Section 3.1 Increasing and Decreasing functions and Relative Maxima and Minima
1-12: Find
a) the open intervals where the function graphed is increasing
b) the open intervals where the function graph is decreasing
c) the coordinates of all relative maxima
d) the coordinates of all relative minima
1)
2)
3)
4)
Section 3.1 Increasing and Decreasing functions and Relative Maxima and Minima
5)
6)
7)
8)
Section 3.1 Increasing and Decreasing functions and Relative Maxima and Minima
9)
10)
11)
12)
Section 3.1 Increasing and Decreasing functions
#13 – 34: For each function find the following:
a) f’(x)
b) the critical numbers
c) the open interval(s) where the function is increasing
d) the open interval(s) where the function is decreasing
e) the coordinates of all relative maxima
f) the coordinates of all relative minima
13) 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 3
14) 𝑓(𝑥) = 2𝑥 2 − 8𝑥 + 1
15) 𝑓(𝑥) = 𝑥 2 − 3
16) 𝑓(𝑥) = 𝑥 2 + 5
17) 𝑓(𝑥) = 𝑥 3 − 12𝑥 + 4
18) 𝑓(𝑥) = 𝑥 3 − 48𝑥 + 18
19) 𝑓(𝑥) = −𝑥 3 − 3𝑥 2 + 45𝑥 − 5
20) 𝑓(𝑥) = −𝑥 3 − 6𝑥 2 + 24𝑥 − 9
21) 𝑓(𝑥) =
𝑥3
3
− 4𝑥
22) 𝑓(𝑥) =
𝑥3
3
− 9𝑥
23) 𝑓(𝑥) = √𝑥 − 3
24) 𝑓(𝑥) = √𝑥 − 5
25) 𝑓(𝑥) = √8 − 2𝑥
26) 𝑓(𝑥) = √12 − 3𝑥
27) 𝑓(𝑥) =
𝑥+2
𝑥−5
28) 𝑓(𝑥) =
𝑥+3
𝑥−7
29) 𝑓(𝑥) =
𝑥−4
𝑥+1
30) 𝑓(𝑥) =
𝑥−6
𝑥+3
31) 𝑓(𝑥) = 𝑥𝑒 3𝑥
32) 𝑓(𝑥) = 𝑥𝑒 2𝑥
33) 𝑓(𝑥) = 𝑥 2 𝑒 𝑥
34) 𝑓(𝑥) = 𝑥 2 𝑒 2𝑥
3.2 Higher Derivatives,
#1 – 6: For each function find the first second and third derivatives. Specifically find
a) f’(x)
b) f’’(x)
c) f’’’(x)
1) f(x) = 12x4 + 5x3 + 3x2 – 6x + 1
2) f(x) = 3x5 – 8x4 + 4x3 + 5x2 + 3x + 5
3) f(x) = -3x3 + 6x2 + 8x + 9
4) f(x) = -2x4 + 3x3 + 4x2 – 2x – 1
5) f(x) = 3x-5
6) f(x) = 2x-3
#7 – 18: For each function find the first and second derivatives. Specifically find
a) f’(x)
b) f’’(x)
7) f(x) = e2x+3
9) f(x) = 𝑒 𝑥
2
8) f(x) = e2x+5
10) f(x) = 𝑒 𝑥
3
11) f(x) = ln(2x)
12) f(x) = ln(3x)
13) f(x) = ln(x2)
14) f(x) = ln(x3)
𝑥+2
15) f(x) = 𝑥−3
2𝑥
17) f(x) = 𝑥+8
𝑥+4
16) f(x) = 𝑥−5
3𝑥
18) f(x) = 𝑥−5
Section 3.3 Concavity and the Second Derivative Test
#1-18:
a) Find the open intervals where the function is concave upward or downward.
b) Find all inflection points
1)
2)
3)
Section 3.3 Concavity and the Second Derivative Test
4)
5)
6)
Section 3.3 Concavity and the Second Derivative Test
7)
8)
Section 3.3 Concavity and the Second Derivative Test
9)
10)
11)
Section 3.3 Concavity and the Second Derivative Test
12)
13)
14)
Section 3.3 Concavity and the Second Derivative Test
15)
16)
17)
Section 3.3 Concavity and the Second Derivative Test
18)
#19-34:
a) Find the open intervals where the function is concave upward or downward.
b) Find all inflection points
19)
21)
23)
25)
27)
29)
f(x) = x3 – 3x + 2
f(x) = 24x – 2x3 -5
f(x) = x3 + 5x2 – 4x + 1
f(x) = x4 – 6x2 + 4
2
f(x) = 𝑒 𝑥
f(x) = e3x
2
20)
22)
24)
26)
28)
30)
f(x) = 2x3 – 24x + 3
f(x) = 3x – x3 + 4
f(x) = x3 + 4x2 – 3x + 2
f(x) = x4 – 12x3 – 5
2
f(x) = 𝑒 2𝑥
f(x) = e5x
5
31) f(x) = 𝑥−5
32) f(x) = 𝑥+1
33) f(x) = 𝑥+3
34) f(x) = 𝑥+4
𝑥−2
𝑥−7
Section 3.4 Curve sketching
#1-12: For each function
a) Find the domain
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find the interval(s) where the graph of the function is increasing
e) Find the interval(s) where the graph of the function is decreasing
f) Find all relative maxima and relative minima
g) Find the interval(s) where the graph of the function is concave up (if any)
h) Find the interval(s) where the graph of the function is concave down (if any)
i) Find all inflection points (if any)
j) Sketch a graph
1) f(x) = x3 – 3x2
2) f(x) = 2x3 – 12x2
3) f(x) = x4 – 4x3
4) f(x) = 2x4 – 24x3
5) f(x) = x3 - 6x2 + 12x - 11
6) f(x) = 2x3 + 3x2 – 12x + 1
7) f(x) = x4 – 24x2 + 40
8) f(x) = x4 – 48x2 + 5
(use your calculator to estimate the x-intercepts for problem 5,6,7 and 8)
9) f(x) = x5 – 15x3
10) f(x) = 45x3 – x5
11) f(x) = x3 – 27x
12) f(x) = x3 – 243x
Section 3.4 Curve sketching
#13-20: For each function
a) Find the domain and vertical asymptotes
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find all horizontal asymptotes
e) Find the interval(s) where the graph of the function is increasing
f) Find the interval(s) where the graph of the function is decreasing
g) Find all relative maxima and relative minima
h) Find the interval(s) where the graph of the function is concave up (if any)
i) Find the interval(s) where the graph of the function is concave down (if any)
j) Find all inflection points (if any)
k) Sketch a graph
13) 𝑓(𝑥) =
3
𝑥−4
𝑥+2
15) 𝑓(𝑥) = 𝑥−3
14) 𝑓(𝑥) =
2
𝑥−3
𝑥+1
16) 𝑓(𝑥) = 𝑥+5
Do not answer concavity questions for #17 – 20
2
17) 𝑓(𝑥) = 𝑥 2 +6𝑥−7
19) 𝑓(𝑥) =
1
𝑥 2 −4
18)
3
𝑥 2 −5𝑥−6
20) 𝑓(𝑥) =
2
𝑥 2 −9
Section 3.4 Curve sketching
#21-32: For each function
a) Find the domain
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find all vertical asymptotes and horizontal asymptotes
e) Find the interval(s) where the graph of the function is increasing
f) Find the interval(s) where the graph of the function is decreasing
g) Find all relative maxima and relative minima
h) Find the interval(s) where the graph of the function is concave up (if any)
i) Find the interval(s) where the graph of the function is concave down (if any)
j) Find all inflection points (if any)
k) Sketch a graph
21) f(x) = 𝑒 𝑥
2
22) f(x) = 𝑒 2𝑥
2
23) f(x) = xex
24) f(x) = 3xex
25) f(x) = x2ex
26) f(x) = 2x2ex
27) f(x) = 2xe-3x
28) f(x) = 3xe-2x
29) f(x) = ln(x-3)
30) f(x) = ln(x-2)
31) f(x) = ln(2x+8)
32) f(x) = ln(3x+6)
3.5 Applications of Extrema
1) A very small company constructs and sells waterproof cell phone cases. The cases sell for $20 each at
the local flea market. The dollar cost of producing x protectors is modeled by the following cost function
𝐶(𝑥) = 𝑥 2 + 8𝑥 + 10 (I kept the numbers in the problem small to make the math easier)
a)
b)
c)
d)
Create a revenue function
Create a profit function
Determine the number of cases the company should produce and sell to achieve maximum profit.
What is the maximum profit?
2) 𝐶(𝑥) = 𝑥 2 + 4𝑥 − 40 is the total daily cost to produce for x bracelets. The bracelets sell for $8
each. (I kept the numbers in the problem small to make the math easier)
a)
b)
c)
d)
Create a revenue function
Create a profit function
Determine the number of bracelets the company should produce and sell to maximize profit.
What is the maximum profit?
3) Boxed greeting cards cost the distributor 60 cents/box. The revenue generated from selling x boxes
1
of greeting cards is modeled by the function: 𝑅(𝑥) = 120𝑥 − 5 𝑥 2 .
a)
b)
c)
d)
Create a cost function
Create a profit function
Determine the number of boxes of greetings cards that should be bought and sold to maximize profit.
What is the maximum profit?
4) A single widget cost the distributor $12.00 to produce. The revenue generated from selling x widgets
is modeled by the function: 𝑅(𝑥) = 240𝑥 − 2𝑥 2 .
a)
b)
c)
d)
Create a cost function
Create a profit function
Determine the number of widgets that should be produced and sold to maximize profit.
What is the maximum profit?
5) The marketing research department of Shank, a quarterly magazine for beginning golfers, has
determined that the price-demand equation for the magazine is approximated by
𝑝 = 2.75 − 0.01𝑥
where x represents the number of magazines printed and sold each quarter, in hundreds, and p is the
price, in dollars, of the magazine. The cost of printing, distributing, and advertising is given by
𝐶(𝑥) = 0.003𝑥 2 + 0.5𝑥 + 5
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
3.5 Applications of Extrema
6) A headphone determines that in order to sell x units of a new headphone, the price demand equation
for the headphones is given by p = 1000 − x. It also determines that the total cost of producing x units is
given by C(x) = 3000 − 20x .
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
7) The daily production cost for a factory to manufacture x deluxe contour chairs is given to be
𝐶(𝑥) = 500 + 14𝑥 +
𝑥2
2
3
2
. The price demand function is 𝑝 = 150 − 𝑥 .
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
8) The Double D Corporation analyzed the production costs for one of its products and determined that
the daily cost function can be given by 𝐶(𝑥) = 0.03𝑥 3 − 2.4𝑥 2 + 73.16𝑥 + 102.27
where x is the number of units produced each day. The price demand function is given by:
𝑝 = 0.14𝑥 2 − 15.52𝑥 + 561.21
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
Chapter 3 Review
1) Consider the graph of the function f(x) below.
a)
b)
c)
d)
the open intervals where the function graphed is increasing
the open intervals where the function graph is decreasing
the coordinates of all relative maxima
the coordinates of all relative minima
#2-4: For each function find the following:
a) f’(x)
b) the critical numbers
c) the open interval(s) where the function is increasing
d) the open interval(s) where the function is decreasing
e) the coordinates of all relative maxima
f) the coordinates of all relative minima
2) 𝑓(𝑥) = 𝑥 3 − 75𝑥
3) 𝑓(𝑥) =
𝑥 2 −6𝑥+9
𝑥−5
4) 𝑓(𝑥) = 𝑥𝑒 𝑥
5) Consider the graph of the function f(x) below.
a) Find the open intervals where the function is concave upward or downward.
b) Find all inflection points
6) 𝑓(𝑥) = 𝑥 3 − 12𝑥 2 + 7
a) Find the open intervals where the function is concave upward or downward.
b) Find all inflection points
Chapter 3 Review
#7-8: Find the following.
a) Find the domain
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find the interval(s) where the graph of the function is increasing
e) Find the interval(s) where the graph of the function is decreasing
f) Find all relative maxima and relative minima
g) Find the interval(s) where the graph of the function is concave up (if any)
h) Find the interval(s) where the graph of the function is concave down (if any)
i) Find all inflection points (if any)
j) Sketch a graph
7) f(x) = x3 – 3x2
8) f(x) = 4xex
9): Find the following.
a) Find the domain and vertical asymptotes
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find all horizontal asymptotes
e) Find the interval(s) where the graph of the function is increasing
f) Find the interval(s) where the graph of the function is decreasing
g) Find all relative maxima and relative minima
h) Find the interval(s) where the graph of the function is concave up (if any)
i) Find the interval(s) where the graph of the function is concave down (if any)
j) Find all inflection points (if any)
k) Sketch a graph
9) 𝑓(𝑥) =
𝑥−2
𝑥+3
10) A headphone determines that in order to sell x units of a new headphone, the price demand
equation for the headphones is given by p = 500 − x. It also determines that the total cost of producing x
units is given by C(x) = 2000 − 20x .
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
Chapter 3 Practice Test Part 1
1) Consider the graph of the function f(x) below.
a)
b)
c)
d)
the open intervals where the function graphed is increasing
the open intervals where the function graph is decreasing
the coordinates of all relative maxima
the coordinates of all relative minima
#2-3: For each function find the following:
a) f’(x)
b) the critical numbers
c) the open interval(s) where the function is increasing
d) the open interval(s) where the function is decreasing
e) the coordinates of all relative maxima
f) the coordinates of all relative minima
2) 𝑓(𝑥) = 𝑥 4 − 2𝑥 2
3) 𝑓(𝑥) = 𝑒 𝑥
2
4) Consider the graph of the function f(x) below.
a) Find the open intervals where the function is concave upward or downward.
b) Find all inflection points
1
3
5) 𝑓(𝑥) = 𝑥 3 − 3𝑥 2
a)Find the open intervals where the function is concave upward or downward.
b) Find all inflection points
Chapter 3 Practice Test Part 2
#6-7: Find the following.
a) Find the domain
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find the interval(s) where the graph of the function is increasing
e) Find the interval(s) where the graph of the function is decreasing
f) Find all relative maxima and relative minima
g) Find the interval(s) where the graph of the function is concave up (if any)
h) Find the interval(s) where the graph of the function is concave down (if any)
i) Find all inflection points (if any)
j) Sketch a graph
6) f(x) = x3- 3x
7) f(x) = 2xex
𝑥−2
8): Find the following. 𝑓(𝑥) = 𝑥+3
a) Find the domain and vertical asymptotes
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find all horizontal asymptotes
e) Find the interval(s) where the graph of the function is increasing
f) Find the interval(s) where the graph of the function is decreasing
g) Find all relative maxima and relative minima
h) Find the interval(s) where the graph of the function is concave up (if any)
i) Find the interval(s) where the graph of the function is concave down (if any)
j) Find all inflection points (if any)
k) Sketch a graph