seventh pages Chapter 3 Curve Sketching How much metal would be required to make a 400-mL soup can? What is the least amount of cardboard needed to build a box that holds 3000 cm3 of cereal? The answers to questions like these are of great interest to corporations that process and package food and other goods. In this chapter, you will investigate and apply the relationship between the derivative of a function and the shape of its graph. You will use derivatives to determine key features of a graph, and you will find optimal values in real situations. By the end of this chapter, you will determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function that is smooth over these intervals, and describe the behaviour of the instantaneous rate of change at and between local maxima and minima solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions, radical functions, and other simple combinations of functions sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function recognize the second derivative as the rate of change of the rate of change, and sketch the graphs of the first and second derivatives, given the graph of a smooth function determine algebraically the equation of the second derivative f\"(x) of a polynomial or simple rational function f (x), and make connections, through investigation using technology, between the key features of the graph of the function and corresponding features of the graphs of its first and second derivatives describe key features of a polynomial function, given information about its first and/or second derivatives, sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible sketch the graph of a polynomial function, given its equation, by using a variety of strategies to determine its key features, and verify using technology solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results 147 seventh pages Prerequisite Skills Factoring Polynomials 1. Factor each polynomial fully. a) x3 2x2 2x 1 Polynomial and Simple Rational Functions 5. State the domain and range of each function using set notation. b) z3 6z 4 a) y 2x 1 c) t3 6t2 7t 60 b) f (x) x2 9 d) b3 8b2 19b 12 c) f (x) x3 5x2 2 e) 3n3 n2 3n 1 f) 2p3 9p2 10p 3 g) 4k3 3k2 4k 3 h) 6w3 11w2 26w 15 Equations and Inequalities 2. Solve each equation. State any restrictions on the variable. a) x2 7x 12 0 b) 4x2 9 0 c) 18v2 36v d) a2 5a 3a 35 e) 4.9t2 19.6t 2.5 0 f) x3 6x2 3x 10 0 g) d) g(x) 1 x 1 e) f (x) x2 4 x2 f) k(x) 3 x2 9 g) p(x) x x2 1 6. For each function in question 5, determine whether the function has any asymptotes. Write the equations of any asymptotes. 7. State the intervals of increase and decrease for each function. y a) x 2 5x 14 0 x2 1 4 2 3. Solve each inequality. State any restrictions on the variable. 2 0 4x 2 2 a) 2x 10 0 b) x(x 5) 0 c) x2(x 4) 0 d) x2 5x 14 0 e) (x 3)(x 2)(x 1) 0 x f) 0 x2 1 4. Determine the x-intercepts of each function. a) f (x) 5x 15 b) g(x) x2 3x 28 c) h(x) x3 6x2 11x 6 d) y 148 x2 9 x2 1 MHR Calculus and Vectors Chapter 3 b) y 6 4 2 2 0 2 x seventh pages 8. Determine the derivative of each function. 11. State whether each function is even, odd, or neither. a) f (x) 5x2 7x 12 b) y x3 2x2 4x 8 1 c) f (x) x x2 9 d) y 2 x 1 y a) y b) 2 2 0 2 2 x 2 2 0 2 x 2 x 2 Modelling Algebraically c) 10. A right cylinder has a volume of 1000 cm3. Express the surface area of the cylinder in terms of its radius. Recall that the formula for the volume of a cylinder is V r2h, and the formula for the surface area of a cylinder is SA 2r2 2rh. d) y y x 2 2 40 cm 60 cm 9. A 40-cm by 60-cm piece of tin has squares cut from each corner as shown in the diagram. The sides are then folded up to make a box with no top. Let x represent the side length of the squares. Write an expression for the volume of the box. 2 0 2 x 2 0 2 2 12. State whether each function is even, odd, or neither. Use graphing technology to check your answers. Symmetry An even function f (x) is symmetrical about the y-axis: f (x) f (x) for all values of x. An odd function f (x) is symmetrical about the origin: f (x) f (x) for all values of x. a) y 2x b) r(x) x2 2x 1 c) f (x) x2 8 d) s(t) x3 27 1 e) h (x) x x f) f (x) x2 x2 1 CHAPTER PROBLEM Naveen bought 20 m of flexible garden edging. He plans to put two gardens in the back corners of his property: one square and one in the shape of a quarter circle. He will use the edging on the interior edges (shown in green on the diagram). How should Naveen split the edging into two pieces in order to maximize the total area of the two gardens? Assume that each border piece must be at least 5 m long. Prerequisite Skills MHR 149 seventh pages 3.1 Increasing and Decreasing Functions In many situations, it is useful to know how quantities are increasing or decreasing. A company might be interested in which factors result in increases in productivity or decreases in cost. By studying population increases and decreases, governments can predict the need for essential services, such as health care. Investigate How can you identify intervals over which a continuous function is increasing or decreasing? Method 1: Use The Geometer's Sketchpad Tools 1. Open The Geometer's Sketchpad. Graph the function f (x) x3 4x. computer with The Geometer's Sketchpad 2. Click the function f (x) x3 4x. From the Graph menu, choose Derivative. Graph the derivative on the same set of axes as the function. 3. a) Over which values of x is f (x) increasing? b) Over which values of x is f (x) decreasing? 4. R e f l e c t Refer to the graphs of f (x) and f (x). a) What is true about the graph of f when f (x) is increasing? (x) b) What is true about the graph of f when f (x) is decreasing? (x) c) What do your answers to parts a) and b) mean in terms of the slope of the tangent to f (x)? 5. Copy or print the graphs on the same grid. For f (x), colour the increasing parts blue and the decreasing parts red. 6. a) Draw a vertical dotted line through the points on f (x) at which the slope of the tangent is zero. Compare the graphs of f (x) and f (x). b) R e f l e c t What is the behaviour of the graph of f when f (x) is (x) increasing? when f (x) is decreasing? 150 MHR Calculus and Vectors Chapter 3 seventh pages Method 2: Use a Graphing Calculator Tools 1. Graph the function f (x) x3 4x. graphing calculator 2. a) Over which values of x is f (x) increasing? b) Over which values of x is f (x) decreasing? 3. Follow these steps to calculate the first differences in list L3. Clear the lists. In list L1, enter 5 to 5, in increments of 0.5. Highlight the heading of list L2. Press 2ND [L1] ALPHA ["] ENTER . ALPHA Highlight the heading of list L3. Press ) ENTER . 2ND ["] 2ND [LIST] [L1] ^ 3 7: List( - 2ND 4 [L2] 4. a) For which values of x are the first differences positive? b) For which values of x are the first differences negative? 5. R e f l e c t Refer to your answers in steps 2 and 4. Describe the relationship between the first differences and the intervals over which a function is increasing or decreasing. 6. How can the TRACE function be used to determine when the function is increasing or decreasing? The first derivative of a continuous function f (x) can be used to determine the intervals over which the function is increasing or decreasing. The function is increasing when f 0 and decreasing when f 0. (x) (x) Example 1 Find Intervals Find the intervals of increase and decrease for the function defined by f (x) 2x3 3x2 36x 5. Solution Method 1: Use Algebra Determine f (x). f 6x2 6x 36 (x) The function f (x) is increasing when 6x2 6x 36 0. 6x2 6x 36 0 x2 x 6 0 (x 3)(x 2) 0 x 3 or x 2 To solve an inequality, first solve the corresponding equation. So, f 0 when x 3 or x 2. (x) 3.1 Increasing and Decreasing Functions MHR 151 seventh pages The values of x at which the slope of the tangent, f is zero divide the (x), domain into three intervals: x 3, 3 x 2, and x 2. Test any number in each interval to determine whether the derivative is positive or negative on the entire interval. x < 3 Test Value f '(x) x 3 3 < x < 2 x2 x>2 x 4 x 3 x0 x2 x3 f'(4) 36 0 Positive f'(0) 36 Negative 0 f'(3) 36 Positive f(x) From the table, the function is increasing on the intervals x 3 and x 2, and decreasing on the interval 3 x 2. This can be confirmed by graphing the function. y 80 60 40 y f (x) 20 4 2 0 2 x 20 40 Method 2: Use the Graph of f'(x) Determine the derivative, f then graph it. (x), f 6x2 6x 36 (x) Use the graph of f to determine the intervals on (x) which the derivative is positive or negative. The graph of f is above the x-axis when x 3 (x) and x 2, so f 0 when x 3 and when x 2. (x) The graph of f is below the x-axis on the interval (x) 3 x 2, so f 0 when 3 x 2. (x) The function f (x) 2x3 3x2 36x 5 is increasing on the intervals x 3 and x 2 and decreasing on the interval 3 x 2. 152 MHR Calculus and Vectors Chapter 3 y y f (x) 80 60 40 20 4 2 0 20 40 2 x seventh pages Example 2 Use the First Derivative to Sketch a Function For each function, use the graph of f to sketch a possible function f (x). (x) y a) b) y 2 2 0 2 2 c) x 2 d) 12 8 x 4 x y 0 y f (x) 2 0 2 y f (x) y 2 4 y f (x) 8 4 2 y f (x) 2 0 x 2 12 Solution a) The derivative f is constant (x) at 2. So, f (x) has a constant slope of 2. Graph any line with slope 2. y b) The derivative f is positive (x) when x 2 and negative when x 2. So, f (x) is increasing when x 2 and decreasing when x 2. 4 2 0 2 2 y 4 f(x) f (x) f(x) 2 x 4 0 y 2 c) The derivative f is never (x) negative. From left to right, it is large and positive, decreases to zero at x 1, and then increases again. y f(x) x d) The derivative f is negative (x) when x 1 and when x 3. It is positive when 1 x 3. So, f (x) is decreasing when x 1, increasing when 1 x 3, and decreasing when x 3. 4 4 f (x) y 2 x f(x) 0 8 0 4 y 12 2 2 f (x) 2 4 y 4 x f (x) 8 3.1 Increasing and Decreasing Functions MHR 153 seventh pages Example 3 Interval of Increasing Temperature The temperature of a person with a certain strain of flu can be approximated 5 15 by the function T (d) d 2 d 37, where 0 d 6, 18 9 T represents the person's temperature, in degrees Celsius, and d is the number of days after the person first shows symptoms. During what interval will the person's temperature be increasing? Solution 5 15 T (d) d 2 d 37 18 9 5 15 T d ) d ( 9 9 Method 1: Use Algebra Solve for T 0. (d) Set T 0. (d) 5 15 d 0 9 9 d 3 d 3 divides the domain into two parts: 0 d 3 and 3 d 6. Test any x-value from each interval: 0 A function is increasing on an interval if the slope of the tangent is positive over the entire interval. y A function is decreasing on an interval if the slope of the tangent is negative over the entire interval. Intervals over which a function f (x) is increasing or decreasing can be determined by finding the derivative, f and then solving (x), the inequalities f 0 and f 0. (x) (x) When the graph of f is positive, or above (x) the x-axis, on an interval, then the function f (x) increases over that interval. Similarly, when the graph of f is negative, or below (x) the x-axis, on an interval, then the function f (x) decreases over that interval. x y x 3.1 Increasing and Decreasing Functions MHR 155 seventh pages Communicate Your Understanding C1 A function increases when 0 x 10. Which is greater: f (3) or f (8)? Explain your reasoning. C2 How can you use the derivative of a function to find intervals over which the function is increasing or decreasing? C3 A linear function is either increasing or decreasing. Is this statement always true, sometimes true, or always false? Explain. A Practise 1. Determine algebraically the values of x for which each derivative equals zero. a) f 15 5x (x) 4. Given each graph of f state the intervals of (x), increase and decrease for the function f (x). b) h x2 8x 9 (x) 2 c) g 3x2 12 (x) d) y a) g) b x3 3x2 4x 12 (x) 2 y c) 2 0 e) 4 iv) f (x) (x2 4)2 v) f (x) 2x x2 vi) f (x) 5x3 6x2 2x 1 vii) f (x) x3 6 x 3 1 viii) f (x) 3x3 x 156 MHR Calculus and Vectors Chapter 3 x y y f (x) 2 y 0 0 2 x 4 y h) 2 y f (x) 50 2 x y f (x) 100 4 2 4 x 4 g) 0 y f) 2 i) f (x) 6x 15 iii) y f (x) 4 2 0 c) Use the graph to determine the intervals of increase and decrease for the function f (x). f (x) x3 3x2 9x 6 8 2 y f (x) b) Use a graphing calculator or other graphing technology to graph the derivative. ii) 2 2 a) Find the derivative. f (x) (x 5)2 y y f (x) 4x 2 4 2. For each derivative in question 1, find the intervals of increase and decrease for the function. d) Verify your response by graphing the function f (x) on the same set of axes. 0 d) 2 h) f x4 x3 x2 x (x) 3. For each function, do the following. 2 8 e) d x2 2x 4 (x) y f (x) 4 4x 2 4 f x3 6x2 (x) f) k x3 3x2 18x 40 (x) 0 y b) y f (x) 2 2x 0 2 2 5. Sketch a possible graph of y f (x) for each graph of y f in question 4. (x) x seventh pages B Connect and Apply 6. Sketch a continuous graph that satisfies each set of conditions. b) Explain how your answer in part a) would change if f x2(x 1)(x 2). (x) a) f 0 when x 3, f 0 when x 3, (x) (x) f (3) 5 11. The table shows the intervals of increase and decrease for a function h(x). b) f 0 when 1 x 3, f 0 when (x) (x) x 1 and when x 3, f (1) 0, f (3) 4 x < 3 x < 3 3 x < 3 h(x) Negative 0 Positive x3 x>3 0 Positive c) f 0 when x 2, f (2) 1 (x) d) f 1 when x 2, f 1 when (x) (x) x 2, f (2) 4 7. Given the graph of k determine which (x), value of x in each pair gives the greater value of k(x). Explain your reasoning. y 8 4 2 0 2 4 6 8 a) Sketch the graph of the function. b) Compare your graph to that of a classmate. Explain why there are an infinite number of graphs possible. c) Write the equation for a function with these properties. y k(x) 6 2 h(x) x 2 a) k(3), k(5) b) k(8), k(12) c) k(5), k(9) d) k(2), k(10) 8. Use each method below to show that the function g(x) x3 x is always increasing. a) Find g then sketch and examine its graph. (x), b) Use algebra to show that g 0 for all x. (x) 9. Given f (x) x2 2x 3 and g(x) x 5, determine the intervals of increase and decrease of h(x) in each case. a) h(x) f (x) g(x) b) h(x) f (g(x)) c) h(x) f (x) g(x) 2 d) h(x) f (x)g(x) 10. The derivative of a function f (x) is (x) f x(x 1)(x 2). a) Find the intervals of increase and decrease of f(x). 12. Chapter Problem Naveen needs to cut 20 m of garden edging into two pieces, each at least 5 m long: one for the quarter circle and one for the square. The total area of Naveen's gardens can be modelled by the function 4 2 A(x) x 10x 100, where x 4 represents the length of edging to be used for the quarter circle. a) Evaluate A(0). Explain what A(0) represents and why your answer makes sense. b) Find A Determine the intervals on (x). which A(x) is increasing and decreasing. c) Verify your result by graphing A(x) using graphing technology. Reasoning and Proving 13. In an experiment, Representing Selecting Tools the number of a Problem Solving certain type of bacteria is given by Connecting Reflecting Communicating n(t) 100 32t2 t4, where t is the time, in days, since the experiment began, and 0 t 5. a) Find the intervals of increase and decrease of the number of bacteria. b) Describe how your answer would be different if no interval were specified. 3.1 Increasing and Decreasing Functions MHR 157 seventh pages 14. The range, R, of a small aircraft, in miles, at engine speed s, in revolutions per minute, is modelled by the function 1 2 R s 2s 1500. 2000 a) Determine the range at engine speed 2100 r/min. C b) The engine speed is restricted to values from 500 r/min to 3100 r/min. Within these values, determine intervals on which the function is increasing and decreasing. c) Verify your answer to part b) using graphing technology. Extend and Challenge 15. Explain why the function defined by f (x) 3x2 bx c cannot be strictly decreasing when a x , where a is any number. 16. For the function defined by f (x) x3 bx2 12x 3, find the values of b that result in f (x) increasing for all values of x. 17. Math Contest Which of these functions is increasing for all positive integers n? A y x2n xn 1 B y x n x n1 ... x 1 C y x 2n x 2n2 ... x 2 1 D y x 2n1 x 2n1 ... x3 x E y x 2n x 2n1 ... x n 18. Math Contest A function f (x) is even if f (x) f (x) for all x; f (x) is odd if f (x) f (x) for all x. Which of these statements is true? i) The derivative of an even function is always even. ii) The derivative of an even function is always odd. iii) The derivative of an odd function is always even. iv) The derivative of an odd function is always odd. A i) and iii) only B i) and iv) only C ii) and iii) only D ii) and iv) only E none of the above CAREER CONNECTION Aisha studied applied and industrial math at University of Ontario Institute of Technology for 5 years. She now works in the field of mathematical modelling, by helping an aircraft manufacturer to design faster, safer, and environmentally cleaner airplanes. With her knowledge of fluid mechanics and software programs that can, for example, model a wind tunnel, Aisha will run experiments. Data from these tests help her to translate physical phenomena into equations. She then analyzes and solves these complex equations and interprets the solutions. As computers become even more powerful, Aisha will be able to tackle more complex problems and get answers in less time, thereby reducing research and development costs. 158 MHR Calculus and Vectors Chapter 3 seventh pages 3.2 Maxima and Minima A favourite act at the circus is the famous Human Cannonball. Shot from a platform 5 m above the ground, the Human Cannonball is propelled high into the air before landing safely in a net. Although guaranteed a safe landing, the feat is not without risk. Launched at the same speed and angle each time, the Human Cannonball knows the maximum height he will reach. The stunt works best when his maximum height is less than the height of the ceiling where he performs! Investigate How can you find maximum or minimum values? Method 1: Compare the Derivative of a Function to the Graph of the Function 1. Consider the function f (x) 2x3 3x2. Tools graphing calculator a) Determine the intervals of increase and decrease for the function. b) For each interval, determine the values of f at the endpoints of the (x) interval. c) Graph the function using a graphing calculator. In each interval, determine if there is a maximum or a minimum. If so, determine the maximum or minimum value. 2. Repeat step 1 for each function. a) f (x) x3 6x b) f (x) 3x4 6x2 4 c) f (x) 2x3 18x2 48x d) f (x) x 4 x3 12 x 2 3 3. R e f l e c t Refer to your answers to steps 1 and 2. Describe how you can use f to determine the local maximum and minimum values of f (x). (x) Method 2: Use The Geometer's Sketchpad Tools 1. Open The Geometer's Sketchpad. Go to www.mcgrawhill. ca/links/calculus12 and follow the links to 3.2. Download the file 3.2 SlidingTangent.gsp, which shows the function f (x) 2x3 3x2 and a tangent that can be dragged along the curve. computer with The Geometer's Sketchpad 3.2 SlidingTangent.gsp 2. Drag the tangent, from left to right, through the highest point on the graph. As you drag the tangent, notice what happens to the magnitude and sign of the slope. 3. R e f l e c t Describe what happens to the slope of the tangent as it moves from left to right through each of the following points. a) the highest point on the graph b) the lowest point on the graph 3.2 Maxima and Minima MHR 159 seventh pages Given the graph of a function f (x), a point is a local maximum if the y-coordinates of all the points in the vicinity are less than the y-coordinate of the point. Algebraically, if f changes from positive to zero to negative as (x) x increases from x a to x a, then (a, f (a)) is a local maximum and a is a local maximum value . Similarly, a point is a local minimum if the y-coordinates of all the points in the vicinity are greater than the y-coordinate of the point. If f changes (x) from negative to zero to positive as x increases from x a to x a, then (a, f (a)) is a local minimum and a is a local minimum value . Local maximum and minimum values of a function are also called local extreme values, local extrema , or turning points. A function has an absolute maximum at a if f (a) f (x) for all x in the domain. The maximum value of the function is f (a). The function has an absolute minimum at a if f (a) f (x) for all x in the domain. The minimum value of the function is f (a). Example 1 Local Versus Absolute Maxima and Minima Consider this graph of a function on the interval 0 x 10. a) Identify the local maximum points. y y f (x) 8 6 b) Identify the local minimum points. c) What do all the points identified in parts a) and b) have in common? d) Identify the absolute maximum and minimum values in the interval 0 x 10. 4 2 2 0 4 6 8 10 x Solution a) The local maxima are at points A and C. b) The local minimum is at point B. c) At each of the local extreme points, A, B, and C, the tangent is horizontal. d) The absolute maximum value occurs at the highest point on the graph. In this case, the absolute maximum is 8 and occurs at the local maximum at C. The absolute minimum value occurs at the lowest point on the graph. In this case, the absolute minimum is 3 and occurs at D. 160 MHR Calculus and Vectors Chapter 3 y C 8 E 6 A B 4 D 2 0 2 4 6 8 10 x seventh pages A critical number of a function is a value a in the domain of the function for which either f 0 or f does not exist. If a is a critical number, the point (a) (a) (a, f (a)) is a critical point . To determine the absolute maximum and minimum values of a function in an interval, find the critical numbers, then substitute the critical numbers and also the x-coordinates of the endpoints of the interval into the function. Example 2 Use Critical Numbers to Find the Absolute Maximum and Minimum Find the absolute maximum and minimum of the function f (x) x3 12x 3 on the interval 3 x 4. Solution Find the critical numbers. f 3x2 12 (x) 3x2 12 0 3(x2 4) 0 3(x 2)(x 2) 0 x 2 or x 2 Find the values of x for which f '(x) 0. Examine the local extrema that occur at x 2 and x 2, and also the endpoints of the interval at x 3 and x 4. Evaluate f (x) for each of these values. f (3) (3)3 12(3) 3 6 f (2) (2)3 12(2) 3 13 f (2) (2)3 12(2) 3 19 f (4) (4)3 12(4) 3 13 y 12 8 4 f (x) x3 12 x 3 2 0 2 x 4 8 12 16 The absolute maximum value is 13. It occurs twice, at a local maximum point when x 2 and at the right endpoint. The absolute minimum value is 19. It occurs at a local minimum point when x 2. Example 3 Maximum Volume The surface area of a cylindrical container is to be 100 cm2. Its volume is given by the function V 50r r3, where r represents the radius, in centimetres, of the cylinder. Find the maximum volume of the cylinder in each case. a) The radius cannot exceed 3 cm. b) The radius cannot exceed 2 cm. 3.2 Maxima and Minima MHR 161 seventh pages Solution a) The radius cannot be less than zero and cannot exceed 3 cm. This means the interval will be 0 r 3. Find the critical numbers on this interval. V 50r r 3 V 50 3 r 2 0 50 3 r 2 50 3 r 2 50 r2 3 50 r 3 r 2 .3 r 0 since V cannot be negative. a There is a critical point when the radius is approximately 2.3 cm. Substitute r 2.3 and the endpoints, r 0 and r 3, into the volume formula, V 50r r3. V(0) 50(0) (0)3 0 V(2.3) 50(2.3) (2.3)3 76.8 V(3) 50(3) (3)3 65.2 If the radius cannot exceed 3 cm, the maximum volume is approximately 76.8 cm3. The radius of the cylinder with maximum volume is approximately 2.3 cm. b) Find the critical numbers on 0 r 2. From part a), there are no critical points between r 0 and r 2. If there are no critical points, and therefore no local extrema, then the maximum volume must be found at one of the endpoints. Test r 0 and r 2. V(0) 50(0) (0)3 0 V(2) 50(2) (2)3 74.9 Technology Tip If the radius cannot exceed 2 cm, the maximum volume is approximately 74.9 cm3. To draw a vertical line on a graphing calculator, press 2ND [DRAW] 4:Vertical. Use the arrow keys to move the line left or right. The results are displayed on the graph. The vertical line marks the endpoint of the interval. The absolute maximum occurs at the intersection of the function and the vertical line. 162 MHR Calculus and Vectors Chapter 3 V 60 V50rr3 40 20 0 r2 2 4 r seventh pages < KEY CONCEPTS > a If f changes from positive to zero to negative as x (x) increases from x a to x a, then (a, f (a)) is a local maximum value. If f changes from negative to zero to positive as x (x) increases from x a to x a, then (a, f (a)) is a local minimum value. a The absolute maximum and minimum values are found at local extrema or at the endpoints of the interval. A critical number of a function is a number a in the domain of the function for which either f 0 or f does not exist. (a) (a) Communicate Your Understanding C1 If f 0, then there must be a local maximum or minimum. Is this (x) statement true or false? Explain. C2 Does the maximum value in an interval always occur when f 0? (x) Explain. C3 Local extrema are often called turning points. Explain why this is the case. Refer to the slope of the tangent in your explanation. C4 A function is increasing on the interval 2 x 5. Where would you find the absolute maximum and minimum values? Explain your reasoning. A Practise 1. Determine the absolute maximum and minimum values of each function. a) y 2. Determine the absolute and local extreme values of each function on the given interval. a) y x 7, 10 x 10 8 b) f (x) 3x2 12x 7, 0 x 4 4 c) g(x) 2x3 3x2 12x 2, 3 x 3 0 b) 2 4 6 x e) y (x 3)2 9, 8 x 3 3. Find the critical numbers for each function. y a) f (x) x2 6x 2 0.4 b) f (x) x3 2x2 3x 0.2 0 d) f (x) x3 x, 0 x 10 c) y x4 3x3 5 2 4 6 x d) g(x) 2x3 3x2 12x 5 e) y x x 3.2 Maxima and Minima MHR 163 seventh pages 4. Find and classify the critical points of each function. Determine whether the critical points are local maxima, local minima, or neither. a) y 4x x2 b) f (x) (x 1)4 c) g(x) 2x3 24x 5 d) h(x) x5 x3 5. Suppose that the function f (t) represents your elevation after riding for t hours on your mountain bike. If you stop to rest, B explain why f 0 at that time. Under (t) what circumstances would you be at a local maximum, a local minimum, or neither? 6. a) Find the critical numbers of f (x) 2x3 3x2 12x 5. b) Find any local extrema of f (x). c) Find the absolute extrema of f (x) in the interval [2, 4]. Connect and Apply 7. Use the critical points to sketch each function. a) f (x) 7 6x x2 b) g(x) x4 8x2 10 c) y x(x 2)2 d) h(x) 27x x3 12. Chapter Problem Recall that the equation representing the total area of Naveen's garden 4 2 x 10x 100, where x is A(x) 4 represents the length of the edging to be used for the quarter circle. 8. On the interval a x b, the absolute minimum of a function, f (x), occurs when x b. The absolute maximum of f (x) occurs when x a. Do you agree with the following statement? Explain. a) What are the critical numbers of A(x)? f (x) is decreasing and there cannot be any extrema on the interval a x b. d) Find the maximum area on the interval 5 x 15. 9. For a particular function f (x 3)2. (x) a) State the coordinates of the vertex and the direction of opening. b) Make a table showing the behaviour of the derivative in the vicinity of the critical value. c) Is the critical point a local maximum or a local minimum? How do you know? 13. A section of roller coaster is in the shape of f (x) x3 2x2 x 15 where x is between 2 and 2. b) Find the maximum and minimum values of f (x) on the interval 3 x 6. a) Find all local extrema and explain what portions of the roller coaster they represent. c) Explain how you could answer part b) without finding the derivative. b) Is the highest point of this section of the ride at the beginning, the end, or neither? 10. For a particular function f x3 2x2. (x) a) For which values of x does f 0? (x) b) Find the intervals of increase and decrease for f (x). c) How can you tell by examining f that (x) there would be only one turning point for f (x)? 11. Consider the function y x3 6x2 11x. a) Find the critical numbers. b) Find the absolute maximum and minimum values on the interval 0 x 4. 164 MHR Calculus and Vectors Chapter 3 14. Use Technology The height of the Human Cannonball is given by h(t) 4.9t2 9.8t 5, where h is the height, in metres, t seconds after the cannon is fired. Graph the function on a graphing calculator. a) Find the maximum and minimum heights during the first 2 s of flight. b) How many different ways can you find the answer to part a) with a graphing calculator? Describe each way. c) Describe techniques, other than using derivatives or graphing technology, that could be used to answer part a). seventh pages Reasoning and Proving 15. The distance, d, in Representing Selecting Tools metres, that a scuba Problem Solving diver can swim at a depth of 10 m and Connecting Reflecting Communicating a speed of v metres per second before her air runs out can be modelled by d 4.8v3 28.8v2 52.8v for 0 v 2. Achievement Check 16. The height, h metres, of a ski ramp over a horizontal distance, x metres, is given by h(x) 0.01x3 0.3x2 60 for the interval 0 x 22. a) Use graphing technology to draw the graph. b) Find the minimum height of the ramp. a) Determine the speed that results in the maximum distance. c) Find the vertical drop from the top of the ramp to the lowest point on the ramp. b) Verify your result using graphing technology. d) Find the vertical rise from the lowest point to the end of the ramp. c) Why does this model not apply if v 2? C Extend and Challenge For questions 22 and 23, refer to this graph of the first derivative, f of a function f (x). (x), 17. For the quartic function defined by f (x) ax4 bx2 cx d, find the values of a, b, c, and d such that there is a local maximum at (0, 6) and a local minimum at (1, 8). y 18. For the cubic function defined by f (x) ax3 bx2 cx d, find the relationship between a, b, and c in each case. 0 y f (x) a b x a) There are no extrema. b) There are exactly two extrema. 19. Explain why a cubic function has either exactly zero or exactly two extrema. 20. Consider the function g(x) x2 9 . a) Graph g(x). How can you use y x2 9 to help? b) Find and classify the critical points. c) How could you find g (x)? 21. Math Contest Which statement is true for the graph of y xn nx, for all integers n, where n 2 A There is a local maximum at x 1. B There is a local minimum at x 1. C There is a local maximum at x 1. 22. Math Contest Which statement is true for the function f (x) at x a? A f (x) is increasing at x a. B f (x) is decreasing at x a. C f (x) has a local maximum at x a. D f (x) has a local minimum at x a. E None of the statements are true. 23. Math Contest Which statement is true for the function f (x) at x b? A f (x) has a local maximum at x b. B f (x) has a local minimum at x b. C f (x) is undefined at x b. D f (x)is undefined at x b. E f (x) has a horizontal tangent at x b. D There is a local minimum at x 1. E There are local extrema at x 1 and x 1. 3.2 Maxima and Minima MHR 165 seventh pages 3.3 Concavity and the Second Derivative Test Two cars are travelling side by side. The cars are going in the same direction, both at 80 km/h. Then, one driver decelerates while the other driver accelerates. How would the graphs that model the paths of the two cars differ? How would they be the same? In this section, you will explore what it means when the slope of the tangent is increasing or decreasing and relate it to the shape of a graph. Investigate A How can you determine the shape of a function? Method 1: Use The Geometer's Sketchpad Tools computer with The Geometer's Sketchpad 3.3 DraggingTangents.gsp 1. Open The Geometer's Sketchpad. Go to mcgrawhill.ca/links/calculus12 and follow the links to 3.3. Download the file 3.3 DraggingTangents.gsp. This file shows the graph of the function f (x) x4 2x3 5 and a tangent to the function at point A. Suppose the graph represents the path of a car, A, as it drives along a road, and the tangent shows the direction the car is steering at any moment. Drag the 'car' from one end of the 'road' to the other. Describe what happens to the tangent line as the car moves. Pick a fixed location for A. a) What is the value of x at point A? b) What is the slope of the tangent at point A? c) Does the tangent lie above or below the curve at point A? d) Describe the shape of f (x) around point A. Is the curve increasing or decreasing? 2. Drag point A to at least eight different locations on the curve. Repeat step 1 for each new location of point A. 166 MHR Calculus and Vectors Chapter 3 seventh pages 3. Describe how the shape of the graph and the slope of the tangent are related. 4. R e f l e c t Describe the shape of the graph of f (x) in each case. a) The tangent lies above the curve. b) The tangent lies below the curve. Tools Method 2: Use Paper and Pencil 1. On grid paper, graph the function g(x) x4 2x3 5. grid paper 2. Choose at least eight different points on the function. Label the points A, B, C, and so on. Draw the tangent to the function at each point. 3. Answer these questions for each of the points. a) What is the value of x at the point? b) What is the slope of the tangent at this point? c) Does the tangent lie above or below the curve at this point? d) Describe the shape of g(x) around the point. Is the curve increasing or decreasing? 4. Describe how the shape of the graph and the slope of the tangent are related. 5. R e f l e c t Describe the shape of the graph of g(x) in each case. a) The tangent lies above the curve. b) The tangent lies below the curve. The graph of a function f (x) is concave up on the interval a x b if all the tangents on the interval are below the curve. The graph curves upward as if wrapping around a point above the curve. The graph of a function f (x) is concave down on the interval a x b if all the tangents on the interval are above the curve. The graph curves downward as if wrapping around a point below the curve. A point at which the graph changes from being concave up to concave down, or vice versa, is called a point of inflection . Concave Down Point of Inflection Point of Inflection Concave Up Concave Down Concave Up 3.3 Concavity and the Second Derivative Test MHR 167 seventh pages Investigate B How can the second derivative be used to classify critical points? 1. Consider the function f (x) x3 4x2 x 6. Tools grid paper Optional computer with The Geometer's Sketchpad a) Graph f (x). b) Determine f Graph f on the same set of axes as f (x). (x). (x) c) Over which intervals is f (x) concave up? concave down? Determine the local maxima and minima of f (x). d) Determine f(x). Graph f(x) on the same set of axes as f (x) and f (x). e) Compare f and f(x) to f (x) over each interval from part c). What (x) do you notice? f) Determine the coordinates of the points of inflection for f (x). What are the values of f and f(x) at these points? (x) 2. Repeat step 1 for each function. i) k(x) x5 x4 4x3 4x2 ii) h(x) x3 2x2 5x 6 iii) b(x) x4 5x2 iv) g(x) x3 5x2 2x 8 3. R e f l e c t Describe how to use the first and second derivatives to determine the intervals over which a function is concave up or concave down. How does the second derivative relate to the concavity of a function? The Second Derivative Test If f 0 and f(a) 0, f (x) is concave up. (a) There is a local minimum at (a, f (a)). Concave Up Concave Down If f 0 and f(a) 0, f (x) is concave (a) down. There is a local maximum at (a, f (a)). CONNECTIONS Points of inflection occur only when f \"(a) = 0 or f \"(a) is undefined, but neither of these conditions is sufficient to guarantee a point of inflection at (a, f (a)). A simple example is f (x) = x4 at x = 0. 168 Point of Inflection If f(x) 0 and f(x) changes sign at a, there is a point of inflection at (a, f (a)). MHR Calculus and Vectors Chapter 3 seventh pages Example 1 Intervals of Concavity For the function f (x) x4 6x2 5, find the points of inflection and the intervals of concavity. Solution Find the first and second derivatives of the function. f 4x3 12x (x) f(x) 12x2 12 Method 1: Use Algebra At a point of inflection, the second derivative equals zero and changes sign from positive to negative or vice versa. 12x2 12 0 12(x2 1) 0 (x 1)(x 1) 0 x 1 or x 1 These values divide the domain into three intervals: x 1, 1 x 1, and x 1. Choose an x-value from each interval to test whether f(x) is positive or negative. Determine the coordinates of the points of inflection by substituting x 1 and x 1 into f (x) x4 6x2 5. x < 1 Test Value f(x) x 1 x 2 f(2) 36 1< x < 1 x0 0 f(0) 12 Positive Concave down Point of inflection (1, 10) x<1 x2 0 Negative Concave up f(x) x1 f(2) 36 Positive Point of inflection (1, 10) Concave up The concavity of the graph changes at (1, 10) and at (1, 10), so these are the points of inflection. The function is concave up to the left of x 1 and to the right of x 1. The function is concave down between these x-values. 3.3 Concavity and the Second Derivative Test MHR 169 seventh pages Method 2: Graph f"(x) Recall that f (x) x4 6x2 5, f 4x3 12x, and f(x) 12x2 12. (x) Graph f(x) 12x2 12. y 32 f (x) 12 x2 12 24 16 8 1.5 1 0.5 0 0.5 1 1.5 x 8 From the graph: f(x) 0 when the graph of f(x) lies above the x-axis, so f(x) 0 when x 1 and when x 1. f(x) 0 when the graph of f(x) lies below the x-axis, so f(x) 0 when 1 x 1. The graph of f(x) intersects the x-axis at x 1 and x 1. At these points, the sign of f(x) changes, so there are points of inflection on f(x) at x 1 and x 1. Substituting x 1 and x 1 into y f (x) to determine the coordinates of the points of inflection gives the points of inflection as (1, 10) and (1, 10). The function is concave up to the left of x 1 and to the right of x 1. The function is concave down between these x-values. Example 2 Classify Critical Points For each function, find the critical points. Then, classify them using the second derivative test. a) f (x) x3 3x2 2 b) f (x) x4 Solution a) f (x) x3 3x2 2 Determine the critical numbers for f (x). f 3x2 6x (x) 3x2 6x 0 3x(x 2) 0 x 0 or x 2 170 MHR Calculus and Vectors Chapter 3 seventh pages The critical numbers are x 0 and x 2. Substitute the critical numbers into f (x) to find the critical points. f (0) (0)3 3(0)2 2 2 f (2) (2)3 3(2)2 2 2 The critical points are (0, 2) and (2, 2). Since f x3 3x2 2, then f(x) 6x 6 (x) x0 x2 f(0) 6 f(2) 6 Negative Positive Concave down Concave up f(x) f (x) The second derivative is negative at x 0, so the graph is concave down when x 0, and there is a local maximum at the point (0, 2). The second derivative is positive at x 2, so the graph is concave up when x 2, and there is a local minimum at the point (2, 2). b) y 3 2 8 f (x) x 3x 2 4 1 0 4 (0, 2) 1 x 2 3 (2, 2) 8 f (x) x4 Determine the critical numbers for f (x). f 4x3 (x) 0 4x3 0x So, f 0 when x 0. (x) Substitute x 0 into f(x) 12x2. f(0) 12(0)2 f(0) 0 y 4 6 f (x) x Since f(0) 0, it appears that this is a point of inflection. However, the second derivative, f(x) 12x2, is always positive, so it does not change sign, and there is no change in concavity. This function is always concave up, because f is (x) always greater than or equal to zero. 4 2 1 0 (0, 0) 1 x 3.3 Concavity and the Second Derivative Test MHR 171 seventh pages Example 3 Interpret the Derivatives to Sketch a Function Sketch a graph of a function that satisfies each set of conditions. a) f(x) 2 for all x, f (3) 0, f (3) 9 b) f(x) 0 when x 1, f 0 when x 1, f (x) (3) 0, f 0 (1) Solution a) f(x) 2 for all x, so the function is concave down. f (3) 0, so there is a local maximum at x 3. The function passes through the point (3, 9). y (3, 9) 8 6 4 2 6 4 2 0 x 2 b) f(x) 0 when x 1, so the function is concave down to the left of x 1. f(x) 0 when x 1, so the function is concave up to the right of x 1. f (3) 0, so there is a local maximum at x 3. f 0, so there is a local minimum at x 1. (1) y 4 2 4 2 0 2 x 2 4 6 Note that this is only one of the possible graphs that satisfy the given conditions. If this graph were translated up by k units, k , the new graph would also satisfy the conditions since no specific points were given. 172 MHR Calculus and Vectors Chapter 3 seventh pages < KEY CONCEPTS > The second derivative is the derivative of the first derivative. It is the rate of change of the slope of the tangent. Intervals of concavity can be found by using the second derivative test or by examining the graph of f(x). A function is concave up on an interval if the second derivative is positive on that interval. If f 0 and (a) f(a) 0, there is a local minimum at (a, f (a)). A function is concave down on an interval if the second derivative is negative on that interval. If f 0 and (a) f(a) 0, there is a local maximum at (a, f (a)). If f(a) 0 and f (x) changes sign at x a, there is a point of inflection at (a, f (a)). Communicate Your Understanding C1 Describe what concavity means in terms of the location of the tangent relative to the function. C2 If a graph is concave up on an interval, what happens to the slope of the tangent as you move from left to right? C3 When there is a local maximum or minimum on a function, the first derivative equals zero and changes sign on each side of the zero. Make a similar statement about the second derivative. Use a diagram to explain. C4 Describe how to use the second derivative test to classify critical points. A Practise 2. Given each graph of f (x), state the intervals of concavity for the function f (x). Also indicate where any points of inflection will occur. 1. For each graph, identify the intervals over which the graph is concave up and the intervals over which it is concave down. y a) y b) 20 4 4 2 0 4 a) 2 10 2x 4 2 0 8 20 16 30 b) y f (x) 2 4 c) 4x 2 2 2 2x y 2 2 0 2 0 y f (x) 4 10 12 y y y f (x) 2 y 4 2 2 0 d) 2 x 2 0 4 2 x y f (x) 3.3 Concavity and the Second Derivative Test MHR 173 seventh pages 3. For each graph of f(x) in question 2, sketch a possible graph of y f (x). 4. Find the second derivative of each function. a) y 6x2 7x 5 b) f (x) x3 x B c) g(x) 2x3 12x2 9 d) y x6 5x4 5. For each function in question 4, find the intervals of concavity and the coordinates of any points of inflection. Connect and Apply 6. Sketch a graph of a function that satisfies each set of conditions. a) f (x) 2 for all x, f 0, f (2) 3 (2) b) f (x) 0 when x 0, f (x) 0 when x 0, f 0, f (0) 0 (0) c) f (x) 0 when x 1, f (x) 0 when x 1, f (1) 1, f (1) 2 d) f (x) 0 when 2 x 2, f (x) 0 when | x | 2, f (2) 1, f (x) is an even function e) f (x) 0 when x 5, f (x) 0 when x 5, f (5) 3, f (5) 2 f) f 0 when 2 x 1, f (x) 0 when (x) x 2 and x 1, f (2) 4, f (0) 0 7. For each function, find and classify all the critical points. Then, use the second derivative to check your results. a) y x2 10x 11 b) g(x) 3x5 5x3 5 c) f (x) x4 6x2 10 d) h(t) 4.9t2 39.2t 2 10. Chapter Problem The equation representing the total area of Naveen's gardens is 4 2 A(x) x 10x 100, where x 4 represents the length of the edging to be used for the quarter circle. a) What are the intervals of concavity for A(x)? How can you tell by looking at the equation? b) Does the graph of A(x) have a local maximum or a local minimum? c) Based on your answers to parts a) and b), what x-value provides the maximum area? Assume 0 x 20. Explain your reasoning. 11. The graph represents the position of a car, moving in a straight line, with respect to time. Describe what is happening at each of the key points shown on the graph, as well as what is happening in the intervals between those points. 8. The shape of a ski ramp is defined by the function h(x) 0.01x3 0.3x2 60 on the interval 0 x 22. b) Find the steepest point on the ski ramp. Distance a) Find the intervals of concavity for the given interval. E D F C 9. Is each statement always true, sometimes true, or never true? Explain. a) f 0 at a local maximum or minimum (x) on f (x). b) At a point of inflection, f(x) 0. 174 MHR Calculus and Vectors Chapter 3 B G A Time seventh pages Reasoning and Proving 12. The body temperature Representing Selecting Tools of female mammals Problem Solving varies over a fixed period. For humans, Connecting Reflecting Communicating the period is about 28 days. The temperature T, in degrees Celsius, varies with time t, in days, and can be represented by the cubic function T(t) 0.0003t3 0.012t2 0.112t 36. c) What kind of point is the point described in part b)? Justify your answer. 13. The second derivative of a function, f (x), is defined by f (x) x2(x 2). a) For what values of x is f (x) 0? b) Determine the intervals of concavity. c) If f (2) 1, sketch a possible graph of f (x). a) Determine the critical numbers of the function. b) The female is most likely to conceive when the rate of change of temperature is a maximum. Determine the day of the cycle when this occurs. C Extend and Challenge 14. Use this graph of f How many points of (x). inflection are on the graph of f (x)? Explain your reasoning. y 4 y f (x) A f 0 (a) B f (a) 0 C f 1) f 2) 0 if x1 a x2 and both f 1) (x (x (x and f 2) exist. (x 2 0 18. Math Contest Which statement is always true for a function f (x) with a local maximum at x a? 2 x 2 4 15. Prove that a polynomial function of degree four has either two points of inflection or no points of inflection. 16. A function is defined by f (x) ax3 bx2 cx d. a) Find the values of a, b, c, and d if f (x) has a point of inflection at (0, 2) and a local maximum at (2, 6). b) Explain how you know there must also be a local minimum. D There exists an interval I containing a, such that f 0 for all x a in I and f 0 (x) (x) for all x a in I. E There exists an interval I containing a, such that f (x) f (a) for all x in I. 19. Math Contest Which statement is always true for a function f (x) with f 0 f (a), (a) where a is in the domain of f (x)? A f (x) has a local maximum at x a. B f (x) has a local minimum at x a. C f (x) has either a local maximum or a local minimum at x a. D f (x) has a point of inflection at x a. E None of the above are true. 17. Assume each function in question 6 is a polynomial function. What degree is each function? Is it possible to have more than one answer? Explain your reasoning. 3.3 Concavity and the Second Derivative Test MHR 175 seventh pages 3.4 Simple Rational Functions Rational functions can be used in a number of contexts. The function 100 relates the velocity, v, in kilometres per hour, required to t k travel 100 km to time, t, in hours. The function T 2 relates temperature, r T, to distance, r, from the sun; in the function, k is a constant. In this section, you will examine the features of derivatives as they relate to rational functions and practical situations. v Investigate A Tools graphing calculator How does the graph of a rational function behave in the vicinity of its vertical asymptotes? Recall that an asymptote is not part of a function, but a boundary that shows where the function is not defined. The line x a is a vertical asymptote if f (x) as x a from the left and/or the right. 1. Use a graphing calculator to graph 1 f (x) . Use the ZOOM or WINDOW x commands to examine the graph in the vicinity of x 0. Describe what you see. Sketch the graph. 2. Press TRACE 0 ENTER . Record the y-value when x 0. 3. Press 2ND [TBLSET]. Begin at 1 and set x to 0.1. Press 2ND [TABLE]. Describe what is happening to the y-values as x approaches zero. Include what happens on both sides of x 0. 4. R e f l e c t Explain why f (x) is not defined at x 0. Explain why the y-value gets very large and positive as x approaches zero from the right, and large and negative as x approaches zero from the left. 1 1 5. Repeat steps 1 to 4 for g(x) . How does it compare to f (x) ? x 1 x 1 2. How does it compare to f (x) 1 ? 6. Repeat steps 1 to 4 for h(x) x3 x 7. R e f l e c t Describe how you could graph h(x) or another similar function without graphing technology. 1 , have x2 vertical asymptotes . These usually occur at x-values for which the denominator is zero and the function is undefined. However, a more precise definition involves examining the limit of the function as these x-values are approached. y Many rational functions, such as y 176 MHR Calculus and Vectors Chapter 3 2 2 0 2 2 4x seventh pages Investigate B How can you determine whether the graph approaches positive or negative infinity on either side of the vertical asymptotes? 1. Open The Geometer's Sketchpad. Tools 1 1 1 , and h(x) Graph f (x) on the same set , g(x) 3 (x 1)5 (x 1) (x 1) of axes. Use a different colour for each function. computer with The Geometer's Sketchpad 2. Describe how the graphs in step 1 are similar and how they are different. 1 1 1 , m(x) 3. Graph k(x) on the , and n(x) 2 4 (x 1) (x 1) (x 1)6 same set of axes. Use a different colour for each function. 4. Compare the graphs and equations in step 3 to each other, and to the graphs and equations in step 1. 5. R e f l e c t Explain how the graphs in step 3 are different from those in step 1. 6. Describe the effect of making each change to the functions in steps 1 and 3. a) Change the numerator to 1. b) Change the numerator to x. 7. R e f l e c t Summarize what you have discovered about rational functions of 1 the form f (x) . (x 1)n One-sided limits occur as x a from either the left or the right. x 3 reads \"x approaches 3 from the left.\" For example, 2.5, 2.9, 2.99, 2.999, ... x 3 reads \"x approaches 3 from the right.\" For example, 3.5, 3.1, 3.01, 3.001, ... x 2 reads \"x approaches 2 from the left.\" For example, 2.1, 2.01, 2.001, ... x 2 reads \"x approaches 2 from the right.\" For example, 1.9, 1.99, 1.999, ... Example 1 Vertical Asymptotes Consider the function defined by f (x) 1 . (x 2)(x 3) a) Determine the vertical asymptotes. b) Find the one-sided limits in the vicinity of the vertical asymptotes. c) Sketch a graph of the function. 3.4 Simple Rational Functions MHR 177 seventh pages Solution a) Vertical asymptotes occur at x-values for which the function is undefined. The function f (x) is undefined when the denominator equals zero. x20 or x30 x 2 x3 The equations of the vertical asymptotes are x 2 and x 3. b) Consider the vertical asymptote defined by x 3. One way to determine the behaviour of the function as it approaches the limit is to consider what happens if x is very close to the limit. 1 (x 2)(x 3) 1 (3 2)(very small negative number) b 1 (5)(very small negative number) lim f (x) lim x 3 x 3 It is important to determine whether the factor that is causing the vertical asymptote is approaching a small positive or negative number when examining the one-sided limits. As x approaches 3 from the left, f (x) approaches a very large negative number. lim f (x) lim x 3 x 3 1 (x 2)(x 3) 1 (3 2)(very small positive number) 1 (5)(very small positive number) Since the exponent on (x 3) is odd, once it is known what occurs as x approaches 3 from the left, you know the opposite occurs when x approaches 3 from the right. As x approaches 3 from the right, f (x) approaches a very large positive number. 0.6 The graph shows the behaviour of f (x) near x 3. 0.4 Now consider the vertical asymptote defined by x 2. Another way to determine the behaviour of the function as it approaches the limit is to substitute values very close to the limit for x, and find the value of the function. lim f (x) lim x 2 x 2 1 (x 2)(x 3) 1 (2.01 2)(2.01 3) 1 (0.01)(5.01) 19.96 178 MHR Calculus and Vectors Chapter 3 y 0.2 0 0.2 0.4 To approximate the limit as x approaches 2 from the left, substitute a number slightly less than 2, such as 2.01. 0.6 0.8 2 4x seventh pages As x approaches 2 from the left, f (x) approaches a large positive number. lim f (x) lim x 2 x 2 1 (x 2)(x 3) 1 (1.99 2)(1.99 3) 1 (0.01)(4.99) 20.04 1.99 is close to, but greater than 2. As x approaches 2 from the right, f (x) approaches a large negative number. The graph shows the behaviour of f (x) near x 2. y 0.6 0.4 0.2 6 4 2 0 x 0.2 0.4 0.6 0.8 c) y 0.6 0.4 0.2 6 4 2 0 2 4 6 x 0.2 0.4 0.6 f (x) 1 (x 2)(x 3) 0.8 3.4 Simple Rational Functions MHR 179 seventh pages Example 2 Derivatives of Rational Functions Consider the function defined by f (x) 1 . x2 1 a) Find the intervals over which the function is increasing and decreasing. b) Find the locations of any points of inflection. c) Explain why the graph never crosses the x-axis and why there are no vertical asymptotes. d) Sketch a graph of the function. Solution a) A function is increasing if the first derivative is positive. Express the function in the form f (x) (x2 1)1, then find f (x). f x) 1(x 2 1)2 (2x) ( 2 x f x) 2 ( ( x 2 1)2 becomes ( x 2 1)2 in the denominator. (x 1)2 Because the exponent is even, the denominator, (x2 1)2, is always positive. The numerator determines whether f is positive. (x) Find the values of x when f 0. (x) Because f is a rational function, f 0 when the numerator equals (x) (x) zero. f 0 when 2x 0 (x) f 0 when x 0 (x) x 0 divides the domain into two parts: x 0 and x 0. In the table, x 1 and x 1 are substituted into f for the two intervals. (x) x<0 test value x0 x 1 2(1) ((1)2 1)2 2 4 positive x1 negative f 1) ( f'(x)>0 0 f (x) From the table, f (x) is increasing when x 0 and decreasing when x 0. 180 MHR Calculus and Vectors Chapter 3 seventh pages b) A function has a point of inflection if the second derivative is zero or undefined and is changing sign at that point. Express the first derivative of the function in the form f (2x)(x2 1)2, then find f(x). (x) d d (2x) (2x) (x 2 1)2 dx dx 2(x 2 1)2 (2 x)[2(x 2 1)3 (2 x)] f (x) (x 2 1)2 2(x 2 1)2 8x 2 (x 2 1)3 (x 2 1)3[2(x 2 1) 8x 2 ] 6x2 2 (x 2 1)3 The value of f(x) is zero when the numerator is zero. 0 6x2 2 1 x2 3 1 x 3 Determine if f(x) is changing sign when f(x) 0. f (1) 1 2 f (0) 2 f (1) 1 2 1 There are points of inflection at x . 3 1 , the numerator is a positive constant, and x2 1 the denominator is positive for all values of x, because x2 1 has a minimum value of 1. y Therefore, the value of f (x) is always positive. As 1 1 f (x) 2 the values of x become large (positive or x 1 negative), the denominator becomes large, and 0.8 1 becomes small and positive. 0.6 x2 1 c) For the function f (x) Since there are no values of x for which the function is undefined, there are no vertical asymptotes. d) Because the y-value approaches 0 as x , the graph must be concave up for large positive or negative values of x. 0.4 0.2 2 0 2 4x 3.4 Simple Rational Functions MHR 181 seventh pages Example 3 Concavity of Rational Functions Find the intervals of concavity for f (x) 1 . Sketch the graph. x2 Solution Rewrite f (x) 1 as f (x) 1(x 2)1. x2 f x) (1)(1)(x 2)2 ( f x) (x 2)2 ( f (x) (2)(x 2)3 2 f (x) (x 2)3 The numerator in f(x) is a constant, so f(x) 0. There is a vertical asymptote at x 2. The denominator changes sign at the vertical asymptote, so f(x) also changes sign. This results in a change of concavity. y 4 2 4 f (x) 2 1 x2 0 2 x 2 4 x < 2 Test Value f(x) Positive x 2 x 3 Concave up f(x) < KEY CONCEPTS x > 2 x0 Undefined Vertical asymptote Negative Concave down > Vertical asymptotes usually occur in rational functions at values of x that make the denominator equal to zero. The line x a is a vertical asymptote if f (x) . as x a from the left and/or the right. Vertical asymptotes must be considered when finding intervals of concavity or intervals of increase or decrease. Use patterns to determine how a function behaves in the vicinity of a vertical asymptote. 182 MHR Calculus and Vectors Chapter 3 seventh pages Communicate Your Understanding C1 Changes in concavity can occur only at points of inflection. Is this statement true or false? Explain. 1 C2 Describe the domain of the function f (x) . x 1 C3 Explain the conditions under which a rational function would have no vertical asymptotes. A Practise 1. For each function, find the equations of any vertical asymptotes that exist. a) f (x) c) k(x) e) h(x) x x5 3 x2 5 x5 x2 2x 4 x2 g) p(x) 4 x 8 b) f (x) d) y x3 x2 4 x2 x 2 3x 2 f) y 2 x 1 x 2x 3 h) g(x) 2 x 6x 9 2. For each function in question 1 that has vertical asymptotes, find the one-sided limits approaching the vertical asymptotes. 3. Find the derivative of each function. Then, determine whether the function has any local extrema. 2 1 a) y 2 b) f (x) x3 x c) g(x) e) y B x x4 x x2 1 d) h(x) 3 (x 2)2 f) t(x) 2x 3x 2 12 x Connect and Apply 4. Consider the function f (x) 2 . (x 1)2 a) Describe how f (x) compares to the function 1 g(x) 2 . x b) Find the intervals of increase and decrease for f (x). c) Find the intervals of concavity for f (x). 5. Consider the function h(x) 1 . x2 4 a) Write the equations of the vertical asymptotes. b) Make a table showing the intervals over which the function is increasing and decreasing. c) How can you use the table from part b) to determine the behaviour of f (x) in the vicinity of the vertical asymptotes? Reasoning and Proving 6. After a chemical spill, Representing Selecting Tools the cost of cleaning Problem Solving up p percent of the Reflecting Connecting contaminants is Communicating represented by the 75 000 . equation C(p) 100 p a) Find the cost of removing 50% of the contaminants. b) Find the limit as p approaches 100 from the left. c) Why is it not feasible to remove all of the contaminants? 7. A function has a vertical asymptote defined by x 2. The function is concave down when x 2. Find lim f (x). Explain your x 2 reasoning. d) Sketch a graph of the function. 3.4 Simple Rational Functions MHR 183 seventh pages 8. A pollutant has been leaking steadily into a river. An environmental group undertook a clean-up of the river. The number of units of the pollutant in the river t years after the clean-up began is given by the equation 1 N(t) 2t . 10t 1 a) How many units of the pollutant were in the river when the clean-up began? b) After how many years is the number of units a minimum? c) What may have happened at this point? C x . x 1 a) State the equation of the vertical asymptote. 9. Consider the function f (x) b) Make a table showing the intervals over which the function is increasing and decreasing. c) How can you use your table from part b) to determine the behaviour of f (x) in the vicinity of the vertical asymptote? d) Are there any turning points? Explain how this might help you graph f (x) for large values of x. Extend and Challenge 10. Use a graphing calculator to graph the x2 4 . Use the ZOOM or function f (x) x2 WINDOW commands to examine the graph near x 2. Use the TABLE feature to examine the y-values at and near x 2. a) Why is x 2 not a vertical asymptote? b) Is the function defined at x 2? Explain. 11. Prove that a function of the form ax f (x) , where a, b, and c are non-zero bx c constants, will never have a turning point. 12. Write the equation of a function f (x) with vertical asymptotes defined by x 2 and x 1, and an x-intercept at 1. 13. Math Contest Which statements are true about the graph of the function y (x 1)2 ? 2 x 2 5x 3 i) The x-intercept is 1. ii) There is a vertical asymptote at x 1. 1 iii) There is a horizontal asymptote at y . 2 184 Achievement Check MHR Calculus and Vectors Chapter 3 A i) only B iii) only C i) and iii) only D ii) and iii) only E i), ii), and iii) 14. Math Contest Consider these functions for positive values of n. For which of these functions does the graph not have an asymptote? A y x2n 1 x2n xn B y x2n 1 xn 1 x2n 1 xn 1 x2n xn D y 2n x xn 2 x 2n1 x 1 E y x2n 1 C y seventh pages 3.5 Putting It All Together Some investors buy and sell stocks as the price increases and decreases in the short term. Analysing patterns in stock prices over time helps investors determine the optimal time to buy or sell. Other investors prefer to make long-term investments and not worry about short-term fluctuations in price. In this section, you will apply calculus techniques to sketch functions. Example 1 Analyse a Function CONNECTIONS Consider the function f (x) x3 6x2 9x. a) Determine whether the function is even, odd, or neither. b) Determine the domain of the function. c) Determine the intercepts. d) Find and classify the critical points. Identify the intervals of increase and decrease, any extrema, the intervals of concavity, and the locations of any points of inflection. Recall that an even function f (x) is symmetrical about the y-axis: f (-x) = f (x) for all values of x. Similarly, an odd function f (x) is symmetrical about the origin: f (-x) -f (x) for all values of x. Solution a) f (x) (x)3 6(x)2 9(x) x3 6x2 9x The function is neither even nor odd. b) The function is defined for all values of x, so the domain is x . c) The y-intercept is 0. The x-intercepts occur when f (x) 0. 0 x3 6x2 9x x(x2 6x 9) x(x 3)2 The x-intercepts are 0 and 3. d) Determine the first and second derivatives, and find the x-values at which they equal zero. f 3x2 12x 9. (x) 0 3x2 12x 9. 0 3(x2 4x 3) 0 3(x 3)(x 1) x 3 and x 1 f (x) 6x 12 0 6x 12 6x 12 x 2 Since f (2) 0, there may be a point of inflection at x 2. 3.5 Putting It All Together MHR 185 seventh pages The critical numbers divide the domain into three intervals: x 3, 3 x 1, and x 1. Test an x-value in each interval. Test x 4 in the interval x 3. f (4) 3(4)2 12(4) 9 f (4) 9 f (4) 6(4) 12 f (4) 12 On the interval x 3, f 0, so f (x) is increasing and f (x) 0, so (x) f (x) is concave down. Test x 2 in the interval 3 x 1. f (2) 3(2)2 12(2) 9 f (2) 3 Since f 0, f (x) is decreasing on the interval 3 x 1. (x) Test x 0 in the interval x 1. f 3(0)2 12(0) 9 (0) f 9 (0) f (0) 6(0) 12 f (0) 12 On the interval x 1, f 0, so f (x) is increasing and f (x) 0, so (x) f (x) is concave up. At x 2, f (x) 0 and is changing sign from negative to positive. So there is a point of inflection at x 2. There are local extrema at x 3 and x 1. Use the second derivative test to classify the local extrema as local maxima or local minima and to determine the concavity of the function. f (3) 6(3) 12 f (3) 6 Since f (x) 0, f (x) is concave down and there is a local maximum at x 3. f (1) 6(1) 12 f (1) 6 Since f (x) 0, f (x) is concave up and there is a local minimum at x 1. 186 MHR Calculus and Vectors Chapter 3 seventh pages Summarize the information in a table. x < 3 x 3 3 < x < 1 x 1 x > 1 Test Value x 4 x 3 x 2 x 1 x0 f (x) Positive 0 Negative 0 Positive Negative Negative 0 Positive Positive Increasing Local maximum Decreasing Local minimum Increasing f(x) Concave down Point of inflection Concave up f (x) Follow these steps to sketch the graph of a polynomial function y f (x): Step 1 Determine the domain of the