Calculus for Business
Lesson- Extrema and Critical Number
Name:____________________________________
Date:_____________________________________
Extreme Value Theorem
If f is continuous on a closed interval I = [a,b] then f has both a minimum and a maximum value on I.
f has a relative maximum at c or f(c) is a relative maximum of f if there is some open interval I containing c
on which f(c) is the maximum.
f has a relative minimum at c or f(c) is a relative minimum of f if there is some open interval I containing c on
which f(c) is the minimum.
Relative Extrema
Let I be any interval (closed or open) containing the x-value c.
The extreme values, or extrema, of a function f on I are defined by:
The absolute maximum of f on I is f(c) if f (c)  f ( x) for all x in I.
The absolute minimum of f on I is f(c) if f (c)  f ( x) for all x in I.
NOTE: A function need not have a maximum value or a minimum value on a given interval I.
Process
 Find the first derivative and plug the x-value of the coordinate of the relative min/max & solve for f’(x)
Or

Find the left- and right-hand side derivatives using lim
x c
f ( x )  f (c )
xc
Examples:
For each: determine any relative extrema and then determine the derivative at each relative extremum.
1.
f ( x)  x on the interval [-2,2].
2.
f ( x) 
1
on the interval [-2, 2].
x
3.
f ( x)  x 2 on the interval (-1, 2).
1
4.
f ( x)  x 3  3 x 2
5.
f ( x)  x
6.
f ( x)  sin x
7.
f ( x) 
2
3
8
x 4
2
Critical Values
A critical value is a number c in the domain of f for which f (c)  0 or f (c ) does not exist.
*Relative Extrema Occur Only at Critical Numbers
* One result of this theorem is that, on a closed interval, absolute extrema must occur at local extrema or at the
endpoints.
Long story short: Find the derivative, set = 0, solve for x, plug x back into original function.
Examples:
Find the critical value(s)
8.
f ( x)  x
10.
f ( x) 
2
3
8
x 4
2
9.
f ( x)  sin x
11.
y  2x 3
2
Method for Finding Absolute Extrema on a Closed Interval [a,b]
(1)
(2)
(3)
(4)
Find the critical numbers of f in (a,b).
Evaluate f at each critical number in (a,b).
Evaluate f(a) and f(b).
Compare: the least of these y-values is the minimum; the greatest is the maximum.
2x  5
on the interval [0, 5].
3
12.
Find the maximum and minimum values of f ( x) 
13.
Find the maximum and minimum values of f ( x)  x 2  2 x  4 on the interval [-2, 1].
14.
Find the maximum and minimum values of f ( x)  x3  12 x on the interval [0, 4].
15.
Find the maximum and minimum values of g ( x)  3 x on the interval [-1, 1].
16.
   
,
Find the maximum and minimum values of h( x)  sec x on the interval 
.
 6 3 
3
Calculus for Business
Lesson- First Derivative Test
Name:____________________________________
Date:_____________________________________
Objective:
Learn about Increasing and Decreasing Functions & the First Derivative Test for Extrema
Graphically
f increasing
f decreasing
Numerically
When x’s go up, y’s go up.
When x’s go up, y’s go down
Sign of the Derivative
f ( x)  0
f ( x)  0
First Derivative Test
If f  changes from – to + at c, f has a relative minimum at (c, f(c)).
If f  changes from + to – at c, f has a relative maximum at (c, f(c)).
If f  doesn’t change sign at c, then f(c) is not a relative extremum for f.
Where “c” is a critical value.
Process:
Step 1 Find all critical numbers of f in the given interval; break the interval into smaller intervals using these
critical values, points of discontinuity, and the endpoints.
Step 2 Create a sign chart, by picking an x-value in each interval and finding the sign of f  there.
Step 3 For each sub-interval state whether f is increasing or decreasing there.
Example
1.
Tell where f ( x)  x3  2 is increasing, decreasing, and identify any relative extrema.
x3
is increasing, decreasing, and identify any relative extrema.
x2
2.
Tell where f ( x) 
3.
Use the graph of f  to tell where f is increasing, decreasing, and identify any relative extrema.
4
Calculus for Business
Lesson- Concavity, Inflection, 2nd Derivative Test
Name:____________________________________
Date:_____________________________________
Objective:
To use derivatives to determine concavity and points of inflection
Concavity-
Function has a hill (concave down) or a valley (concave up)
Points of Inflection- Location at which a function goes from being concave down to concave up (or vice versa)
Concave Up
Concave Down
Inflection
Second Derivative Test for Concavity
Process:
1. Find the 1st derivative
2. Set = 0 and solve
3. Find the 2nd derivative
4. Set = 0 and solve (result could be a point of inflection)
5. Create sign chart
Guided Example
1 3 1 2
x  x
3
2
2
f ' ( x)  x  x
f " ( x)  2 x  1
0x x
0  2x  1
x  0,1
x  1 / 2
f ( x) 
2
Intervals
Test Values
Sign of f’(x)
Sign of f”(x)
Conclusion
 ,1
-2
+
Increasing
Concave down
(-1, -1/2)
-3/4
+
Increasing
Concave down
(-1/2, 0)
-1/4
+
+
Increasing
Concave up
0, 
2
+
+
Increasing
Concave up
Since the function went from concave down to concave up at x = -1/2, that must be the inflection point.
5
Examples
1.
Find the intervals where f ( x) 
6
is concave upward and concave downward.
x 3
2
Determine relative extrema, concavity, points of inflection.
2.
g ( x)  x 3  2 .
3.
  
h( x)  2 x  tan x ;   ,  .
 2 2
Let f be a function with f (c)  0 and the second derivative f  exists on an open interval containing c.
Then
if f”(c)>0, then f has a relative minimum at (c, f(c)),
if f”(c)<0, then f has a relative maximum at (c, f(c)),
if f”(c)=0, then test fails; must use the 1st Derivative Test for Extrema (could be max, min, or neither)
Find all the relative extrema
x
.
x 1
4.
f ( x)  x3  9 x 2  27 x.
5.
f ( x) 
6.
f ( x)  x 2  1.
7.
f ( x)  sin x  cos x.
6
Calculus for Business
Lesson- Curve Sketching
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to sketch a graph using calculus and without using a graphing calculator
Analyzing and Sketching the Graph of a Function
1. Decide domain and (where possible) range.
2. Determine x- and y-intercepts.
3. Note symmetry where applicable. (even function: f(-x) = f(x) -- symmetric across y-axis
odd function: f(-x) = -f(x) – symmetric across origin)
4. Find any points of discontinuity.
5. Find any vertical and horizontal asymptotes.
6. Find x-values where f and f are 0 or non-existent. Use these to get relative extrema and pts. of inflection.
7. Find end behaviors that are infinite.
Example 1 Polynomial Function
f(x) = x4 - 12x3 + 48x2 - 64x
Example 2
Rational Function
2
2( x  9)
f ( x)  2
 x  4
7
Calculus for Business
Lesson- Optimization
Name:____________________________________
Date:_____________________________________
Optimization Problems—Applied Minimum and Maximum Problems
Method for Solving:
Step 1 Define variables needed to describe the quantity to be optimized.
Step 2 Write a primary equation for the quantity to be optimized.
Step 3 Use other info given in the problem to write a secondary equation that relates the variables used in the
primary equation. Solve this equation to get one variable in terms of the other. Use this equality to rewrite the primary equation in terms of one variable.
Step 4 Find a feasible domain for the function you wrote in Step 3, i.e., upper and lower bounds on the variable
that make sense for your problem.
Step 5 Find absolute or relative max or min of your function on the feasible domain.
Step 6 Write your answer in an English sentence.
Example 1
Sr. Karlien wants to create a rectangular garden. She has available 100 feet of fencing to make
the border. What are the dimensions of the plot with the largest area that can be enclosed with
this fence?
Example 2
Find 2 positive numbers whose product is 192 and whose sum is a minimum.
Example 3
A manufacturer wants to design an open box having a square base and a surface area of 108
square inches. What dimensions will produce a box of maximum volume?
Example 4
A box with a square base and open top must have a volume of 32,000 cubic cm.
Find the dimensions of the box that minimize the amount of material used
8
Mixed Problem Set- Optimization
1.
A box with an open top and a square base is to have a surface area of 108 square inches. Determine the
dimensions that will maximize the volume of the box.
2.
Two posts, one 12 feet high and the other 28 feet high, are 30 feet apart. They are anchored by two wires
running from the top of each pole to a single stake in the ground at a point between the two posts. Where
should the stake be placed so that the minimum amount of wire is used?
3.
A bus stop shelter has two square sides, a back, and a roof. The volume is 256 cubic feet. What
dimension will allow for the least amount of material to be used?
4.
Scrumptious Soup Company makes a soup can with a volume of 250 cm3. What dimensions will allow
for the minimum amount of metal to produce the can?
5.
A jumbo-size can of baked beans has a volume of 600 cm3. What dimensions will allow for the
minimum amount of metal to produce the can?
6.
The surface area of a can of chunked chicken requires 60 square inches of material. What dimensions
allow for maximum volume?
7.
A large can of tuna requires a surface area of 100 square inches. What dimensions provide the maximum
volume?
8.
A wire of length 12 inches can be bent into a circle, a square, or cut to make both a circle and a
square. How much wire should be used for the circle if the total area enclosed by the figure(s) is
to be a minimum? A maximum?
|
circle
9.
square
A window consisting of a rectangle topped by a semicircle is to have an outer perimeter P. Find the
radius of the semicircle if the area of the window is to be a maximum.
9
10.
A rectangular field as shown is to be bounded by a fence. Find the dimensions of the field with
maximum area that can be enclosed with 1000 feet of fencing. You can assume that fencing is not
needed along the river and building.
11.
A company manufactures cylindrical barrels to store nuclear waste. The top and bottom of the barrels
are to be made with material that costs $10 per square foot and the rest is made with material that costs
$8 per square foot. If each barrel is to hold 5 cubic feet, find the dimensions of the barrel that will
minimize the total cost.
12.
The operating cost of a truck is 12 
13.
A furniture business rents chairs for conferences. A contract is drawn to rent and deliver up to 400 chairs
for a particular meeting. The exact number would be determined by the customer later. The price will be
$90 per chair up to 300 chairs. If the order goes above 300 chairs, the price would be reduced by $0.25
per chair for every additional chair ordered above 300. This reduced price would be applied to the entire
order. Determine the largest and smallest revenues this business can make under this contract.
14.
The speed of traffic through the Lincoln Tunnel depends on the density of the traffic. Let S be the speed
in miles per hour and D be the density in vehicles per mile. The relationship between S and D is
D
approximately S  42 
for D  100 . Find the density that will maximize the hourly flow.
3
15.
A commercial cattle company currently allows 20 steer per acre of grazing land. On average a steer
weighs 2000 pounds at the market. Estimates by the Department of Agriculture indicate that the average
weight per steer will be reduced by 50 pounds for each additional steer added per acre of grazing land.
How many steer per acre should be allowed in order to optimize the total market weight of the cattle?
x
cents per mile when the truck travels x miles per hour. If the
6
driver earns $6 per hour, what is the most economical speed to operate the truck on a 400 mile turnpike?
Due to construction, the truck can only travel between 35 and 60 miles per hour.
10
16.
Catching Rainwater. A 1125-ft3 open-top rectangular tank with a square base x ft on a side and
y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with
the tank involve not only the material from which the tank is made but also an excavation charge
proportional to the product xy. If the total cost is c = 5(x2 + 4xy) + 10xy, what values of x and y will
minimize it?
17.
Designing a Poster. You are designing a rectangular poster to contain 50 in2 of printing with a 4-in.
margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the
amount of paper used?
18.
Vertical Motion. The height of an object moving vertically is given by s = -16t2 + 96t + 112,
with s in ft and t in sec. Find (a) the object's velocity when t = 0, (b) its maximum height and when it
occurs, and (c) its velocity when s = 0.
19.
Finding an Angle. Two sides of a triangle have lengths a and b, and the angle between them is  .
What value of  will maximize the triangle's area? (Hint: A = ½ ab sin  .)
20.
Designing a Can.
What are the dimensions of the lightest open-top right circular cylindrical can that
will hold a volume of 1000 cm3?
21.
Designing a Can.
You are designing a 1000-cm3 right circular cylindrical can whose manufacture
will take waste into account. There is no waste in cutting the aluminum for the side, but the top and
bottom of radius r will be cut from squares that measure 2r units on a side. The total amount of
aluminum used up by the can will therefore be A = 8r2 + 27 rh the ratio of h to r for the most
economical can is?
22.
Designing a Box with Lid A piece of cardboard measures 10- by 15-in. Two equal squares are
removed from the comers of a 10-in. side as shown in the figure. Two equal rectangles are removed
from the other corners so that the tabs can be folded to form a rectangular box with lid.
(a)
Write a formula V(x) for the volume of the box.
(b)
Find the domain of V for the problem situation and graph V over this domain.
(c)
Use a graphical method to find the maximum volume and the value of x that gives it.
(d)
Confirm your result in (c) analytically.
11
23.
Designing a Suitcase.
A 24- by 36-in. sheet of cardboard is folded in half to form a 24- by 18-in.
rectangle as shown in the figure. Then four congruent squares of side length x are cut from the corners
of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and
a lid.
(a)
Write a formula V(x) for the volume of the box.
(b)
Find the domain of V for the problem situation and graph V over this domain.
(c)
Use a graphical method to find the maximum volume and the value of x that gives it.
(d)
Confirm your result in (c) analytically.
(e)
Find a value of x that yields a volume of 1120 in3.
24.
Quickest Route.
Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a
straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where
should she land her boat to reach the village in the least amount of time?
25.
Inscribing Rectangles.
A rectangle is to be inscribed under the arch of the curve y = 4cos(0.5x)
from x = -  to x =  . What are the dimensions of the rectangle with largest area, and what is the largest
area?
12
Calculus for Business
Lesson- Business Applications
Name:____________________________________
Date:_____________________________________
Objective-
To learn about maximizing yield and revenue while minimizing costs of inventory control.
Cobb-Douglas productivity models
Recall:
Revenue (R(x))- amount received in sale of units produced.
Maximizing Revenue involves determining the price at which a quantity should be sold in order to
obtain maximum revenue.
Profit (P(x))- revenue minus cost: R(x) – C(x); where R is revenue and C is cost
Maximizing Profit involves determining the price at which a quantity should be sold in order to obtain
maximum profit.
Example 1:
Rosie’s Discount Mart sells paperback books. At price $p, Rosie can sell q( p)  5 p 2  55 p  60 books. What
price would give Rosie the greatest revenue?
Example 2:
Mark’s restaurant can produce one chicken sandwich for $2. The sandwiches sell for $5 each and at this price,
his customers buy 1200 sandwiches each month. Because of rising costs from suppliers, the restaurant is
planning on raising the price of the sandwich. Based on the results of the previous price increases, Mark
estimates that he will sell 120 fewer sandwiches each month for every $1 he increases the price. At what price
should the sandwiches be sold to maximize Mark’s profit? What is the maximum profit?
13
Maximizing yield involves optimization situations in which an increase in one variable causes a decrease in
another related variable.
Example 3:
Taylor’s Orchard has always planted 40 trees per acre, with a yield of 300 apples per tree. For each additional
tree planted per acre, the yield drops by 5 apples per tree. How many trees should be planted per acre for
maximum yield?
Inventory Control:
deciding the most appropriate time to produce x quantity of an item.
Possibilities:
1.
Produce all of a given item at the beginning of a year.
Advantages: All items are on hand for immediate sale. Cheaper to produce at bulk rate.
Disadvantage: Need to store unsold items
2.
Produce items throughout the year as needed.
Advantage: No cost to store items
Disadvantage: Will cost more to produce in smaller quantities
Example:
MAC Boats anticipates a demand for 12,000 fishing boats over the next year. The start-up costs
of each production run are $5000, and it costs the company $40 to store each boat during the year. How many
boats should be made during each production run to minimize total costs? How many production runs should
there be?
14
Cobb-Douglas Productivity Model
The productivity of a plant or factory is given by: q  Kx a y1a where q is the number of units produced, x
is the number of employees, and y is the operating budget or capital. K and a are constants that are determined
by each individual factory or plant, with 0 < a < 1.
Example:
Brian’s Beach Shop manufactures surfboards. Daily operating costs are $80 per employee and
$25 per machine. The number of surfboards produced each day is given by q  4.5x 0.8 y 0.2 , where x is the
number of employees and y is the number of machines. If Brian wants to produce 90 surfboards each day at
minimum cost, how many employees and how many machines should he use?
Point of Diminishing Returns
the point at which the rate of growth of the profit function begins to decline.
The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on
1 3
s  6 s 2  400 . Find the amount of money the company should spend on advertising in
advertising is P ( s ) 
10
order to yield the maximum profit. Find the point of diminishing returns.
15
Mixed Problem Set- Business Applications
1.
Marge is planning a casino bus trip. If 100 people sign up, the cost is $300 per person. For each
additional person above 100, the cost per person is reduced by $2 per person. To maximize Marge's
revenue, how many people should go on the trip? What is the cost per person?
2.
A charter dinner cruise boat holds 50 people. The company will charter the boat for 35 or more people.
If 40 people are on board, the cost per person is $150. For each additional person, the cost per person is
reduced by $3. How large a group should be on the cruise to maximize the revenue? What is the cost per
person?
3.
A peach orchard has an average yield of 90 bushels per tree if there are 20 trees per acre. For each
additional tree per acre, the yield decreases by 3 bushels per tree. How many trees should the orchard
plant per acre to maximize the yield? What is the total yield?
4.
An orange grove plants 25 trees per acre and gets a yield of 116 bushels of oranges per tree. For each
additional tree planted per acre, the yield decreases by 4 bushels per tree. How many trees should be
planted per acre to maximize the yield? What is the total yield?
5.
Matt's Top 40 rents movies. If the rental fee is $4 each, Matt knows he can rent 100 movies per week.
For each additional $1 increase in the rental fee, Matt loses 10 rentals per week. What rental fee should
Matt charge for a movie to maximize his revenue? If each movie costs Matt $2, what should his rental
price be to maximize profit?
6.
Barb's Babysitting charges $8 per hour and, at that rate, averages 20 jobs each week. For each additional
$1 charge per hour, the number of jobs per week declines by two. What should Barb charge per hour to
maximize revenue? If Barb spends $2 per job on supplies, what should she charge per hour to
maximize her profits?
7.
When Jerry's Jalopies charges $20 to do an oil change, there are 80 customers per month. For each
additional $1 charge, the number of customers per month drops by four. If it costs Jerry $5 per customer
for the supplies, what should he charge for an oil change to maximize profits? How many customers
will there be each month?
8.
Missy's Tutoring charges $35 per hour for a tutoring session and has 60 clients each week. For each
additional $1 charge, there are two fewer clients each week. It costs $12 per client for supplies. What
should Missy charge per hour to maximize profits? How many clients will there be each week?
9.
A bicycle plant assembles 2000 bicycles per month. Each production run costs $1200, and it costs $20
to store a bicycle for a month. How many production runs should the plant use to minimize inventory
costs? How many bicycles are assembled in each production run?
10.
A soda bottling company bottles 20,000 cases of lime soda each year. Each production run costs $1400,
plus a storage cost of $18 per case. How many production runs should the company use to minimize
inventory costs? How many cases are bottled in each production run?
11.
A textbook publisher estimates that the demand for a new calculus book will be 6000 copies. Each book
costs $12 to print, and set-up costs are $1800 for each printing run. Storage costs $3 per book per year.
How many books should be printed per printing run and how many printings should there be to
minimize inventory costs?
12.
A car dealer expects to sell 500 new convertibles tllis year. Each convertible costs the dealer $16,000
plus a fixed $5000 delivery fee per order. It costs $500 to store each car for a year. How many orders
should be made and how many cars should there be in each order to minimize inventory costs?
13.
A golf club manufacturer finds that production of golf clubs follows the model q. = 25xo.25l.75, where
x is the number of employees and y is the number of machines. If the manufacturer must produce 2000
clubs and it costs $70 per employee and $20 per machine, how many employees and how many
machines will minimize costs? If production is increased to 3000 clubs and costs increase to $30 per
machine, how many employees and how many machines will minimize costs?
16
Calculus for Business
Lesson- Implicit Differentiation
Name:____________________________________
Date:_____________________________________
So far we have been working with functions in explicit form (equations solved for y in terms of x). Now we
will learn how to work with implicit forms of equations (equations not solved for y or not easily solved for y).
Explicit Form of y = function of x;
y written explicitly in terms of x
y
Implicit Form of y = function of x;
an equation that relates y to x but where y cannot
necessarily be isolated
1
x
xy  1
Implicit Differentiation
Step 1
Step 2
Step 3
Step 4
Note:
Differentiate both sides of the equation with respect to x.
dy
Collect all terms which contain
on one side of the equation and everything else on the other side.
dx
dy
Factor
out of all the terms on the one side.
dx
dy
Solve for
dx
dy
will be in terms of x and y.
dx
Example 1
Graph x 2  y 2  16 and find
dy
implicitly.
dx
Find all points where the graph has a horizontal tangent.
dy
Using implicit form of
:
dx
Using explicit form of
dy
:
dx
Using explicit form of
dy
:
dx
Find all points where the graph has a vertical tangent.
Using implicit form of
dy
:
dx
17
Example 2
a.
Find the first derivative of x 2 y  y 2 x  2 by implicit differentiation.
b.
Find f’(2)
Example 3
x2 y 2

1
6
8
a.
Graph the hyperbola:
b.
Find an equation of the tangent line to the graph of the hyperbola
Example 4
Find an equation of the tangent line to the astroid x
Example 5
Differentiate implicitly:
2sin x cos y  1
a.
b.
2
3
y
x2 y 2

 1 at the point (3,-2).
6
8
2
3
 5 at the point (8,1).
cot y  x  y
18
Calculus for Business
Lesson- Related Rates
Name:____________________________________
Date:_____________________________________
Objectives:
Identify a mathematical relationship between quantities that are each changing.
Use one or more rates to determine another rate.
Process:
Step 1: Draw a diagram.
Step 2: Determine which quantities and rates are given, and which to be found.
Step 3: Identify the primary function to use. (Often this is a formula from geometry.)
Step 4: Differentiate with respect to the independent variable
Step 5: Write a related rates equation
Step 6: Substitute known quantities and solve for desired rate.
[NOTE: Do not substitute known quantities before this last step!]
Type 1: Explicit Function of One Variable
Examples
1.
Air is being blown into a sphere at the rate of 6 cubic inches per minute. How fast is the radius
changing when the radius of the sphere is 2 inches?
2.
The edge of a cube is increasing at a rate of 2 inches per minute. At the instant the edge is 3 inches, how
fast is the volume increasing?
19
3.
A point moves along the curve y   x  3 such that its x-coordinate is increasing at 4 units per second.
(a)
At the moment x = 1, how fast is the y-coordinate changing? Interpret your answer based on the
shape of the graph and the location of the point.
2
(b)
At the moment x = 1, how fast is the point’s distance from the origin changing?
Type 2 - Implicit Function of One Variable
4.
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at
a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is
6 ft from the wall?
Type 3: Functions of Two Variables—2 rates given
5.
The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at
a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the
area is 100 cm2?
Type 4: Functions of Two Variables—1 rate given--secondary equation needed
6.
A water tank has the shape of an inverted circular cone with base radius 2m and height 4 m. If water is
being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the
water is 3 m deep.
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Mixed Problem Set- Related Rates
1.
A conical tank is being filled with water. The tank has height 4 ft and radius 3 ft. If water is being
pumped in at a constant rate of 2 cubic inches per minute, find the rate at which the height of the cone
changes when the height is 26 inches. Note the difference in units.
2.
A searchlight is positioned 10 meters from a sidewalk. A person is walking along the sidewalk at a
constant speed of 2 meters per second. The searchlight rotates so that it shines on the person. Find the
rate at which the searchlight rotates when the person is 25 meters from the searchlight.
3.
A person 5 feet tall is walking toward an18 foot pole. A light is positioned at the top of the pole. Find
the rate at which the length of the person’s shadow is changing when the person is 30 feet from the pole
and walking at a constant speed of 6 feet per second.
4.
The length of a rectangle increases by 3 feet per minute while the width decreases by 2 feet per minute.
When the length is 15 feet and the width is 40 feet, what is the rate at which the following changes:
a. area
b. perimeter
c. diagonal
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5.
6.
7.
1
C 2 h where C is the circumference of the tree in meters at
12
ground level and h is the height of the tree in meters. Both C and h are functions of time t in years.
The volume of a tree is given by V 
dV
. What does it represent in practical terms?
dt
a.
Find a formula for
b.
Suppose the circumference grows at a rate of 0.2 meters/year and the height grows at a rate of 4
meters/year. How fast is the volume of the tree growing when the circumference is 5 meters and
the height is 22 meters?
a.
When the radius of a spherical balloon is 10 cm, how fast is the volume of the balloon changing
with respect to change in its radius?
b.
If the radius of the balloon is increasing by 0.5 cm/sec, at what rate is the air being blown into
the balloon when the radius is 6 cm?
c.
When the volume of the balloon is 50 cubic cm, at what rate is the radius of the balloon
changing?
When hyperventilating, a person breathes in and out very rapidly. A spirogram is a machine that
draws a graph of the volume of air in a person’s lungs as a function of time. During hyperventilation, the
person’s spirogram trace might be represented by V  3  0.05 cos200t  where V is the volume of air
in liters in the lungs at time t minutes.
a.
Sketch a graph of one period of this function.
b.
What is the rate of flow of air in liters/minute? Sketch a graph of this function.
c.
Mark the following on each of the graphs above.
i) the interval when the person is breathing in
ii) the interval when the person is breathing out
iii) the time when the rate of flow of air is a maximum when the person is breathing in
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