) Find P’(x)
b) Find P’(800) and interpret your answer
30) The profit from sale of x cell phones is given by the formula: P(x)= 450x - 0.055x2 - 300000
a) Find P’(x)
b) Find P’(1000) and interpret your answer
31) The cost of manufacturing x chairs is given by the function: C(x) = x2 + 40x + 800
a) Find C’(x)
b) Find C’(30) and interpret your answer
32) The cost of manufacturing x books is given by the function: C(x) = x2 + 30x + 50
a) Find C’(x)
b) Find C’(20) and interpret your answer
Chapter 1 Review
1)
a) f(2)
d)
lim 𝑓(𝑥)
𝑥→ −2−
g) lim− 𝑓(𝑥)
𝑥→3
b) f(3)
e)
lim 𝑓(𝑥)
𝑥→ −2+
h) lim+ 𝑓(𝑥)
𝑥→3
2)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
c) f(1)
f) lim 𝑓(𝑥)
𝑥→ −2
i) lim 𝑓(𝑥)
𝑥→3
Chapter 1 Review
3)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
4) Find the following limits using Algebra.
a) lim (𝑥 2 − 3𝑥 + 5)
𝑥→2
𝑥 2 −7𝑥+10
b) lim 𝑥 2 +4𝑥−12
c) lim
√𝑥−5
𝑥→25 𝑥−25
𝑥→2
5) Find the following limits using Algebra.
6𝑥 2 −2𝑥+1
𝑥→∞ 3𝑥 2 +4𝑥−5
a) lim
b) lim
12𝑥+6
𝑥→∞ 3𝑥 2 −4𝑥+2
6) Find all values of x = a where the function is discontinuous. For each point of discontinuity state
which of the following three reasons for the point of discontinuity (state all reasons that are applicable).
I) The function is not defined at the point of discontinuity
II) The limit does not exist at the point of discontinuity
III) The limit does not equal the function value at the point of discontinuity
6a)
6b)
Chapter 1 Review
7) Find all values of x=a where the function is discontinuous. There are a few problems that do not have
any values of x = a where the function is discontinuous, for these problems simply state that the
function is continuous everywhere and that there are no values of x = a where the function is
discontinuous. State the reason for each point of discontinuity.
𝑥+1
7a) 𝑓(𝑥) = 𝑥 2 +4𝑥+3
7b) 𝑓(𝑥) = 3𝑥 − 12
8) Find the average rate of change for each function over the given interval. Sketch a graph to model
your answer. (You may use your calculator obtain the graph, be sure to label the necessary points.)
f(x) = x2 - 2x between x = 1 and x = 4
9) For each problem complete the following.
I) Use the definition of the derivative to find f’(x)
II) Find f’(3)
9a) 𝑓(𝑥) = 𝑥 2 − 3𝑥 + 1
7
9b) 𝑓(𝑥) = 𝑥
#10-11: For each problem complete the following:
a) Find a formula to find the slope of a tangent line.
b) Find the equation of the tangent line through the given value of x.
10) 𝑓(𝑥) = 𝑥 2 + 5𝑥 − 4; 𝑥 = 3
11) 𝑓(𝑥) =
−2
;𝑥
𝑥
=4
12) A toy rocket is launched straight up so that its height s, in meters, at time t, in seconds, is given by
s(t)=−2t2+10t+3.
a) Find s’(t)
b) Find s’(2) and interpret your answer
13) The profit from sale of x cell phones is given by the formula: P(x)= 350x - 0.015x2 - 200000
a) Find P’(x)
b) Find P’(500) and interpret your answer
Chapter 1 Practice test Part 1
1) Below is a graph of the function f(x).
Find the following.
a) f(0)
b) f(1)
c) f(-2)
d) lim− 𝑓(𝑥)
e) lim+ 𝑓(𝑥)
f) lim 𝑓(𝑥)
g) lim− 𝑓(𝑥)
h) lim+ 𝑓(𝑥)
i) lim 𝑓(𝑥)
𝑥→ 0
𝑥→2
𝑥→ 0
𝑥→2
2)
a) lim 𝑓(𝑥) =
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
𝑥→ 0
𝑥→2
Chapter 1 Practice Test Part 1
3)
a) lim 𝑓(𝑥)
𝑥→∞
b) lim 𝑓(𝑥)
𝑥→−∞
4) Find the following limits using Algebra.
a) lim (𝑥 2 + 4𝑥 − 3)
𝑥→2
𝑥 2 +5𝑥+6
𝑥→−2 𝑥 2 +8𝑥+12
b) lim
c) lim
√𝑥−7
𝑥→49 𝑥−49
5) Find the following limits using Algebra.
8𝑥 2 +1
𝑥→∞ 2𝑥 2 +4𝑥
a) lim
b) lim
5𝑥−4
𝑥→∞ 2𝑥 2 −𝑥+2
6) Find all values of x = a where the function is discontinuous. For each point of discontinuity state
which of the following three reasons for the point of discontinuity (state all reasons that are applicable).
I) The function is not defined at the point of discontinuity
II) The limit does not exist at the point of discontinuity
III) The limit does not equal the function value at the point of discontinuity
Chapter 1 Practice Test Part 1
7) Find all values of x=a where the function is discontinuous. There are a few problems that do not have
any values of x = a where the function is discontinuous, for these problems simply state that the
function is continuous everywhere and that there are no values of x = a where the function is
discontinuous. State the reason for each point of discontinuity.
𝑥+3
7a) 𝑓(𝑥) = 𝑥 2 +4𝑥−5
7b) 𝑓(𝑥) = 5𝑥 + 10
Chapter 1 Practice Test Part 2
8) Find the average rate of change for each function over the given interval. It is not necessary to
sketch a graph to model the average rate of change.
f(x) = x3 + 5x between x = 1 and x = 2
9) f(x) = x2 + 3
a) Use the definition of the derivative to find f’(x)
b) Find f’(5)
2
10) 𝑓(𝑥) = 𝑥
a) Find a formula to find the slope of a tangent line.
b) Find the equation of the tangent line when x = 3
11) A toy rocket is launched straight up so that its height s, in meters, at time t, in seconds, is given by
s(t)=−2t2+20t.
a) Find s’(t)
b) Find s’(2) and interpret your answer