Unit V Applications of Derivatives
|
|
- Dwight Carr
- 5 years ago
- Views:
Transcription
1 Unit V Applications of Derivatives Curve Sketching Inequalities (Not covered in 3205) Polynomial Rational Functions Related Rate Problems Optimization (Max/Min) Problems 1
2 Inequalities Linear Polynomial Rational
3 Linear Inequalities Solve the following: (get x by itself) Give answer in interval notation. A) 2x 3 > 7 B) 4x 3 2x 5
4 C) 5 2x 12 D) 2 3x 4 8
5 E) 2x 1 4x 1 7x 2
6 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.
7 Solve: x 2 2x 3 = 0 Graph Where is the function (y-values) positive? The solution to x 2 2x 3 > 0 is:
8 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:
9 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.
10 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.
11 Solve the following: A) 2 2x 3 7x
12 B) 3 2 x x x 6 0
13 C) 3 2 2x 6x x 3 0
14 D) 3 2 x 13x 15 x
15 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) which is also
16 x 1 2x 0 Find zeros Find undefined values
17
18 Solve: 1. 2 x x 4x Find zeros Find undefined values
19
20
21 2. x 2 x 1 Find zeros Find undefined values
22
23 3. x x 2 2 3x 2 0 4x 3 Find zeros Find undefined values
24
25 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
26 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
27 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
28 Example: 1. Determine the intervals of increase and decrease for the following: 28
29 29
30 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
31 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?
32 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
33 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
34 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
35 Examples 1. Find the intervals of increase and decrease for f x x x 2 Find fx 0 Test each interval 35
36 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
37 2. Find the intervals of increase and decrease for 2 2 f x x 4 3 Find fx 0 Test each interval 37
38 38
39 #1, 4acdefgh 39
40 40
41 Extreme Values (Extrema) What is the highest mountain in Newfoundland? 41
42 Highest Peaks of the Long Range Mountains of Newfoundland Rank Name m ft 1 The Cabox Gros Morne Blue Mountain Big Level Round Hill Rocky Harbour Hill Mount Saint Gregory Gros Paté Big Hill Old Crow
43 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
44 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
45 Example: What is the absolute extrema for this function on [-2, 1]? Absolute Min Absolute Max 45
46 Example 2: Find the Absolute Extrema Absolute Min Absolute Max 46
47 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
48 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
49 Example: What is the relative extrema for this function on [-2, 1]? Relative Min Relative Max 49
50 Example 2: Find the Relative Extrema Local Min Local Max 50
51 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
52 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
53 53
54 Note: Functions need not have maximum or minimum values Example: y = x 3 This graph has no finite maximum or minimum 54
55 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
56 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 is a critical number but this function does not have a local maximum or minimum. 56
57 So, critical numbers do not always give a maximum or minimum value, but they give a place to start looking for them!! 57
58 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
59 Example: Find the absolute extrema for 1. y = f(x) on the indicated intervals. f x x x x on , [ 3, 4] Find the critical numbers f x x x 2 6x 6x 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
60 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
61 TEXT Page # 5,6, 7, 9, 12, 23, 29, 31, 35, 38, 41, 42, 47, 49 61
62 5 p150
63 6 p150
64 Checking for Local Extrema First Derivative Test (checking slopes) Using Intervals of Increase and Decrease a b c d 64
65 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
66 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
67 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
68 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
69 Sketch the graph 69
70 TEXT Page 164 # 1, 4 (A and B only) 70
71 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
72 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
73 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
74 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
75 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
76 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
77 Determine the intervals of concavity and find any points of inflection 1. f(x) = x 4 77
78 2. f(x) = x 4 + x 3 3x
79 3. f( x) 2 6 x 3 79
80 80
81 LETS do this one in class 81
82 2 3 y 3x 2x A) Find the intervals of increase and decrease. B) Find the local extrema. 82
83 2 3 y 3x 2x C) Find the intervals of concavity. D) Find the points of inflection. 83
84 2 3 y 3x 2x E) Sketch the graph 84
85 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
86 Find the local extrema of f(x) = x 3-6x Find the critical numbers Use the 2 nd derivative test f x x x 3 xx ( 4) 0 x 0 x f x 6x 12 f 0 6(0) Thus f(x) is concave down at x = 0 Therefore f(0) = 5 is a local max 4 6(4) f Thus f(x) is concave up at x = 4 Therefore f(4) = -27 is a local min 86
87 TEXT Page 164 # 1, 4 (C and D only) 7, 8. 10, 11 87
88 DO #11 y x 88
89 CURVE SKETCHING 89
90 Warm UP #12 Page
91 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
92 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
93 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
94 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
95 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
96 Examples. Sketch the following 1. y = x 3 3x 2 + 3x
97 97
98 2. y = 2x 3 3x 2-12x
99 99
100 3. f( x) 2x 2 x 1 100
101 101
102 4. f( x) 2 x 2 x 3 102
103 103
104 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
105 TEXT Page 172 # 1, 5, 9, 12, 14, 17,
106 Related Rate Problems
107 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
108 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
109 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
110 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
111 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.
112 Most related rate problems are one of 2 categories. Distance Problems Size problems (Volume or area)
113 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
114 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 dr r dt r 2 dt 3 dr 100 dt 4 2 Solve for dr dt cm s cm cm s cm 2 dr B) dt r dr dt 100 4cm 3 cm s 4 4cm cm s 2
115 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 ) s 2 da cm 100 dt s da cm B) 84cm dt s 2 da cm 50 dt s
116 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 cm cm s 3 cm Note: 20 dt s 3 cm 3 v (3) 20 (3 s ) 60cm s cm r r cm 3 4 r 2.429cm
117 (ii) da dt when t = 3.0 s A 4 r 2 da dr 8r dt dt da cm cm dt s 2 da cm dt s
118 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
119 A 4 r 2 V 4 3 V A 3 4 3V A Differentiate wrt time 4 3 r Solve for V for r 3 3 V r 3 4 da 2 3V 3 dv 4 dt dt da 4 dv 2 dt 3 V dt 119
120 What is the volume when t =10 s? V 3 cm 420 *10s s cm da dt t da 4 dv 2 dt 3 V dt cm cm s cm s 120
121
122
123
124 Other Problems 1. The population of a bacteria culture after t hours is given by: 2 3 N t t 5t A) Find the rate of growth after 4 hours.
125 1. The population of a bacteria culture after t hours is given by: 2 3 N t t 5t B) Find the interval when the growth is decreasing. C) What is the minimum number of bacteria? when does this occur?
126 2. A) Consider x 2 + y = 3 Find dy dt when x = 2 if dx dt 5
127 2. B) Consider xy + y 2 = x Find dx dt when y = 2 if dy 3 dt
128
129 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?
130 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
131 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
132 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
133 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
134 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
135 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?
136
137 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?
138 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
139 Text Page 133 #11-14, 20,
140 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?
141 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?
142 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?
143 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?
144 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?
145 Text Page #3-6, 24-27, 33,
146 146
147 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)
148 Optimization Problems MAX/MIN PROBLEMS
149 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
150 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.
151 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,
152 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
153 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
154 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
155 In this next example we must use calculus techniques since the Primary (or Optimizing) function is not quadratic. 155
156 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?
157 157
158 158
159 3. Find the maximum y value of where x 0,5 3 2 y x 6x 9x
160 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.
161 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 t find: a) the maximum height of the rock b) The minimum height of the rock.
162 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.
163 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?
164 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
165 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.
166 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.
167 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? Printed Material
168 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
169 Project E What point on the curve is closest to the origin? y = x 2 + 4x + 3
170 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)
Unit V Applications of Derivatives
Unit V Applications of Derivatives Curve Sketching Inequalities (Covered in 2200) Polynomial Rational Functions Related Rate Problems Optimization (Max/Min) Problems 1 Inequalities Linear Polynomial Rational
More informationAP Calculus Free-Response Questions 1969-present AB
AP Calculus Free-Response Questions 1969-present AB 1969 1. Consider the following functions defined for all x: f 1 (x) = x, f (x) = xcos x, f 3 (x) = 3e x, f 4 (x) = x - x. Answer the following questions
More informationFree Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom
Free Response Questions 1969-010 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions
More informationAP Calculus BC Chapter 4 AP Exam Problems A) 4 B) 2 C) 1 D) 0 E) 2 A) 9 B) 12 C) 14 D) 21 E) 40
Extreme Values in an Interval AP Calculus BC 1. The absolute maximum value of x = f ( x) x x 1 on the closed interval, 4 occurs at A) 4 B) C) 1 D) 0 E). The maximum acceleration attained on the interval
More informationChapter 3: The Derivative in Graphing and Applications
Chapter 3: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems.
More information4.1 & 4.2 Student Notes Using the First and Second Derivatives. for all x in D, where D is the domain of f. The number f()
4.1 & 4. Student Notes Using the First and Second Derivatives Definition A function f has an absolute maximum (or global maximum) at c if f ( c) f ( x) for all x in D, where D is the domain of f. The number
More informationApplications of Derivatives
Applications of Derivatives Extrema on an Interval Objective: Understand the definition of extrema of a function on an interval. Understand the definition of relative extrema of a function on an open interval.
More informationChapter 2 Polynomial and Rational Functions
SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear
More informationSee animations and interactive applets of some of these at. Fall_2009/Math123/Notes
MA123, Chapter 7 Word Problems (pp. 125-153) Chapter s Goal: In this chapter we study the two main types of word problems in Calculus. Optimization Problems. i.e., max - min problems Related Rates See
More informationChapter 8: Radical Functions
Chapter 8: Radical Functions Chapter 8 Overview: Types and Traits of Radical Functions Vocabulary:. Radical (Irrational) Function an epression whose general equation contains a root of a variable and possibly
More informationMath 2413 General Review for Calculus Last Updated 02/23/2016
Math 243 General Review for Calculus Last Updated 02/23/206 Find the average velocity of the function over the given interval.. y = 6x 3-5x 2-8, [-8, ] Find the slope of the curve for the given value of
More informationApplications of Derivatives
Applications of Derivatives Related Rates General steps 1. Draw a picture!! (This may not be possible for every problem, but there s usually something you can draw.) 2. Label everything. If a quantity
More informationCalculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016
Name: Class: Date: Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016 Short Answer 1. Decide whether the following problem can be solved using precalculus, or whether calculus is required.
More informationCalculus I 5. Applications of differentiation
2301107 Calculus I 5. Applications of differentiation Chapter 5:Applications of differentiation C05-2 Outline 5.1. Extreme values 5.2. Curvature and Inflection point 5.3. Curve sketching 5.4. Related rate
More informationMath Fall 08 Final Exam Review
Math 173.7 Fall 08 Final Exam Review 1. Graph the function f(x) = x 2 3x by applying a transformation to the graph of a standard function. 2.a. Express the function F(x) = 3 ln(x + 2) in the form F = f
More informationAP Calculus AB Semester 1 Practice Final
Class: Date: AP Calculus AB Semester 1 Practice Final Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the limit (if it exists). lim x x + 4 x a. 6
More informationSection 3.1 Extreme Values
Math 132 Extreme Values Section 3.1 Section 3.1 Extreme Values Example 1: Given the following is the graph of f(x) Where is the maximum (x-value)? What is the maximum (y-value)? Where is the minimum (x-value)?
More informationMath3A Exam #02 Solution Fall 2017
Math3A Exam #02 Solution Fall 2017 1. Use the limit definition of the derivative to find f (x) given f ( x) x. 3 2. Use the local linear approximation for f x x at x0 8 to approximate 3 8.1 and write your
More information1. Determine the limit (if it exists). + lim A) B) C) D) E) Determine the limit (if it exists).
Please do not write on. Calc AB Semester 1 Exam Review 1. Determine the limit (if it exists). 1 1 + lim x 3 6 x 3 x + 3 A).1 B).8 C).157778 D).7778 E).137778. Determine the limit (if it exists). 1 1cos
More informationApplications of Differentiation
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Module9 7 Introduction Applications of to Matrices Differentiation y = x(x 1)(x 2) d 2
More informationAP Calculus AB Chapter 4 Packet Implicit Differentiation. 4.5: Implicit Functions
4.5: Implicit Functions We can employ implicit differentiation when an equation that defines a function is so complicated that we cannot use an explicit rule to find the derivative. EXAMPLE 1: Find dy
More informationAP CALCULUS AB Study Guide for Midterm Exam 2017
AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed
More informationCHAPTER 3 APPLICATIONS OF THE DERIVATIVE
CHAPTER 3 APPLICATIONS OF THE DERIVATIVE 3.1 Maxima and Minima Extreme Values 1. Does f(x) have a maximum or minimum value on S? 2. If it does have a maximum or a minimum, where are they attained? 3. If
More informationMultiple Choice. Circle the best answer. No work needed. No partial credit available. is continuous.
Multiple Choice. Circle the best answer. No work needed. No partial credit available. + +. Evaluate lim + (a (b (c (d 0 (e None of the above.. Evaluate lim (a (b (c (d 0 (e + + None of the above.. Find
More informationWorkbook for Calculus I
Workbook for Calculus I By Hüseyin Yüce New York 2007 1 Functions 1.1 Four Ways to Represent a Function 1. Find the domain and range of the function f(x) = 1 + x + 1 and sketch its graph. y 3 2 1-3 -2-1
More informationU of U Math Online. Young-Seon Lee. WeBWorK set 1. due 1/21/03 at 11:00 AM. 6 4 and is perpendicular to the line 5x 3y 4 can
U of U Math 0-6 Online WeBWorK set. due //03 at :00 AM. The main purpose of this WeBWorK set is to familiarize yourself with WeBWorK. Here are some hints on how to use WeBWorK effectively: After first
More informationAbsolute and Local Extrema. Critical Points In the proof of Rolle s Theorem, we actually demonstrated the following
Absolute and Local Extrema Definition 1 (Absolute Maximum). A function f has an absolute maximum at c S if f(x) f(c) x S. We call f(c) the absolute maximum of f on S. Definition 2 (Local Maximum). A function
More informationAnalyzing Functions. Implicit Functions and Implicit Differentiation
Analyzing Functions Implicit Functions and Implicit Differentiation In mathematics, an implicit function is a generalization of the concept of a function in which the dependent variable, say, has not been
More informationU3L2: Sec.3.1 Higher Order Derivatives, Velocity and Acceleration
U3L1: Review of Prerequisite Skills for Unit # 3 (Derivatives & their Applications) Graphing polynomial and simple rational functions Working with circles in standard position Solving polynomial equations
More informationDays 3 & 4 Notes: Related Rates
AP Calculus Unit 4 Applications of the Derivative Part 1 Days 3 & 4 Notes: Related Rates Implicitly differentiate the following formulas with respect to time. State what each rate in the differential equation
More informationTest 3 Review. fx ( ) ( x 2) 4/5 at the indicated extremum. y x 2 3x 2. Name: Class: Date: Short Answer
Name: Class: Date: ID: A Test 3 Review Short Answer 1. Find the value of the derivative (if it exists) of fx ( ) ( x 2) 4/5 at the indicated extremum. 7. A rectangle is bounded by the x- and y-axes and
More informationRelated Rates Problems. of h.
Basic Related Rates Problems 1. If V is the volume of a cube and x the length of an edge. Express dv What is dv in terms of dx. when x is 5 and dx = 2? 2. If V is the volume of a sphere and r is the radius.
More information5. Find the intercepts of the following equations. Also determine whether the equations are symmetric with respect to the y-axis or the origin.
MATHEMATICS 1571 Final Examination Review Problems 1. For the function f defined by f(x) = 2x 2 5x find the following: a) f(a + b) b) f(2x) 2f(x) 2. Find the domain of g if a) g(x) = x 2 3x 4 b) g(x) =
More informationCALCULUS I: FIU FINAL EXAM PROBLEM COLLECTION: VERSION WITHOUT ANSWERS
CALCULUS I: FIU FINAL EXAM PROBLEM COLLECTION: VERSION WITHOUT ANSWERS FIU MATHEMATICS FACULTY NOVEMBER 2017 Contents 1. Limits and Continuity 1 2. Derivatives 4 3. Local Linear Approximation and differentials
More informationReview for the Final Exam
Calculus Lia Vas. Integrals. Evaluate the following integrals. (a) ( x 4 x 2 ) dx (b) (2 3 x + x2 4 ) dx (c) (3x + 5) 6 dx (d) x 2 dx x 3 + (e) x 9x 2 dx (f) x dx x 2 (g) xe x2 + dx (h) 2 3x+ dx (i) x
More informationUnit 5 ICM/AB Applications of the Derivative Fall Nov 10 Learn Velocity and Acceleration: HW p P ,103 p.
Unit 5 ICM/AB Applications of the Derivative Fall 2016 Nov 4 Learn Optimization, New PS up on Optimization, HW pg. 216 3,5,17,19,21,23,25,27,29,33,39,41,49,50 a,b,54 Nov 7 Continue on HW from Nov 4 and
More informationAll quadratic functions have graphs that are U -shaped and are called parabolas. Let s look at some parabolas
Chapter Three: Polynomial and Rational Functions 3.1: Quadratic Functions Definition: Let a, b, and c be real numbers with a 0. The function f (x) = ax 2 + bx + c is called a quadratic function. All quadratic
More informationChapter 3 Prerequisite Skills. Chapter 3 Prerequisite Skills Question 1 Page 148. a) Let f (x) = x 3 + 2x 2 + 2x +1. b) Let f (z) = z 3 6z 4.
Chapter 3 Curve Sketching Chapter 3 Prerequisite Skills Chapter 3 Prerequisite Skills Question 1 Page 148 a) Let f (x) = x 3 + 2x 2 + 2x +1. f (1) = 6 f (Ğ1) = 0 (x +1) is a factor. x 3 + 2x 2 + 2x +1
More informationApplications of Derivatives
ApplicationsDerivativesII.nb 1 Applications of Derivatives Now that we have covered the basic concepts and mechanics of derivatives it's time to look at some applications. We will start with a summary
More informationCollege Calculus Final Review
College Calculus Final Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine the following limit. (Hint: Use the graph to calculate the limit.)
More informationAbsolute Extrema and Constrained Optimization
Calculus 1 Lia Vas Absolute Extrema and Constrained Optimization Recall that a function f (x) is said to have a relative maximum at x = c if f (c) f (x) for all values of x in some open interval containing
More informationBonus Homework and Exam Review - Math 141, Frank Thorne Due Friday, December 9 at the start of the final exam.
Bonus Homework and Exam Review - Math 141, Frank Thorne (thornef@mailbox.sc.edu) Due Friday, December 9 at the start of the final exam. It is strongly recommended that you do as many of these problems
More informationApplications of Derivatives
Applications of Derivatives Big Ideas Connecting the graphs of f, f, f Differentiability Continuity Continuity Differentiability Critical values Mean Value Theorem for Derivatives: Hypothesis: If f is
More informationReading Mathematical Expressions & Arithmetic Operations Expression Reads Note
Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ
More informationÏ ( ) Ì ÓÔ. Math 2413 FRsu11. Short Answer. 1. Complete the table and use the result to estimate the limit. lim x 3. x 2 16x+ 39
Math 43 FRsu Short Answer. Complete the table and use the result to estimate the it. x 3 x 3 x 6x+ 39. Let f x x.9.99.999 3.00 3.0 3. f(x) Ï ( ) Ô = x + 5, x Ì ÓÔ., x = Determine the following it. (Hint:
More informationMath 125: Exam 3 Review
Math 125: Exam 3 Review Since we re using calculators, to keep the playing field level between all students, I will ask that you refrain from using certain features of your calculator, including graphing.
More informationWeBWorK demonstration assignment
WeBWorK demonstration assignment The main purpose of this WeBWorK set is to familiarize yourself with WeBWorK. Here are some hints on how to use WeBWorK effectively: After first logging into WeBWorK change
More informationMath 75B Practice Problems for Midterm II Solutions Ch. 16, 17, 12 (E), , 2.8 (S)
Math 75B Practice Problems for Midterm II Solutions Ch. 6, 7, 2 (E),.-.5, 2.8 (S) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual
More information4.1 Analysis of functions I: Increase, decrease and concavity
4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval
More informationConcepts of graphs of functions:
Concepts of graphs of functions: 1) Domain where the function has allowable inputs (this is looking to find math no-no s): Division by 0 (causes an asymptote) ex: f(x) = 1 x There is a vertical asymptote
More informationExam Review Sheets Combined
Exam Review Sheets Combined Fall 2008 1 Fall 2007 Exam 1 1. For each part, if the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital
More informationMEMORIAL UNIVERSITY OF NEWFOUNDLAND
MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS FINAL EXAMINATION Solutions Mathematics 1000 FALL 2010 Marks [12] 1. Evaluate the following limits, showing your work. Assign
More informationWW Prob Lib1 Math course-section, semester year
Young-Seon Lee WW Prob Lib Math course-section, semester year WeBWorK assignment due /4/03 at :00 PM..( pt) Give the rational number whose decimal form is: 0 7333333 Answer:.( pt) Solve the following inequality:
More informationUnit IV Derivatives 20 Hours Finish by Christmas
Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one
More informationUnit IV Derivatives 20 Hours Finish by Christmas
Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one
More informationAPPLICATION OF DERIVATIVES
94 APPLICATION OF DERIVATIVES Chapter 6 With the Calculus as a key, Mathematics can be successfully applied to the explanation of the course of Nature. WHITEHEAD 6. Introduction In Chapter 5, we have learnt
More information1 The Derivative and Differrentiability
1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped
More informationUnit 5: Applications of Differentiation
Unit 5: Applications of Differentiation DAY TOPIC ASSIGNMENT 1 Implicit Differentiation (p. 1) p. 7-73 Implicit Differentiation p. 74-75 3 Implicit Differentiation Review 4 QUIZ 1 5 Related Rates (p. 8)
More informationA.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3
A.P. Calculus BC Test Three Section Two Free-Response No Calculators Time 45 minutes Number of Questions 3 Each of the three questions is worth 9 points. The maximum possible points earned on this section
More informationFoothill High School. AP Calculus BC. Note Templates Semester 1, Student Name.
Foothill High School AP Calculus BC Note Templates Semester 1, 2011-2012 Student Name Teacher: Burt Dixon bdixon@pleasanton.k12.ca.us 2.1 Limits Chap1-2 Page 1 Chap1-2 Page 2 Chap1-2 Page 3 Chap1-2 Page
More informationSecond Midterm Exam Name: Practice Problems Septmber 28, 2015
Math 110 4. Treibergs Second Midterm Exam Name: Practice Problems Septmber 8, 015 1. Use the limit definition of derivative to compute the derivative of f(x = 1 at x = a. 1 + x Inserting the function into
More informationdx dt = x 2 x = 120
Solutions to Review Questions, Exam. A child is flying a kite. If the kite is 90 feet above the child s hand level and the wind is blowing it on a horizontal course at 5 feet per second, how fast is the
More informationReview Questions, Exam 3
Review Questions, Exam. A child is flying a kite. If the kite is 90 feet above the child s hand level and the wind is blowing it on a horizontal course at 5 feet per second, how fast is the child paying
More informationMA 125 CALCULUS I SPRING 2007 April 27, 2007 FINAL EXAM. Name (Print last name first):... Student ID Number (last four digits):...
CALCULUS I, FINAL EXAM 1 MA 125 CALCULUS I SPRING 2007 April 27, 2007 FINAL EXAM Name (Print last name first):............................................. Student ID Number (last four digits):........................
More information4.1 Implicit Differentiation
4.1 Implicit Differentiation Learning Objectives A student will be able to: Find the derivative of variety of functions by using the technique of implicit differentiation. Consider the equation We want
More informationAP Calculus Chapter 4 Testbank (Mr. Surowski)
AP Calculus Chapter 4 Testbank (Mr. Surowski) Part I. Multiple-Choice Questions 1. Let f(x) = x 3 + 3x 2 45x + 4. Then the local extrema of f are (A) a local minimum of 179 at x = 5 and a local maximum
More information5.5 Linearization & Differentials (NO CALCULATOR CALCULUS 30L OUTCOME)
5.5 Linearization & Differentials (NO CALCULATOR CALCULUS 30L OUTCOME) Calculus I CAN USE LINEAR APPROXIMATION TO ESTIMATE THE VALUE OF A FUNCTION NEAR A POINT OF TANGENCY & FIND THE DIFFERENTIAL OF A
More informationChapter 6 Notes, Applied Calculus, Tan
Contents 4.1 Applications of the First Derivative........................... 2 4.1.1 Determining the Intervals Where a Function is Increasing or Decreasing... 2 4.1.2 Local Extrema (Relative Extrema).......................
More informationFind the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3)
Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x
More informationFinal Exam Review / AP Calculus AB
Chapter : Final Eam Review / AP Calculus AB Use the graph to find each limit. 1) lim f(), lim f(), and lim π - π + π f 5 4 1 y - - -1 - - -4-5 ) lim f(), - lim f(), and + lim f 8 6 4 y -4 - - -1-1 4 5-4
More information4. Solve for x: 5. Use the FOIL pattern to multiply (4x 2)(x + 3). 6. Simplify using exponent rules: (6x 3 )(2x) 3
SUMMER REVIEW FOR STUDENTS COMPLETING ALGEBRA I WEEK 1 1. Write the slope-intercept form of an equation of a. Write a definition of slope. 7 line with a slope of, and a y-intercept of 3. 11 3. You want
More informationSemester 1 Review. Name. Period
P A (Calculus )dx Semester Review Name Period Directions: Solve the following problems. Show work when necessary. Put the best answer in the blank provided, if appropriate.. Let y = g(x) be a function
More informationUNIT 2 SIMPLE APPLICATION OF DIFFERENTIAL CALCULUS
Calculus UNIT 2 SIMPLE APPLICATION OF DIFFERENTIAL CALCULUS Structure 2.0 Introduction 2.1 Objectives 2.2 Rate of Change of Quantities 2.3 Increasing and Decreasing Function 2.4 Maima and Minima of Functions
More informationMAX-MIN PROBLEMS. This guideline is found on pp of our textbook.
MA123, Chapter 7: Word Problems (pp. 125-153, Gootman) Chapter Goals: In this Chapter we learn a general strategy on how to approach the two main types of word problems that one usually encounters in a
More informationNovember 13, 2018 MAT186 Week 8 Justin Ko
1 Mean Value Theorem Theorem 1 (Mean Value Theorem). Let f be a continuous on [a, b] and differentiable on (a, b). There eists a c (a, b) such that f f(b) f(a) (c) =. b a Eample 1: The Mean Value Theorem
More informationDifferential Calculus Average Rate of Change (AROC) The average rate of change of y over an interval is equal to
Differential Calculus Average Rate of Change (AROC) The average rate of change of y over an interval is equal to change in y y y1 f ( x) f ( x1 ) f ( b) f ( a). changein x x x x x b a 1 Example: Find the
More informationWORKBOOK. MATH 31. CALCULUS AND ANALYTIC GEOMETRY I.
WORKBOOK. MATH 31. CALCULUS AND ANALYTIC GEOMETRY I. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributors: U. N. Iyer and P. Laul. (Many problems have been directly taken from Single Variable Calculus,
More informationMath 111 Calculus I Fall 2005 Practice Problems For Final December 5, 2005
Math 111 Calculus I Fall 2005 Practice Problems For Final December 5, 2005 As always, the standard disclaimers apply In particular, I make no claims that all the material which will be on the exam is represented
More informationMATH 3A FINAL REVIEW
MATH 3A FINAL REVIEW Guidelines to taking the nal exam You must show your work very clearly You will receive no credit if we do not understand what you are doing 2 You must cross out any incorrect work
More information3.Applications of Differentiation
3.Applications of Differentiation 3.1. Maximum and Minimum values Absolute Maximum and Absolute Minimum Values Absolute Maximum Values( Global maximum values ): Largest y-value for the given function Absolute
More informationSection 1.1: A Preview of Calculus When you finish your homework, you should be able to
Section 1.1: A Preview of Calculus When you finish your homework, you should be able to π Understand what calculus is and how it compares with precalculus π Understand that the tangent line problem is
More informationFinal Examination 201-NYA-05 May 18, 2018
. ( points) Evaluate each of the following limits. 3x x + (a) lim x x 3 8 x + sin(5x) (b) lim x sin(x) (c) lim x π/3 + sec x ( (d) x x + 5x ) (e) lim x 5 x lim x 5 + x 6. (3 points) What value of c makes
More informationPart I: SCIENTIFIC CALCULATOR REQUIRED. 1. [6 points] Compute each number rounded to 3 decimal places. Please double check your answer.
Chapter 1 Sample Pretest Part I: SCIENTIFIC CALCULATOR REQUIRED 1. [6 points] Compute each number rounded to 3 decimal places. Please double check your answer. 3 2+3 π2 +7 (a) (b) π 1.3+ 7 Part II: NO
More informationRelated Rates. 2. List the relevant quantities in the problem and assign them appropriate variables. Then write down all the information given.
Calculus 1 Lia Vas Related Rates The most important reason for a non-mathematics major to learn mathematics is to be able to apply it to problems from other disciplines or real life. In this section, we
More informationSB CH 2 answers.notebook. November 05, Warm Up. Oct 8 10:36 AM. Oct 5 2:22 PM. Oct 8 9:22 AM. Oct 8 9:19 AM. Oct 8 9:26 AM.
Warm Up Oct 8 10:36 AM Oct 5 2:22 PM Linear Function Qualities Oct 8 9:22 AM Oct 8 9:19 AM Quadratic Function Qualities Oct 8 9:26 AM Oct 8 9:25 AM 1 Oct 8 9:28 AM Oct 8 9:25 AM Given vertex (-1,4) and
More information5 t + t2 4. (ii) f(x) = ln(x 2 1). (iii) f(x) = e 2x 2e x + 3 4
Study Guide for Final Exam 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its expression to be well-defined. Some examples of the conditions are: What
More informationCalculus AB Topics Limits Continuity, Asymptotes
Calculus AB Topics Limits Continuity, Asymptotes Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam. Consider f x 2x 1 x 3 1 x 3
More informationName: Date: Block: Quarter 2 Summative Assessment Revision #1
Name: Date: Block: Multiple Choice Non-Calculator Quarter Summative Assessment Revision #1 1. The graph of y = x x has a relative maximum at (a) (0,0) only (b) (1,) only (c) (,4) only (d) (4, 16) only
More informationR1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member
Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers
More informationMath 210 Midterm #2 Review
Math 210 Mierm #2 Review Related Rates In general, the approach to a related rates problem is to first determine which quantities in the problem you care about or have relevant information about. Then
More informationReview Sheet 2 Solutions
Review Sheet Solutions 1. If y x 3 x and dx dt 5, find dy dt when x. We have that dy dt 3 x dx dt dx dt 3 x 5 5, and this is equal to 3 5 10 70 when x.. A spherical balloon is being inflated so that its
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationAP Calculus Related Rates Worksheet
AP Calculus Related Rates Worksheet 1. A small balloon is released at a point 150 feet from an observer, who is on level ground. If the balloon goes straight up at a rate of 8 feet per second, how fast
More informationCalculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ), c /, > 0 Find the limit: lim 6+
More informationFinal Exam Review Packet
1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics
More informationFinal Exam Review Packet
1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics
More informationPurdue University Study Guide for MA Credit Exam
Purdue University Study Guide for MA 16010 Credit Exam Students who pass the credit exam will gain credit in MA16010. The credit exam is a two-hour long exam with multiple choice questions. No books or
More informationMATH 2053 Calculus I Review for the Final Exam
MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x
More informationlim 2 x lim lim sin 3 (9) l)
MAC FINAL EXAM REVIEW. Find each of the following its if it eists, a) ( 5). (7) b). c). ( 5 ) d). () (/) e) (/) f) (-) sin g) () h) 5 5 5. DNE i) (/) j) (-/) 7 8 k) m) ( ) (9) l) n) sin sin( ) 7 o) DNE
More information1. Find all critical numbers of the function. 2. Find any critical numbers of the function.
1. Find all critical numbers of the function. a. critical numbers: *b. critical numbers: c. critical numbers: d. critical numbers: e. no critical numbers 2. Find any critical numbers of the function. a.
More information