Section 4.1 Absolute Extrema
#1-9: Find the absolute maximum and absolute minimum
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Section 4.1 Absolute Extrema
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Section 4.1 Absolute Extrema
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Section 4.1 Absolute Extrema
#10-27: Find the absolute maximum and absolute minimum of the function under the given interval.
10) 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 5; [−3,3]
11) 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 4; [−5,5]
12) 𝑓(𝑥) = 𝑥 3 − 6𝑥 2 ; [−2, 2]
13) 𝑓(𝑥) = 𝑥 3 + 6𝑥 2 ; [−2,1]
14) 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 ;
15) 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 2; [−1,5]
[−1,3]
16) 𝑓(𝑥) = 𝑥 4 − 𝑥 3 + 5; [-2,2]
17) 𝑓(𝑥) = 3𝑥 4 − 4𝑥 3 ; [-2,3]
18) 𝑓(𝑥) = (𝑥 2 − 9)4 ; [0,2]
19) 𝑓(𝑥) = (𝑥 2 − 16)3 ; [-2,2]
3
5
20) 𝑓(𝑥) = √𝑥 ; [−1,2]
21) 𝑓(𝑥) = √𝑥 ; [−3,2]
22) 𝑓(𝑥) = 𝑥𝑒 𝑥 ; [0,3]
23) 𝑓(𝑥) = 2𝑥𝑒 𝑥 ; [0,3]
2
2
24) 𝑓(𝑥) = 𝑒 3𝑥 ; [−1,1]
25) 𝑓(𝑥) = 𝑒 𝑥 ; [−2,1]
26) 𝑓(𝑥) = 𝑥 3 𝑒 𝑥 ; [-3,1]
27) 𝑓(𝑥) = 𝑥 2 𝑒 𝑥 ; [-3,1]
Section 4.2 Applications of Extrema
1) A campground owner has 1000 meters of fencing. He wants to enclose a rectangular field with the
fence that he has. Let W represent the width of the field and L represent the length of the field.
a) Write an equation for the length of the field.
b) Write an equation for the area of the fenced in field.
c) Find the domain of the area equation that was created in part b.
(This domain will be of the form: # ≤ 𝑊 ≤ #)
d) Find the value of w leading to the maximum area
e) Find the value of L leading to the maximum area
f) Find the maximum area.
2) A campground owner has 5000 meters of fencing. He wants to enclose a rectangular field with the
fence that he has. Let W represent the width of the field and L represent the length of the field.
a) Write an equation for the length of the field.
b) Write an equation for the area of the fenced in field.
c) Find the domain of the area equation that was created in part b.
(This domain will be of the form: # ≤ 𝑊 ≤ #)
d) Find the value of w leading to the maximum area
e) Find the value of L leading to the maximum area
f) Find the maximum area.
3) A campground owner has 1000 meters of fencing. He wants to enclose a rectangular field bordering
a river, with no fencing needed along the river, and let W represent the width of the field and L
represent the length of the field. Make W be the side of the fence that is perpendicular to the river so
that two widths and one length will need to be constructed.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the domain of the area equation that was created in part b.
(This domain will be of the form: # ≤ 𝑊 ≤ #)
d) Find the value of w leading to the maximum area
e) Find the value of L leading to the maximum area
f) Find the maximum area.
Section 4.2 Applications of Extrema
4) A campground owner has 4000 meters of fencing. He wants to enclose a rectangular field bordering
a river, with no fencing needed along the river, and let W represent the width of the field and L
represent the length of the field. Make W be the side of the fence that is perpendicular to the river so
that two widths and one length will need to be constructed.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the domain of the area equation that was created in part b.
(This domain will be of the form: # ≤ 𝑊 ≤ #)
d) Find the value of w leading to the maximum area
e) Find the value of L leading to the maximum area
f) Find the maximum area.
e) Find the value of L leading to the minimum cost
f) Find the minimum cost.
5) An open box with a square base is to be made from a square piece of cardboard 10 inches on a side
by cutting out a square ( x inches by x inches) from each corner and turning up the sides.
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Find the domain of the volume equation created in part b.
(This domain will be of the form: # ≤ 𝑥 ≤ #)
d) Find the value of x that makes the volume the largest.
e) Find the maximum volume.
6) An open box with a square base is to be made from a square piece of cardboard 12 inches on a side
by cutting out a square ( x inches by x inches) from each corner and turning up the sides.
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Find the domain of the volume equation created in part b.
(This domain will be of the form: # ≤ 𝑥 ≤ #)
d) Find the value of x that makes the volume the largest.
e) Find the maximum volume.
Section 4.2 Applications of Extrema
7) An open box is to be made by cutting a square corner of a 20 inch by 20 inch piece of metal then
folding up the sides. What size square should be cut from each corner to maximize volume?
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Find the domain of the volume equation created in part b.
(This domain will be of the form: # ≤ 𝑥 ≤ #)
d) Find the value of x that makes the volume the largest.
e) Find the maximum volume.
8) An open box is to be made by cutting a square corner of a 30 inch by 30 inch piece of metal then
folding up the sides. What size square should be cut from each corner to maximize volume?
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Find the domain of the volume equation created in part b.
(This domain will be of the form: # ≤ 𝑥 ≤ #)
d) Find the value of x that makes the volume the largest.
e) Find the maximum volume.
Section 4.3 Elasticity of Demand
(Round all answers to 2 decimals when appropriate.)
1) A clothing company manufactures t-shirts that sell for $15. The price demand equation for the tshirts is: 𝑑(𝑝) = 60 − 3𝑝 where d(p) is the number of t-shirts sold at price p.
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(15) (round to 2 decimals)
c) At a price of $15 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the t-shirts was raised by 25% to $18.75?
2) Suppose that the price demand function for a certain electronic cigarette is:
𝑑(𝑝) = −0.375𝑝 + 7.87 Where p is the price of the cigarette and d(p) is the number of cigarettes that
will be demanded at price p.
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(5) (round to 2 decimals)
c) At a price of $5 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the cigarettes was raised by 20% to $6.00?
3) Currently about 180 people take the shuttle between ASU main and ASU west. The cost of a trip is
$4. The number of people d(p) willing to take the shuttle at a price p is given by the price demand
function:
𝑑(𝑝) = 60(5 − 𝑝1⁄2 )
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(4) (round to 2 decimals)
c) At a price of $4 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the shuttle ride was raised by 10% to $4.40?
Section 4.3 Elasticity of Demand
4) A company produces fit bracelets. The price demand function is given by:
𝑑(𝑝) = 20(4 − 𝑝1⁄2 )
Where d(p) is the number of bracelets sold at price p.
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(15) (round to 2 decimals)
c) At a price of $15 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the fit bracelets was raised by 10% to $16.50?
5) A company sells ice cream cones. The demand function for the cones is given by: 𝑑(𝑝) = 50 − 2𝑝.
Where p is the price of an ice cream cone and d(p) is the number of cones sold at price p.
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(4) (round to 2 decimals)
c) At a price of $4 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the cone was raised by 25% to $5.00?
6) An store has started selling a new backpack. The price demand equation for the backpack is given by:
𝑑(𝑝) = −𝑝2 + 800. Where p is the price of a backpack and d(p) is the number of backpacks sold.
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(25) (round to 2 decimals)
c) At a price of $25 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the backpack was raised by 20% to $30.00?
Section 4.4 Implicit Differentiation
𝑑𝑦
#1-24: Use implicit differentiation to determine 𝑑𝑥 .
1) 3x + y = 12
2) 4x2 + y = 3x – 2
3) 5x2 + 3y = 2x – 4
4) 4x3 + 7y = x2 + 5x – 3
5) 2x + 3y2 = 5x2 – 2x3
6) 5y3 + 6x2 = 4 – 3x
7) 𝑥 1⁄2 + 𝑦 1⁄3 = 3
8) 𝑥 3⁄4 + 𝑦 1⁄2 = 4
9) 2x2 + xy = 5
10) x3 + xy = 3
11) 2x2 + xy2 = 5
12) x3 + xy3 = 3
13) 2x2 + xy = 5y
14) x3 + xy = 3y
15) 2x2 + xy2 = 5y3
16) x3 + xy3 = 3y3
17) e4y + 3x = 12x2
18) 3e2y + x2 = 5x3
19) e4y + 3x = 12y2
20) 3e2y + x2 = 5y3
21) 3√𝑦 + 3𝑥𝑦 = 2𝑥 2
22) √𝑦 + 4𝑥𝑦 = 3𝑥 2
23) xey + 2y = 3x
24) x2e2y + 3y2 = 5x
#25- 34: Find the equation of the line tangent to the graph at the indicated point.
25) x2 + y2 = 25; (4,3)
26) x3 + y3 = 9; (2,1)
27) 3x2 + 2y = 25; (3,-1)
28) 2x3 + 3y2 = 57; (3,1)
29) 2𝑦 2 − √𝑥 = 4; (16,2)
30) 3𝑦 3 + √𝑥 = 26; (8,2)
31) y2 + 2y =11 + 4x; (1, 3)
32) y3 - 7y = 40 – 2x; (2,4)
33) xey + 2y =3x - 4; (2,0)
34) x2e2y + 3y2 = 5x - 6 (3,0)
3
#35-42: Use implicit differentiation to determine
35) 𝐴 = 2𝜋𝑟
36) 𝐴 = 𝜋𝑟 2
37) 𝑉 = 5 + 6𝑟
38) 𝑉 = 2𝑟 2 + 8
39) 𝑟 = 6𝑥 − 4
40) r = 12x + 10
41) r = y2 + 3
42) r = 5y2 + 6
𝑑𝑟
.
𝑑𝑡
Section 4.5 Related Rates
1) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of
the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what
rate is the total area of the disturbed water changing?
2) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of
the outer ripple is increasing at a constant rate of 2 feet per second. When the radius is 8 feet, at what
rate is the total area of the disturbed water changing?
3) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of
the outer ripple is increasing at a constant rate of 3 feet per second. When the radius is 5 feet, at what
rate is the total area of the disturbed water changing?
4) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of
the outer ripple is increasing at a constant rate of 4 feet per second. When the radius is 7 feet, at what
rate is the total area of the disturbed water changing?
5) Air is being pumped into a spherical balloon at 10 cm3/minute. Calculate the rate at which the radius
of the balloon is changing when the radius of the balloon is 6 cm.
6) Air is being pumped into a spherical balloon at 2 cm3/second. Calculate the rate at which the radius
of the balloon is changing when the radius of the balloon is 3 cm.
7) Air is being pumped into a spherical balloon at 3 cm3/minute. Calculate the rate at which the radius
of the balloon is changing when the radius of the balloon is 8 cm.
8) Air is being pumped into a spherical balloon at 9 cm3/second. Calculate the rate at which the radius
of the balloon is changing when the radius of the balloon is 12 cm.
Chapter 4 Review
1) Find the absolute maximum and absolute minimum
#2-3: Find the absolute maximum and absolute minimum of the function under the given interval.
2) 𝑓(𝑥) = 𝑥 3 − 6𝑥 2 ; [-1,5]
3) 𝑓(𝑥) = 2𝑥 2 𝑒 𝑥 ; [-3,1]
4) A campground owner has 500 meters of fencing. He wants to enclose a rectangular field bordering a
river, with no fencing needed along the river, and let W represent the width of the field and L represent
the length of the field. Make W be the side of the fence that is perpendicular to the river so that two
widths and one length will need to be constructed.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the domain of the area equation that was created in part b.
(This domain will be of the form: # ≤ 𝑊 ≤ #)
d) Find the value of w leading to the maximum area
e) Find the value of L leading to the maximum area
f) Find the maximum area.
5) An open box is to be made by cutting a square corner of a 8 inch by 8 inch piece of metal then folding
up the sides. What size square should be cut from each corner to maximize volume?
a) Sketch a diagram that models the problem.
b) Write an equation for the volume of the box.
c) Find the domain of the volume equation created in part b.
(This domain will be of the form: # ≤ 𝑥 ≤ #)
d) Find the value of x that makes the volume the largest.
e) Find the maximum volume.
Chapter 4 Review
6) A clothing company manufactures t-shirts that sell for $20. The price demand equation for the tshirts is: 𝑑(𝑝) = 70 − 2𝑝 where d(p) is the number of t-shirts sold at price p.
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(20)
c) At a price of $20 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the t-shirts was raised by 25% to $25?
𝑑𝑦
#7-9: Use implicit differentiation to determine 𝑑𝑥 .
7) x3 + 2xy = 3
8) e4y - 2x = x2
9) 𝑦 2 + 2𝑥𝑦 = 3𝑥 2
10) Find the equation of the line tangent to the graph at the indicated point.
x2 + y2 = 13; (2,3)
11) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The
radius r of the outer ripple is increasing at a constant rate of 2 feet per second. When the radius is 6
feet, at what rate is the total area of the disturbed water changing?
12) Air is being pumped into a spherical balloon at 2 cm3/minute. Calculate the rate at which the radius
of the balloon is changing when the radius of the balloon is 6 cm.
Chapter 4 Practice Test
1) Find the absolute maximum and absolute minimum
#2-3: Find the absolute maximum and absolute minimum of the function under the given interval.
2) 𝑓(𝑥) = 2𝑥 3 − 54𝑥 ; [0,4]
3) 𝑓(𝑥) = 𝑥𝑒 𝑥 ; [-2,2]
4) A campground owner has 100 meters of fencing. He wants to enclose a rectangular field bordering a
river, with no fencing needed along the river, and let W represent the width of the field and L represent
the length of the field. Make W be the side of the fence that is perpendicular to the river so that two
widths and one length will need to be constructed.
a) Write an equation for the length of the field
b) Write an equation for the area of the field.
c) Find the domain of the area equation that was created in part b.
(This domain will be of the form: # ≤ 𝑊 ≤ #)
d) Find the value of w leading to the maximum area
e) Find the value of L leading to the maximum area
f) Find the maximum area.
5) A clothing company manufactures t-shirts that sell for $20. The price demand equation for the tshirts is: 𝑑(𝑝) = 40 − 3𝑝 where d(p) is the number of t-shirts sold at price p.
a) Compute the price elasticity function, 𝐸(𝑝) =
−𝑝∗𝑑 ′ (𝑝)
.
𝑑(𝑝)
b) Compute E(10)
c) At a price of $10 is demand relatively elastic or inelastic?
d) How much would the demand change if the price of the t-shirts was raised by 25% to $25?
𝑑𝑦
#6-7: Use implicit differentiation to determine 𝑑𝑥 .
6) x2 + 4xy = 5
7) 3𝑦 2 + 𝑥𝑦 = 8
8) Find the equation of the line tangent to the graph at the indicated point.
xy = 10; (2,5)
9) A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of
the outer ripple is increasing at a constant rate of 3 feet per second. When the radius is 6 feet, at what
rate is the total area of the disturbed water changing?