Unit V Applications of Derivatives

Unit V Applications of Derivatives Curve Sketching Inequalities (Not covered in 3205) Polynomial Rational Functions Related Rate Problems Optimization (Max/Min) Problems 1

Inequalities Linear Polynomial Rational

Linear Inequalities Solve the following: (get x by itself) Give answer in interval notation. A) 2x 3 > 7 B) 4x 3 2x 5

C) 5 2x 12 D) 2 3x 4 8

E) 2x 1 4x 1 7x 2

Polynomial Inequalities Consider x 2 2x 3 > 0 Here we are looking for values of x where the function is greater than zero (aka positive). On the graph this will occur when the parabola is above the x-axis. To determine where this occurs we need to find the x-intercepts.

Solve: x 2 2x 3 = 0 Graph Where is the function (y-values) positive? The solution to x 2 2x 3 > 0 is:

These polynomial inequalities can be solved without sketching the complete graph by using only a number line. Using a number line. o Find the x-intercepts of the polynomial and place them on a number line. Put an open dot if the inequality is not equal to zero. For the last example:

This breaks the number line into different intervals. In each interval the function will either be entirely above or below the x-axis. o How do we know that the function does not cross the x-axis in the interval? o There would be another x-intercept in the interval, if it crossed the x-axis.

x 3x 1 Once the interval are established we then test each interval to determine if the function is positive or negative. o This is done by taking a number in each interval and subbing it in the factored form of the polynomial.

Solve the following: A) 2 2x 3 7x

B) 3 2 x x x 6 0

C) 3 2 2x 6x x 3 0

D) 3 2 x 13x 15 x

Rational Inequalities x 1 Consider 2x Here we look for zeros AND for undefined values Zeros occur when the numerator is zero o (Roots) Undefined values occur when the denominator is zero. o (Vertical Asymptotes) Occasionally values give undefined. o (Holes in the graph) 0 0 0 which is also

x 1 2x 0 Find zeros Find undefined values

Solve: 1. 2 x 1 0 2 x 4x Find zeros Find undefined values

2. x 2 x 1 Find zeros Find undefined values

3. x x 2 2 3x 2 0 4x 3 Find zeros Find undefined values

Curve Sketching Use derivatives to sketch the graph of polynomial and rational functions. o The first derivative will identify the intervals of increase and decrease, critical numbers, and relative and absolute extrema. o The second derivative will identify the hypercritical numbers, intervals of concavity and points of inflection. 25

Monotonicity In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing, or non-decreasing), if when moving left to right along the graph, the y- coordinate increases in value o (ie. if f (b) > f (a) whenever b > a). 26

Likewise, a function is called monotonically decreasing (also decreasing, or nonincreasing) if when moving left to right along the graph the y-coordinate decreases in value o (ie. if f (b) < f (a) whenever b > a). 27

Example: 1. Determine the intervals of increase and decrease for the following: 28

Example 2. Sketch f ( x ) ( x 2) x 1 Where is f(x) increasing?,0 U2, Where is f(x) decreasing? 2 0,2 What do you notice about the slopes of the tangents when f(x) is: o Increasing Positive o Decreasing Negative 30

Determine the first derivative of f ( x ) ( x 2) x 1 2 Determine the intervals where f( x) 0 f( x) 0 Does this match where the graph is increasing and 31 decreasing?

What is occurring on the graph of the function when f( x) 0? o There is a horizontal tangent. o It may also be a turning point in the graph as it is in both of these cases. o The graph has a maximum value at o The graph has a minimum value at 32

Basically as you go from left to right on a graph: If f(x) is rising then f(x) is increasing If f(x) is dropping then f(x) is decreasing The slope of the tangents are positive The slope of the tangents are negative f x 0 fx 0 33

Test for Monotonicity (Increasing or Decreasing) 1. If fx 0 for all x in an interval then f is increasing on that interval. 2. If fx 0 for all x in an interval then f is decreasing on that interval. 34

Examples 1. Find the intervals of increase and 3 3 2 decrease for f x x x 2 Find fx 0 Test each interval 35

Draw a rough sketch of y = f(x) Why are the x- values, 0 and 1, are not included in the intervals for increase and decrease? The function is not increasing nor decreasing when there is a horizontal tangent. 36

2. Find the intervals of increase and decrease for 2 2 f x x 4 3 Find fx 0 Test each interval 37

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Extreme Values (Extrema) What is the highest mountain in Newfoundland? 41

Highest Peaks of the Long Range Mountains of Newfoundland Rank Name m ft 1 The Cabox 814 2671 2 Gros Morne 807 2644 3 Blue Mountain 800 2625 4 Big Level 795 2608 5 Round Hill 763 2653 6 Rocky Harbour Hill 756 2480 7 Mount Saint Gregory 686 2251 8 Gros Paté 673 2208 9 Big Hill 659 2162 10 Old Crow 649 2129 42

So, is The Cabox the absolute highest mountain in the province? No, Mount Caubvick is!! All of the high mountains in Newfoundland are local maximums whereas Mount Caubvick is the Absolute Maximum for our province! 43

Maximum and Minimum Values (Extrema) in Mathematics Absolute Extrema: Definition: A function f(x) has an absolute maximum at x = c if f ( c) f ( x ) for all x in the domain of f(x). y = f(c) is called the maximum value of f A function f(x) has an absolute minimum at x = c if f ( c) f ( x ) for all x in the domain of f(x). y = f(c) is called the minimum value of f 44

Example: What is the absolute extrema for this function on [-2, 1]? Absolute Min Absolute Max 45

Example 2: Find the Absolute Extrema Absolute Min Absolute Max 46

Maximum and Minimum Values (Extrema) Relative (or Local) Extrema: Definition: A function f(x) has a relative(or local) maximum at x = c if f ( c) f ( x ) when x is close to c. (on both sides of c) y = f(c) is called a relative maximum value of f A function f(x) has a relative(or local) minimum at x = c if f ( c) f ( x ) when x is close to c. (on both sides of c) y = f(c) is called a relative minimum value of f 47

Unlike absolute extrema there may be more than one relative max or min. A relative max or min could also be an absolute max or min. 48

Example: What is the relative extrema for this function on [-2, 1]? Relative Min Relative Max 49

Example 2: Find the Relative Extrema Local Min Local Max 50

Summary The local maximum or minimum are the highest or lowest points on the graph within a certain interval on the graph. The absolute maximum and minimum values are found at local extrema or at the endpoints on the interval. 51

Analyze the following graph with a focus on the following: Point A is an absolute maxiumum. Point B is both a local and absolute minimum. Point C is a local maximum value. Point D is a local minimum. Point E is a local maximum value Point F is neither a local nor an absolute minimum 52

Note: Functions need not have maximum or minimum values Example: y = x 3 This graph has no finite maximum or minimum 54

Finding Extrema Critical Number Definition: If c is in the domain of f(x) and fc 0 or f c is undefined then c is called a critical number. Fermat s Theorem: If f(x) has a local maximum or minimum at c then c must be a critical number. 55

Note: Critical numbers do not always give a maximum or minimum value Example: y = x 3 What is the critical number for this function? Solve f x 0 f x x x 0 2 3 0 0 is a critical number but this function does not have a local maximum or minimum. 56

So, critical numbers do not always give a maximum or minimum value, but they give a place to start looking for them!! 57

Finding Absolute Extrema of a continuous function on a closed interval [a, b] 1. Find critical numbers, c, and evaluate the critical numbers ( find f(c) )that are in [a, b]. 2. Find the values of f(x) at the endpoints. Find f(a) and f(b). 3. A) The largest value from steps above is the absolute maximum. B) The smallest value from steps above is the absolute minimum. 58

Example: Find the absolute extrema for 1. y = f(x) on the indicated intervals. f x x x x on 3 2 2 3 36 62, [ 3, 4] Find the critical numbers 2 6 6 36 f x x x 2 6x 6x 36 0 2 x x 6 0 x 2 x 3 0 x 2 x 3 Evaluate the critical numbers 2 f f 3 Evaluate the endpoints f 3 f 4 Critical Numbers The Absolute Maximum is. The Absolute Minimum is. 59

2. 1 2 3 f x x 1, on [ 2,2] Find the critical numbers Evaluate the critical numbers Evaluate the endpoints The Absolute Maximum is. The Absolute Minimum is. 60

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Checking for Local Extrema First Derivative Test (checking slopes) Using Intervals of Increase and Decrease a b c d 64

First Derivative Test Let c be a critical number of a continuous function f. 1. If fx changes from positive to negative (f(x) goes from Inc to ) Dec at c then there is a local Max at c. 2. If fx changes from negative to positive (f(x) goes from Dec to ) Inc at c then there is a local Min at c. 3. If fxdoes not change sign. (f(x) goes from inc to inc or from dec to dec) then f(x) has no local max or min at c. 65

Example: Find the local extrema of f(x) 1. f x x x 3 ( ) 3 1 Find the critical numbers. Check intervals of Increase and Decrease 66

Example: Find the local extrema of f(x) 2. f( x) 2 x x 1 Find the critical numbers. Check intervals of Increase and Decrease 67

Example: Find the local extrema of f(x) 3. f x 2 x 4 ( ) 3 1 Find the critical numbers. Check intervals of Increase and Decrease 68

Sketch the graph 69

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Concavity and the Second Derivative Definition of Concavity: Let f(x) be a differentiable function on (a, b). We say that the graph of f(x) is: 1. Concave Up on (a, b) if fx is increasing on (a, b). 2. Concave Down on (a, b) if fx is decreasing on (a, b). 71

Concave Up Concave Down a f x is getting larger (increasing). Note: If f is concave up the curve lies above its tangent lines. (Holds Water) If f x is increasing then f x b a f x is getting smaller (decreasing). Note: If f is concave up the curve lies below its tangent lines. (DOES NOT Hold Water) If f x is decreasing then f x 0 0 b TEST FOR CONCAVITY 72

Definition: Point of Inflection If a graph of a continuous function has a point where its concavity changes from upward to downward (or vise versa) we call that point a Point of Inflection. 73

Example: Point of Inflection Point of Inflection Point of Inflection NOTE: 1. At each point of inflection the curve crosses its tangent line, (if it exists) 74

2.It follows from the test from concavity that there will be a point of inflection where the second derivative changes sign. Concave Up to Concave Down f( x) 0 f( x) 0 Concave Down to Concave Up f( x) 0 f( x) 0 75

Property of Points of Inflection If the point (c, f(c)) is a point of inflection of f(x) then either f( c) 0 or f( c) is undefined c is also called a hypercritical number. Hypercritical numbers occur at x = c if f( c) 0 or if f( c) is undefined. NOTE: A point of inflection occurs at a hypercritical number but not all hypercritical numbers give a point of inflection 76

Determine the intervals of concavity and find any points of inflection 1. f(x) = x 4 77

2. f(x) = x 4 + x 3 3x 2 + 1 78

3. f( x) 2 6 x 3 79

LETS do this one in class 81

2 3 y 3x 2x A) Find the intervals of increase and decrease. B) Find the local extrema. 82

2 3 y 3x 2x C) Find the intervals of concavity. D) Find the points of inflection. 83

2 3 y 3x 2x E) Sketch the graph 84

Second Derivative Test for finding local extrema 1. If fc 0 and f c 0 f is concave up then f has a local minimum at c. 2. If fc 0 and f c 0 f is concave down then f has a local maximum at c. 3. If fc 0 and f c 0 then you must use the first derivative test to find the local max or min 85

Find the local extrema of f(x) = x 3-6x 2 + 5 Find the critical numbers Use the 2 nd derivative test f x x x 3 xx ( 4) 0 x 0 x 4 2 3 12 0 f x 6x 12 f 0 6(0) 12 12 Thus f(x) is concave down at x = 0 Therefore f(0) = 5 is a local max 4 6(4) 12 12 f Thus f(x) is concave up at x = 4 Therefore f(4) = -27 is a local min 86

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CURVE SKETCHING 89

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CURVE SKETCHING Many texts present curve sketching as a rather complicated process. In the text curve sketching is an eight step procedure. We prefer to look at the function and its first and 2 nd derivative and write down all of the information that can be found from them. This information is used to sketch the graph 91

Consider f(x). What does the function give you? Domain Intercepts - Symmetry - Asymptotes: x- int y = 0 y-int x = 0 Odd Even f ( x ) f ( x ) f ( x ) f ( x ) Horizontal Vertical Slant lim f( x) x Places where f(x) has division by zero Check limit of both left and right sides Occur when f(x) has a numerator one degree larger than denominator. 92

Consider What does the 1 st derivative give you? Critical Numbers f x 0 f x or is undefined Intervals of Increase and Decrease f x 0 f x 0 Local Extrema Use the 1 st or 2 nd derivative test 93

Consider What does the 2 nd derivative give you? HyperCritical Numbers 0 f Intervals of Concave UP and Concave Down x f x or is undefined f x 0 fx 0 Determine the points of inflection 94

Asymptotes Plot all points Intercepts Max/min points Points of inflection Tie ALL together Sketch the Graph Draw the curve (ie connect the dots) paying attention to concavity 95

Examples. Sketch the following 1. y = x 3 3x 2 + 3x - 1 96

2. y = 2x 3 3x 2-12x + 9 98

3. f( x) 2x 2 x 1 100

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4. f( x) 2 x 2 x 3 102

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5. Sketch the graph of y = f(x) based on the following information: lim f f(0) = 1, f(-2) f(2) are undefined x 1 x f 0 0, f (0) 0 lim f x 0 x lim f x, lim f x, lim f x x2 x2 x 2 104

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Related Rate Problems

Rates of Change Here we look at how one thing changes with respect to another thing changing. Example: We are inflating a balloon We can find how fast the Volume changes wrt the radius or time Surface area changes wrt the radius or time In Physics Velocity is the changing of displacement wrt time Acceleration is the changing of velocity wrt time

In biology, there is a rate of change of population with respect to time. In thermal dynamics, there is the expansion of metal with respect to temperature

Example: 1. A spherical balloon is being inflated. A) Find the rate of change of the volume wrt the radius when the radius is 5 cm. Find dv V 4 3 r dv 4 dr r dv 45cm 2 dr r 5cm 3 2 dr 2 100cm

B) Find the rate of change of the surface area wrt the radius when the radius is 5 cm. A Find da dr 4 r 2 da 8 dr r da 85cm dr r 5cm 40cm

Related Rates In related rate problems we are given the rate of change of one quantity and we are asked to find the rate of change of a related quantity. To do these problems we determine the equation which relates the 2 quantities and use the Chain Rule to differentiable wrt time.

Most related rate problems are one of 2 categories. Distance Problems Size problems (Volume or area)

For Example: If you blow air into a balloon at a certain rate what are you changing: Volume of balloon Radius of balloon da Surface Area dt dp Pressure dt dv dt dr dt

Example 2 Suppose air is going into a spherical balloon at a rate of 100 cm 3 /s. = dv dt Determine how fast the radius is changing when the radius is A) 2 cm B) 4 cm Find dr dv dr dt dt 2 4 Differentiate dt 4 3 r V r wrt time 3 dr A) dv 4 dt 3 3 2 dr r dt r 2 dt 3 dr 100 dt 4 2 Solve for dr dt 1.989 cm s cm cm s cm 2 dr B) dt r dr dt 100 4cm 3 cm s 4 4cm 0.497 cm s 2

Note: As the balloon gets larger the rate at which it gets larger becomes less. 3. For the same situation determine the rate at which the surface area changes when A) 2 cm B) 4 cm 2 A 4r da dr 8r dt dt da A cm dt cm ) 8 2 1.989 s 2 da cm 100 dt s da cm B) 84cm 0.497 dt s 2 da cm 50 dt s

4. Suppose the volume is increasing at a rate of 20 cm 3 /s. Find dv (i) dr dt dr dt dv dt dr dt V when t = 3.0 s 4 3 r 3 dr dt 2 4r dv dt 4 r 2 Differentiate wrt time Solve for dr dt Determine r when t=3 3 cm 20 s 4 2.429cm 2 0.270 cm s 3 cm Note: 20 dt s 3 cm 3 v (3) 20 (3 s ) 60cm s 3 4 3 3 60cm r r 3 3 60cm 3 4 r 2.429cm

(ii) da dt when t = 3.0 s A 4 r 2 da dr 8r dt dt da cm 82.429cm 0.270 dt s 2 da cm 16.483 dt s

5. Suppose the volume is increasing at a rate of 420 cm 3 /s. Find the rate the surface area is increasing after 10 seconds. Note: This problem can be solved like dr example 4 where we find first, but this dt time we are going to take a more direct way. First step is to find the relationship between volume and area. 118

A 4 r 2 V 4 3 V A 3 4 3V A 4 4 2 2 3 Differentiate wrt time 4 3 r Solve for V for r 3 3 V r 3 4 da 2 3V 3 dv 4 dt 3 4 4 dt 1 3 1 3 da 4 dv 2 dt 3 V dt 119

What is the volume when t =10 s? V 3 cm 420 *10s s 3 4200cm da dt t 10 1 3 da 4 dv 2 dt 3 V dt 1 3 3 4 cm 2 420 3 3 4200cm s 2 83.925 cm s 120

Other Problems 1. The population of a bacteria culture after t hours is given by: 2 3 N t 2000 20t 5t A) Find the rate of growth after 4 hours.

1. The population of a bacteria culture after t hours is given by: 2 3 N t 2000 20t 5t B) Find the interval when the growth is decreasing. C) What is the minimum number of bacteria? when does this occur?

2. A) Consider x 2 + y = 3 Find dy dt when x = 2 if dx dt 5

2. B) Consider xy + y 2 = x Find dx dt when y = 2 if dy 3 dt

Distance Problems 1. Suppose a 6 m ladder is leaning against a wall and the bottom of the ladder is sliding away from the wall at a rate of 0.5 m/s. At what rate is the ladder sliding down the wall at the instant when the bottom of the ladder is 3 m from the wall? http://keithnabb.com/animations

Draw an appropriate diagram to represent the situation. This is the most helpful step in related rates problems since it allows you to visualize the problem. 130

Assign variables to each quantity in the problem that is changing. t = time (seconds) x = the horizontal distance (metres) from the wall to the bottom of the ladder at time t y = the vertical distance (metres) from the top of the ladder to the ground at time t 131

List the given information and identify the unknown quantity in terms of the variables. o length of the ladder = 6 m o dx m 0.5 dt s o dy dt x 3m? 132

Write an equation representing the relationship between the variables. o Applying the Pythagorean theorem to the right triangle results in : x 2 + y 2 = 6 2 Implicitly differentiate the relationship with respect to time. 133

What is the significance of the negative result? o The ladder is falling down the wall and the vertical distance is getting smaller Also Note: o It is important to wait until the equation has been differentiated before substituting information into the equation. o If values are substituted too early, it can lead to an incorrect answer. 134

Distance Problems 2. Two airplanes in Horizontal flight cross over a town at 1 pm. One plane travels east at 300 km/h, while the other goes north at 400 km/h. At what rate is the distance between them changing at 3:00 pm?

Triangle Problems 3. A man 2 m tall walks away from a lamppost where the light is 6 m above the ground. If he walks at 2.0 m/s, at what rate is the shadow growing when he is 10 m from the lamppost?

4. A water-skier skis over a ski jump as shown below. If she is moving at a speed of 12 m/s, how fast is she rising when she leaves the jump? 138

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Volume/Area Problems 5. The length of a rectangle is increasing at a rate of 9 cm/s and its width is increasing at a rate of 5 cm/s. When the length is 27 cm and the width is 15 cm, how fast is the area of the rectangle increasing?

6. A) Water is flowing into a trough at a rate of 100 cm 3 /s. The trough has a length of 3 m and a cross-section in the form of a rectangle, whose height is 50 cm, and base is 25 cm. At what rate is the water rising when the depth is 25 cm?

6. b) Same problem as before except the trough a crosssection in the form of an isosceles triangle, whose height is 1 m, and top is 2 m. At what rate is the water rising when the depth is 25 cm?

6. C) Same problem as before except the trough a crosssection in the form of an isosceles trapezoid, whose height is 1 m, top is 2 m and base is 50 cm. At what rate is the water rising when the depth is 25 cm?

7. Snow is being dumped from a conveyor belt at a rate of 1.5 m 3 /min and forms a pile in the shape of a cone whose diameter and height are always equal. How fast is the height of the pile growing when the pile is 3 m high?

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Sketch the graph of a function f that satisfies all the following conditions. a) f(0) = 0, f ' (0) = f ''(0) = 0 b) f has three points of inflection, one of which is (-1, -1) c) f has a relative minimum of -2 at x = -3. d) One of the roots is x = 4 e) f is an even function. (symmetric about the y-axis)

Optimization Problems MAX/MIN PROBLEMS

Guidelines for Solving Max/Min Problems 1. Make a sketch, when feasible; assign meaningful symbols to the quantities that are given and those that are required to be found. i.e. acceleration height Area, etc. 2. Write a primary equation for the quantity to be optimized. 3. Reduce the primary equation so that it contains only one independent variable. This is done by the use of secondary (or constraint) equations. These are found in the restrictions of the problem. 4. Determine the desired maximum or minimum. This is done by finding the absolute max and min of the primary equation

Examples: 1. Eleanor Abernathy wishes to make a pen for her cats behind her house. If she has 60 m of fencing and her house is used as one side of the pen what are the dimensions of the largest pen that can be constructed? There are two quantities being discussed here o perimeter (P) and area (A). These can be written as : o P 60 2x y and A xy Express the area in terms of one variable,

First, lets determine the dimensions for the maximum area using non-calculus techniques. Since this is a quadratic function, you should make the connection between the maximum or minimum value, and the y-value of the vertex. So how do we find the vertex of a parabola? x b 2a 2 A x (60 2 x ) 2x 60x x 60 2( 2) 15 m y 60 2x y 60 2(15) 30 m Therefore the dimensions of the pen with the maximum area is 15 m by 30 m. 152

From a calculus perspective, remember that extreme values occur at critical points (or endpoints) and critical numbers occur where the first derivative equals zero or is undefined. A '(x) = 60-4x 0 = 60-4x x = 15 You can verify that the critical number x = 15 will produce a maximum value by either using the first or second derivative test. 153

In this example, it is easy to identify the second derivative (i.e., A" (x) = -4 < 0). Since this parabola is concave downward the critical point will be the Absolute Maximum. o x = 15 m, y = 30 m o and the maximum area = 450 m 2. 154

In this next example we must use calculus techniques since the Primary (or Optimizing) function is not quadratic. 155

2. Homer wishes to make an open box from a square piece of tin of dimensions 30 cm by 42 cm by cutting out equal squares in each corner and folding up the sides. Homer will get different shape boxes depending the size of the square removed from the corner. Click on image for a web site that calculates the max volume of the box How can we find the volume and the dimensions of the largest capacity box that can be made in this manner without using the internet?

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3. Find the maximum y value of where x 0,5 3 2 y x 6x 9x

4. A Norman window has a shape of a rectangle with the top edge replaced by a semi-circle. If the window has a perimeter of 8 m, determine the dimension of the window so that the greatest possible amount of light is admitted.

5. A rock is thrown upwards from a 200 m cliff. If the height of the rock above the bottom of the cliff is given by s(t) = -5t 2 + 20t + 200 find: a) the maximum height of the rock b) The minimum height of the rock.

6. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of length 3 cm and 4 cm, if the 2 sides of the inscribed rectangle lie along the legs.

7. A cola company wishes to introduce a new can size that will contain 355 ml and be more environmentally friendly. What would the dimensions of the can be such that the amount of aluminium used is a minimum?

GROUP PROJECT Each Group gets a random Optimization Project. The task is to solve the problem and present the findings to the class. You have one Week to complete the task. 164

Project A A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank costs $10 per square meter for the base and $5 per square meter for the sides, what is the cost of the least expensive tank? Justify your answer.

Project B A person has 340 yards of fencing for enclosing two separate fields, one of which is to be a rectangle twice as long as it is wide and the other a square. The square field must contain at least 100 square yards and the rectangular one must contain at least 800 square yards. a) If x is the width of the rectangular field, what are the maximum and minimum possible values of x? b) Set up a function A(x) that represents the total area enclosed by the two fields. c) What is the greatest number of square yards that can be enclosed in the two fields? Justify your answer.

Project C The top and bottom margins of a page are each 1.5 inches and the side margins are each 1 inch. The area of the printed material is 30 square inches. What are the dimensions of the page of smallest possible area? 1.5 1 Printed Material

Project D Determine the total number of local maximum and minimum points of the function whose derivative is given by f (x) = x(x 3) 2 (x + 1) 4

Project E What point on the curve is closest to the origin? y = x 2 + 4x + 3

Answers A $330 B a) b) 20 x 50 A(x) 2x 2 (85 1.5x) 2 c) 5100 C Width 6.47 inches height 9.71 inches D one E (-0.835,0.357)