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 continuous on the closed interval [ ab, ] and differentiable on the open interval ( ab, ) Conclusion: then there exists a number c on the open interval ( ab, ) such that f () c = f( b) f( a) b a What is the difference between a secant line and a tangent line drawn to a curve? How can the slope of both be found? Under what circumstances will a function have a tangent line that is parallel to a secant line on a given interval? Why are hypotheses necessary when applying theorems? Increasing and decreasing on a function occurs on intervals, not at a point Concavity on a function occurs on intervals, not at a point First derivative sign and increasing, decreasing Second derivative sign and concave up, concave down Where is a function increasing / decreasing? Where does a function have extrema (maximum or minimum; local or global)? Where is the graph concave up or concave down? Where does the graph have a point of inflection? Justify your answer Extreme Value Theorem Extreme Value Problems: Local or Global FDT for extrema SDT for extrema Concavity: Definition page 11 FDWK Second derivative test for concavity on an interval Optimization Related Rates Mike Koehler - 1 Applications of Derivatives
Mike Koehler - Applications of Derivatives
Introduction to the Mean Value Theorem for Derivatives Part 1 For each graph, draw the secant line through the two points on the graph corresponding to the endpoints of the indicated interval. On the indicated interval, draw any tangent lines to the graph of the function that are parallel to the secant line. For each tangent line, estimate the x-coordinate of the point of tangency. 1. hx ( ) on the interval [0,] x-value(s). hx ( ) on the interval [1,] x-value(s). hx ( ) on the interval [-1,] x-value(s) 4. hx ( ) on the interval [0,5] x-value(s) Mike Koehler - Applications of Derivatives
Part For each of the following graphs, draw the secant line through the two points corresponding to the endpoints for the indicated interval. Then draw any tangent lines that are parallel to this secant line. Indicate the number of tangents that were drawn. gx [ ] hx [ ] 1. ( ) on the interval 1,.. ( ) on the interval 1,. Number of tangents in (1,) Number of tangents in (1,) [ ]. f( x) on the interval 0,. Number of tangents in (0,) [ ] 4. f( x) on the interval 0,1. Number of tangents in (0,1) Mike Koehler - 4 Applications of Derivatives
gx [ ] hx [ ] 5. ( ) on the interval 1, 4. 6. ( ) on the interval 0,. Number of tangents in (1,4) Number of tangents in (0,) [ ] 7. gx ( ) on the interval 0,. Number of tangents in (0,) [ ] 8. gx ( ) on the interval,4. Number of tangents in (,4) Mike Koehler - 5 Applications of Derivatives
hx [ ] f x [ ] 9. ( ) on the interval,. 10. ( ) on the interval 4,. Number of tangents in (-,) Number of tangents in (-4,) 1. Which graphs are continuous on the indicated interval [ ab, ]?. Which graphs are not continuous on the indicated interval [ ab, ]?. Which graphs are differentiable on the open interval ( ab, )? 4. Which graphs are not differentiable on the open interval ( ab, )? 5. If a function is continuous on a closed interval [ ab, ], is there a tangent line that is parallel to the secant line through the points with x-coordinates x = a and x = b? 6. If a function is differentiable on an open interval ( ab, ), is there a tangent line that is parallel to the secant line through the points with x-coordinates x = a and x = b? 7. In the examples above, in order to ensure that the graph of a function has a tangent line that is parallel to the ab,. For which functions do these secant line, the function must be continuous on [a, b] and differentiable on ( ) two conditions hold? Mike Koehler - 6 Applications of Derivatives
Part For each function and interval, determine if the Mean Value Theorem applies. If it does apply, explain what conclusions you can draw from it; if it does not apply, explain why not. 1. gx= x x + [ ] ( ) 4 4 on the interval 1,1. 1 ( ) = on the interval 0,. x 1. f x [ ] 1 ( ) = on the interval 0,. x + 1. hx [ ] 4. f( x ) on the interval [ 0, 6 ]. 5. gx ( ) on the interval [, ]. 6. The differentiable function R gives the rate at which fuel flows into a tank over time, where the rate Rt () is measured in gallons per hour and time t is measured in hours. The table below shows the rates measured every two hours for the given 1- hour period. Is there some time t,0 < t < 1, such that R () t = 0? Justify your answer. 7. A cyclist rides on a straight road with positive velocity vt (), in miles per minute, at time t minutes, where v is a differentiable function of t. Selected values vt () are given in the table on the right for 0 t 0. Is there a time t, 0 < t < 0, when the cyclist has a negative acceleration? t 0 4 6 8 10 1 Rt () 76 84 88 9 94 9 88 t 0 5 10 15 0 5 0 Rt () 0. 0.4 0.6 0.5 0.5 0.4 0. Mike Koehler - 7 Applications of Derivatives
Mike Koehler - 8 Applications of Derivatives
AP Multiple Choice Questions 008 AB Multiple Choice Problems 11 18 5 6 76 84 88(related rate) 00 AB Multiple Choice 1. The graph of a function f is shown on the right. At which value of x is f continuous but not differentiable? A) a B) b C) c D) d E) e 16. If the line tangent to the graph of the function f at the point (1, 7) passes through the point (, ), then f (1) is A) -5 B) 1 C) D) 7 E) undefined 17. x Let f be the function given by f ( x) = xe. The graph of f is concave down when A) x < B) x > C) x < 1 D) x > 1 E) x < 0 78. The radius of a circle is increasing at a constant rate of 0. meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 0 π meters. A) D) (related rate) 0.04π m sec B) 0π m sec E) 0.4π m sec C) 100π m sec 4π m sec 80. The function f is continuous for x 1and differentiable for < x < 1. If f ( ) = 5 and f (1) = 4, which of the following statements could be false? A) There exists c, where < c< 1, such that f( c) = 0. B) There exists c, where < c< 1, such that f ( c) = 0. C) There exists c, where < c< 1, such that f( c) =. D) There exists c, where < c< 1, such that f ( c) =. E) There exists c, where c 1, such that f( c) f( x) for all x on the closed interval x 1. Mike Koehler - 9 Applications of Derivatives
1998 AB Multiple Choice 1. 1 What is the x-coordinate of the point of inflection on the graph of y = x + 5x + 4? 10 A) 5 B) 0 C) D) -5 E) -10 17. Graph of f( x ). The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true? A) f(1) < f (1) < f (1) B) f(1) < f (1) < f (1) C) f (1) < f(1) < f (1) D) f (1) < f(1) < f (1) E) f (1) < f (1) < f(1) 18. An equation of the line tangent to the graph of y = x+ cos x at the point (0,1) is A) y = x+ 1 B) y = x+ 1 C) y = x D) y = x 1 E) y = 0 77. let f be the function given by ( ) x f x = e and let g be the function given by gx ( ) = 6x. At what value of x do the graphs of f and ghave parallel tangent lines? A) -0.701 B) -0.567 C) -0.91 D) -0.0 E) -0.58 Mike Koehler - 10 Applications of Derivatives
78. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second? (related rate) A) ( 0.) πc B) ( ) D) ( 0.1) C E) ( 0.1) π C 0.1 C C) ( 0.1) π C 90. If the base b of a triangle is increasing at a rate of inches per minute while its height h is decreasing at a rate of inches per minute, which of the following must be true about the area A of the triangle? A) A is always increasing. B) A is always decreasing. C) A is decreasing only when b< h. D) A is decreasing only when b > h. E) A remains constant. (related rate) 1997 AB Multiple Choice 5. The graph of y x x x 4 = 16 + 4 + 48 is concave down for A) x < 0 B) x > 0 C) D) x < or x> E) < x < x< or x > Mike Koehler - 11 Applications of Derivatives
11. The graph of the derivative of f is shown in the figure on the right. Which of the following could be the graph of f? A) B) C ) D) E) 1. At what point on the graph of A) 1 1, B) y 1 1, 8 1 = x is the tangent line parallel to the line x 4y C) 1 1, 4 D) 1 1, =? E) (, ). What are all values for which the function defined by ( ) ( ) f f x = x e x is increasing? A) There are no such values of x. B) x< 1 and x > C) < x < 1 D) 1< x < E) All values of x. 77. The graph of y = x + 6x + 7x cos xchanges concavity at x = A) -1.58 B) -1.6 C) -1.67 D) -1.89 E) -. 80. Let f be the function given by 4 f( x) = e x. For what value of x is the slope of the line tangent to the graph of f at ( x, f( x )) equal to? A) 0.168 B) 0.76 C) 0.18 D) 0.4 E) 0.551 Mike Koehler - 1 Applications of Derivatives
81. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection? A) 57.60 B) 57.88 C) 59.0 D) 60.00 E) 67.40 (related rate) 1997 BC Multiple Choice 1. The graph of f, the derivative of f, is shown in the figure on the right. Which of the following describes all relative extrema of f on the open interval ( ab, )? A) One relative maximum and two relative minima B) Two relative maxima and one relative minimum C) Three relative maxima and one relative minimum D) One relative maximum and three relative minima E) Three relative maxima and two relative minima. In the triangle shown on the right, if θ increases at a constant rate of radians per minute, at what rate is x increasing in units per minute when x equals units? (related rate) A) B) 15 4 C) 4 D) 9 E) 1 Mike Koehler - 1 Applications of Derivatives
AP Free Response Questions 011 AB4 d The continuous function f is defined on the interval 4 x. The graph of f consists of two quarter circles and one line segment, as shown in the figure on the right. d) Find the average rate of change of f on the interval 4 x. There is no point c, 4 < c<, for which f () c is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem. 010 AB5 b The function g is defined and differentiable on the closed interval [ 7,5] and satisfies g (0) = 5. The graph of y = g ( x), the derivative of g, consists of a semicircle and three line segments, as shown in the figure on the right. b) Find the x-coordinate of each point of inflection of the graph of y= gx ( ) on the interval 7< x < 5. Explain your reasoning. 007 AB a b d The functions f and g are differentiable for all real numbers and g is strictly increasing. The table on the right gives values of the functions and their first derivatives at selected values of x. his given by hx ( ) = f gx ( ) 6. The function ( ) a) Explain why there must be a value rfor 1 < r< such that hr ( ) = 5. b) Explain why there must be a value c for 1 < c< such that h ( c) = 5. d) 1 If g is the inverse of g, write an equation for the line tangent to the graph of y = g 1 ( x) at x =. Mike Koehler - 14 Applications of Derivatives
007 AB5 a b The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0 t 1, the graph of r is concave down. The table on the right gives selected values of the rate of change, r () t, of the radius of the balloon over the time interval 0 t 1. The radius of the balloon is 4 0 feet when t = 5. Vsphere = π r a) Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer. b) Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure. 005 AB5 b c d A car is traveling on a straight road. For 0 t 4 seconds, the car's velocity vt (), in meters per second, is modeled by the piecewise-linear function defined by the graph on the right. b) For each of v (4) and v (0), find the value or explain why it does not exist. Indicate units of measure. c) Let at () be the car's acceleration at time t, in meters per second per second. For 0 < t < 4, write a piecewise-defined function for at (). d) Find the average rate of change of v over the interval 8 t 0. Does the Mean Value Theorem guarantee a value of c, for 8 < c< 0, such that v ( c) is equal to this average rate of change? Why or why not? 004 AB4/BC4 Consider the curve given by x + 4y = 7+ xy. a) dy y x Show that =. dx 8y x b) Show that there is a point P with x-coordinate at which the line tangent to the curve at P is horizontal. c) d y Find the value of at the point P found in part (b). Does the curve have a local maximum, a local dx minimum, or neither at the point P? Justify your answer. Mike Koehler - 15 Applications of Derivatives
00 AB a b t ( ) = + 1 sin A particle moves along the x-axis so that its velocity at time t is given by vt ( t ). At time t = 0, the particle is at position x = 1. a) Find the acceleration of the particle at time t =. Is the speed of the particle increasing at t =? Why or why not? b) Find all times t in the open interval 0< t < when the particle changes direction. Justify your answer. 00 AB4 a b Let f be a function defined on the closed interval x 4 with f(0) =. The graph of f, the derivative of f, consists of one line segment and a semicircle, as shown on the right. a) On what intervals, if any, is f increasing? Justify your answer. b) Find the x-coordinate of each point of inflection of the graph of f on the open interval < x < 4. Justify your answer. 00 AB5 (related rate) A container has the shape of an open right circular cone, as shown in the figure on the right. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth h is changing at the constant rate of cm hr. 10 Note: The volume of a cone of height h and radius r is given by 1 V= π rh. a) Find the volume of the water in the container when h = 5 cm. Indicate units of measure. b) Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm. Indicate units of measure. c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality. Mike Koehler - 16 Applications of Derivatives
1999 AB6 In the figure on the right, line is tangent to the graph of 1 1 y = at point P, with coordinates w, x w, where w > 0. Point Q has coordinates ( w, 0). Line crosses the x-axis at point R, with coordinates ( k,0). a) Find the value of k when w=. b) For all w> 0, find k in terms of w. c) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is the rate of change of k with respect to time. d) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is the rate of change of the area of PQR with respect to time? Determine whether the area is increasing or decreasing at this instant. 1996 AB1 The figure below shows the graph of f, the derivative of a function f. The domain of f is the set of all real numbers x such that < x < 5. a) For what values of x does f have a relative maximum? Why? b) For what values of x does f have a relative minimum? Why? c) On what intervals is the graph of f concave up? Use f to justify your answer. d) Suppose that f (1) = 0. In the xy-plane provided, draw a sketch that shows the general shape of the graph of the function f on the open interval 0< x <. Mike Koehler - 17 Applications of Derivatives
1991 AB5 Let f be a function that is even and continuous on the closed interval [-.]. The function f and its derivative have the properties indicated in the table below. x 0 0 < x < 1 1 1 < x < < x < f( x ) 1 Positive 0 Negative -1 Negative f ( x) Undefined Negative 0 Negative Undefined Positive f ( x) Undefined Positive 0 Negative Undefined Negative a) Find the x-coordinate of each point at which f attains an absolute maximum value or an absolute minimum value. For each x-coordinate you give, state whether f attains an absolute maximum or an absolute minimum. Justify your answer. b) Find the x-coordinate of each point of inflection on the graph of f. Justify your answer. c) In the xy-plane provided, sketch the graph of a function with all the given characteristics of f. Mike Koehler - 18 Applications of Derivatives
Textbook Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, l01 Section Questions 5.1 5 6 7 8 9 10 5. 11 1 1 14 4 8 5. 4 9 0 45 46 47 49 50 51 5 QQ p 1 4 5.4 19 7 46 50 5.5 1 4 5 6 5.6 1 19 0 QQ p 59 4 Handouts Mike Koehler - 19 Applications of Derivatives
Mike Koehler - 0 Applications of Derivatives
AP Calculus Chapter 4 Section 1. A particle moves along the x-axis so that its acceleration at any time t is given by at ( ) = 6t 18. At time t = 0 the velocity of the particle is v (0) = 4, and at time t = 1its position is xt ( ) = 0. a) Write an expression for the velocity vt () of the particle at any time t. b) For what values of t is the particle at rest? c) Write an expression for the position xt () of the particle at any time t. d) Find the total distance traveled by the particle from t = 1 to t =.. Let f be a function such that f ( x) = 6x+ 8. Find f( x) if the graph of f is tangent to the line y = x at the point ( 0, ).. Let f be a function that is defined for all real number x and that has the following properties: f ( x) = 4x 18 f (1) = 6 f() = 0 a) Find each x such that the line tangent to the graph of f at ( x, f( x) ) is horizontal. b) Write an expression for f( x ). 4. Let f be the function given by f( x) = x 7x+ 6. a) Write an equation of the line tangent to the graph of f at x = 1. b) Find the number c that satisfies the conclusion of the Mean Value Theorem for f on the closed interval [ 1, ]. 5. Consider the curve y = 4 + x and the chord AB joining points A( 4,0) and B(0,) on the curve. Find the x- and y-coordinates of the point on the curve where the tangent line is parallel to chord AB. 6. Let f be the function defined by ( ) f( x) = x e x for all numbers x. For what values of x is f increasing? 7. Let f be a differentiable function, defined for all real numbers x, with the following properties: f ( x) = ax + bx f (1) = 6 f (1) = 18 f (1) = 11 Find f( x ). Show your work. 8. Suppose f ( x) = ( x+ 1)( x ) gx ( ) where g is a continuous function and gx< ( ) 0for all x. On what interval(s) is f decreasing? 9. A particle, initially at rest, moves along the x-axis so that its acceleration at any time t 0 is given by at ( ) = 1t 4. The position of the particle when t = 1 is x(1) =. a) Find the values of t for which the particle is at rest. b) Write an expression for the position xt () of the particle at any time t 0. c) Find the total distance traveled by the particle from t = 0 to t =. Mike Koehler - 1 Applications of Derivatives
AP Calculus Chapter 4 Section Answers 1. a) vt ( ) = t 18t+ 4 b) t =, t = 4 c) xt ( ) = t 9t + 4t+ 4 d) Total distance = 6.. f x x x x ( ) = + 4 +. a) x = 0, b) f x x x ( ) = 4 9 + 4 4. a) y 1 = 4( x+ 1) b) c = 1 5. Tangent parallel at (-,1). 6. f is increasing when x or x 1 7. f x x x ( ) = 4 + 10 8. ( 1, ) 9. a) vt ( ) = 4t 4t = 0 when t= 0, 1 4 b) xt () = t t + 4 c) Total distance traveled = 10. Mike Koehler - Applications of Derivatives
AP Calculus Chapter 4 Section 1 1. A particle moves along the x-axis with velocity at time t 0 given by vt () = 1+ e t. a) Find the acceleration of the particle at time t =. b) Is the speed of the particle increasing at time t =? Give a reason for your answer. c) Find all values of t at which the particle changes direction. Justify your answer. d) Find the total distance traveled by the particle over the time interval 0 t.. Let f be the function defined by f ( x) = xe x for all real number x. a) Write an equation for the horizontal asymptote for the graph of f. b) Find the x-coordinate of each critical point of f. For each such x, determine whether f( x) is a relative maximum, relative minimum or neither.. A particle moves along the x-axis in such a way that its acceleration at time t for t 0 is given by at ( ) = 4cos( t). At time t = 0, the velocity of the particle is v (0) = 1 and its position is x (0) = 0. a) Write an equation for the velocity vt () of the particle. b) Write an equation for the position xt () of the particle. c) For what values of t, 0 t π, is the particle at rest? 4. A test plane flies in a straight line with positive velocity vt (), in miles per minute at time t minutes, where v is a differentiable function of t. Selected values of vt () are shown in the table below. t (min) 0 5 10 15 0 5 0 5 40 vt ( ) (mpm) 7.0 9. 9.5 7.0 4.5.4.4 4. 7. a) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could be zero on the open interval 0 < t < 40? Justify your answer. t 7t b) The function f, defined by f( t) = 6 + cos + sin, is used to model the velocity of the plane in 10 40 miles per minute, for 0 t 40. According to this model, what is the acceleration of the plane at t =? Indicate units of measure. c) According to the model given in part (b), for what values of t will the acceleration of the plane equal the average acceleration of the plane over the interval 0 t 40? x 5. Let f be the function given by f ( x) = xe. a) Find the lim f( x) and lim f( x) x x b) Find the absolute minimum value of f. Justify that your answer is an absolute minimum. c) What is the range of f? bx d) Consider the family of function defined by y = bxe, where b is a nonzero constant. Show that the bx absolute minimum value of bxe is the same for all nonzero values of b. Mike Koehler - Applications of Derivatives
AP Calculus Chapter 4 Section Answers 1. a) a() = e b) Speed is increasing since v() < 0 and a() < 0. c) Particle changes direction at t = 1. d) Total distance traveled equals e+ e 1 or 1.856075. a) y = 0 b) Critical point at x = 1. f (1) is a relative maximum (also an absolute maximum). a) vt ( ) = sin( t) + 1 b) xt ( ) = cos( t) + t+ 1 7π 11π c) t = or 1 1 4. a) The acceleration will equal 0 for at least two values of t. b) f '() = 0.407 or -0.408 miles per minute c) At t = 8.09799 sec or 7.4674 sec (Use part b only, not part a) 5. a) lim f( x) = 0 and lim f( x) = x x 1 b) is the absolute minimum value. e 1 e, or 0.67, or -0.68, 1 1 d) The absolute minimum value is y = at x = e b c) Range of the function is [ ] [ ] [ ] Mike Koehler - 4 Applications of Derivatives
AP Calculus Chapter 4 Section Find a point c satisfying the conclusion of the Mean Value Theorem for the given function and interval. π 1. y = cos( x) 0, [ ] x. 0, y = e [ ]. y = xln( x) 1, 4. Determine the intervals on which f ( x) is positive or negative assuming that the figure on the right is the graph of f( x ). 5. Determine the intervals on which f( x) is increasing or decreasing assuming that the figure on the right is the graph of f ( x). 6. Sketch the graph of a function f( x) whose derivative f ( x) is negative on ( ) else. Find the critical points and the intervals on which the function is increasing and decreasing. 1, and positive everywhere x 7. f( x) = 8. gx ( ) = x+ e x + 1 x 1 hx x x y x x x ( ) 1 9. ( ) = tan ( ) 10. = ln( ) > 0 Find the general antiderivative of f ( x). 5 11. f ( x) = x 1. f ( x) = 9x+ 15x ( ) = = 1. f ( x) 1e x 4sin( x) 14. f ( x) sec 7x 5 Mike Koehler - 5 Applications of Derivatives
Solve the initial value problems. dy dx dy x x y x y dx 15. = 8 + () = 0 16. = cos(5 ) ( π ) = dy dx π dy x y e y 4 dx 1 x 17. = sec ( ) = 18. = 9 (4) = 7 Find f ( x) and then find f( x) 19. f ( x) = x x+ 1 f (1) = 0 f(1) = 4 0. f ( x) = x cos( x) f (0) = f(0) = ANSWERS 1. c = 0.690107 11. 5 5 f( x) = x + C. 6 1 1 e 9 1 c = ln 1. f( x) = x 15x + C 6. 4 x c = 1. f( x) = 1e + 4cos( x) + C e 1 4. Positive on ( 0,1) (,5) Negative on ( 1, ) ( 5, 6 ) 14. f( x) = tan ( 7x 5) + C 7 4 y = x + x 40 5. Increasing on ( 0, ) ( 4,6 ) Decreasing on (, 4 ) 15. 6. Increasing, deceasing from 1 to, increasing 16. 7. 8. 9. 10. Critical points -, -1, and 0. Decreasing [, 1) ( 1, 0] Increasing (, ] [ 0, ) Critical point 0.,0 Decreasing ( ] Increasing [ 0, ) Critical point -1, 1., 1 1, Decreasing ( ] [ ) Increasing [ 1,1] Critical point 1. 0,1 Decreasing ( ] Increasing [ 1, ) 17. 18. 19. 0. 1 y = sin(5 x ) + 5 1 7 y = tan( x) + 1 x e 10 y = + 1 4 1 f ( x) = x x + x 4 4 1 5 1 1 1 1 f( x) = x x + x x+ 4 0 4 0 1 f ( x) = x sin( x) + 1 f( x) = x + cos( x) + x 6 Mike Koehler - 6 Applications of Derivatives
AP Calculus Chapter 4 Section 1. The graph below is the graph of the derivative of a function f. Use this graph to answer the following questions about f on the interval [-5, 5]. Justify your answers. Graph of f ( x) a. On what subinterval(s) is f increasing? c. Find the x-coordinates of all relative minima of f. e. On what subinterval(s) is f concave up? g. Find the x-coordinates of all points of inflection of f. b. On what subinterval(s) is f decreasing? d. Find the x-coordinates of all relative maxima of f. f. On what subinterval(s) is f concave down?. The figure shows the graph of f, the derivative of the function f, on the closed interval 1 x 5. The graph of f has horizontal tangent lines at x = 1 and x =. The function f is twice differentiable with f () = 6. a. Find the x-coordinate of each of the points of inflection of the graph of f. Give a reason for your answer. b. At what value of x does f attain its absolute minimum value on the closed interval [ 1, 5]? Show the analysis that leads to your answer. c. Let g be the function defined by g( x) = xf ( x). Find an equation for the line tangent to the graph of g at x =. Mike Koehler - 7 Applications of Derivatives
. Determine the constants a, b, c, d so that the graph of and a point of inflection at the origin. y = ax + bx + cx + d has a relative maximum at (, 4) 4. Determine the coefficient c so that the curve a point of inflection at ( 0,6 ). f ( x) ax bx cx d = + + + has a relative minimum at (, 10) and 5. What is the value of k such that the curve y = x has a point of inflection at x = 1? x k Answers, because f ( x) > 0 a [ ] 5,, 5 because f ( x) < 0 b [ ] [ ] 1 because f ( x) changes sign from negative to positive c 5 because f ( x) is negative to the left of 5 because f ( x) changes sign from positive to negative d 5 because f ( x) is negative to the right of 5 5, 0 4, 5 because f ( x) is increasing e ( ) ( ) f ( ) 0, 4 because f ( x) is decreasing g 0, 4 because f ( x) changes from increasing to decreasing or decreasing to increasing a x = 1, x = b 1 a = b = 0 c = d = 0 4 4 a = 1 b = 0 c = 1 d = 6 absolute minimum at x = 4 c y 1 = 4( x ) 5 k = Mike Koehler - 8 Applications of Derivatives
AP Calculus Chapter 4 Problems 1. The graph of y = f ( x) over the interval [ 1, 6] is shown on the right. Write and answer your own questions. Give reasons for your answers. Questions should be of the following: Find the points or intervals where the graph of y = f( x)... Be sure to address maxima and minima, increasing, decreasing, concavity, and inflection points. Question and Answer Reason. Let f be the function given by f( x) ln( x ) x with domain [, 5] = +. a) Find the x coordinate of each relative maximum point and each relative minimum point of f. Justify your answer. b) Find the x coordinate of each inflection point of f. Justify your answer. c) Find the absolute maximum value of f. Mike Koehler - 9 Applications of Derivatives
f( x) = ln + sin( x) for π x π.. Let f be the function defined by ( ) a) Find the absolute maximum value and the absolute minimum value of f. Show the analysis that leads to your conclusion. b) Find the x coordinate of each inflection point of f. Justify your answer. 4. Let f be the function given by f( x) = x 7x+ 6. a) Find the linearization to the curve at x = 1. b) Use the linearization to approximate the value of the function at x = 1.. c) Find the number c that satisfies the conclusion of the Mean Value theorem for [ 1, ]. f on the closed interval Answers a. rel min at x = 1 and x = 5 rel max at x = and x = b. inflection points at x = ± c. absolute max value is ln(1) + a. f has abs min of ln1 = 0 f has abs max of ln 0.69 b. 7π 11π inflection points at x =, 6 6 4a. Lx ( ) = 1-4( x + 1) b. 1.8 c. 1 Mike Koehler - 0 Applications of Derivatives
AP Calculus Chapter 4 Sections 1 5 Problems 1. Let f be the function given by f( x) = x 5x + x+ k, where k is a constant. On what intervals is f increasing? On what intervals is the graph of f concave downward? Find the value of k for which f has 11 for its relative minimum.. Let f be the function defined by f( x) = ln( + sin x) for π x π. Find the absolute maximum value and the absolute minimum value of f. Show the analysis that leads to your conclusion. Find the x-coordinate of each inflection point on the graph of f. Justify your answer. 4. Let f be the function given by f( x) = x + x 1x. Write an equation of the line tangent to the graph of f at the point (, 8). Find the absolute minimum value of f. Show the analysis that leads to your conclusion. Find the x-coordinate of each point of inflection on the graph of f. Show the analysis that leads to your conclusion. 4. A particle moves along the x-axis so that its velocity at any time t is given by vt () = t t 1. The position xt ( ) is 5 for t=. Write a polynomial expression for the position of the particle at any time t. For what values of t,0 t, is the particle's instantaneous velocity the same as its average velocity over the closed interval [0,]? Find the total distance traveled by the particle from time t = 0 until time t =. 5. The graph shown on the right is the graph of the derivative of a function f. Use this graph to answer the following questions about f on the interval [,4]. Give answers to the nearest tenth. Justify your answers. On what subinterval(s) is f increasing? On what subinterval(s) is f decreasing? On what subinterval(s) is f concave up? On what subinterval(s) is f concave down? Find the x-coordinate(s) of all relative minima of f? Find the x-coordinate(s) of all relative maxima of f? Find the x-coordinate(s) of all inflection point(s) of f? Mike Koehler - 1 Applications of Derivatives
6. Find an equation for the differential dy for the equation y = sin(ln( x)) +. Use the differential to approximate the function value at x = 1.. 7. After applying the brakes, the stopping distance of an automobile is approximately f( s) = 1.1s+ 0.054s feet, where s is the speed in mph. If the stopping distance when a car is traveling 55 mph is approximately 5 feet, use linearization to estimate the stopping distance when s = 57 mph. 8. Suppose that the linearization of f( x) at a= is Lx ( ) = x+ 4. What are f() and f ()? 9. If the price of a monthly bus pass is set at x dollars, a bus company takes in monthly revenue of Rx ( ) = 1.5x 0.01x (in thousands of dollars). Estimate the change in revenue if the price rises from $50 to $5. Suppose that x = 80. How will revenue be affected by a small increase in price? Answers 1. (, ][, ) 1 (, ) 5 k = 0. Abs Min 0 Abs Max ln().69 7π 11π x =, 6 6. y+ 8 = 4( x ) 44 7 x =,1 6 xt = t t t+ 4. ( ) t = 1.786 17 units 5. [ 1,1] [, 1] [1, 4] (,.5) (1.8,) (.5,1.8) (,4) x = 1 and x = 4 x = 1 and x = x =.5, 1. 8, 6. cos(ln( x)) dy = dx x dy = 0.6 y(1.).6 7. 14.08 additonal feet appoximately 49 feet 8. f() = 8 f () = 9. Revenue will increase by $1500. Revenue will decrease. Mike Koehler - Applications of Derivatives
AP Calculus Chapter 4 Sections 1 5 Problems 1. A function f is continuous on the closed interval [-1,] and its derivatives have the values indicated in the table below. Sketch the graph of y = f( x). x=-1-1<x<1 x=1 1<x< x= <x< x= f - 0-1 -.5 f ' + DNE - DNE + f '' + DNE - DNE 0. A function is defined by f ( x) = x + ax + bx + c, where a, b, and c are constants. f ( 1) = 4 is a local maximum, and the graph of f has a point of inflection at x =. Find the values of a, b, and c.. Find the interval(s) on which f( x) = x x + 5is increasing. f ( x) = x+ 1 x gx ( ), where g is a continuous function and gx< ( ) 0for all x. On what 4. Suppose ( )( ) interval(s) is f decreasing? 5. Find the minimum value of f( x) = x x+ over the interval,. 6. For what values of x is the graph of y = ln(1 + x ) concave down? 7. The curve y x x x 5 4 = 5 + has a point of inflection at x =? 1 8. For what values of x is the curve f( x) = both increasing and concave down? x + 1 9. Find all values of c on the closed interval [ 0,1] that satisfy the Mean Value Theorem for explain why no such values exist. f( x) 1 = x, or 10. Find the linear approximation L(x) of f( x) = 5 x at x =. Use the linearization to approximate f (.). x 11. Suppose that f is a function such that ( ) Estimate the value of f (.4) using a linear approximation. f ( x) = cos x and f(0) =. (Note: No explicit formula for f is given.) Mike Koehler - Applications of Derivatives
1. Use differentials to approximate dy, the change in sin( θ ) if θ changes from Use the change dy to approximate 61 π. 180 π 60 61 = π to π. 180 180 1. A pizza shop claims its pizzas are circular with a diameter of 18 inches. Estimate the amount of pizza lost or gained if the shop errs in the diameter by at most 0.4 inches. Answers 1.. a = 6 b = 15 c = 4. [,0] 4. [ 1, ) 5. f ( ) = 16 6. (, 1) ( 1, ) 7. x = 1 8. 1,0 9 4 10. Lx = + ( x ) ( ) 10.85 4 8 11. f (0) = 1 Lx ( ) = + 1( x 0) f(.4) L(.4) = + 1(.4 0) =.4 1. 1. y = sin( θ) dy = cos( θ) dθ π π π 1 π π sin = dy = cos = = 180 180 60 61π π sin + =.87475 180 60.6 π or 11.097 in 9. x = 1 Mike Koehler - 4 Applications of Derivatives
AP Calculus Chapter 4 Section 4 Guidelines for solving optimization problems 1. Read the problem. What is given? What quantity is to be found? What are the units?. Draw an appropriate diagram and label. Write down known facts and any relationships involving the variables.. Determine what is to be optimized and express this variable as a function of one of the other variables. 4. Find the critical numbers. Use first or second derivative test to determine extrema. Clearly justify the value as a maximum or minimum by stating the appropriate reason. Check the endpoints if appropriate. 5. Answer the question using proper units. 1. A rancher has 000 ft. of fencing to enclose three adjacent rectangular corrals as shown in the figure at the right. What dimensions should be used so that the enclosed area will be a maximum? x y. A rectangular plot of ground is to be enclosed by a fence and then divided down the middle by another fence. If the fence down the middle cost $ per running foot and the other fence costs $.50 per running foot, find the dimensions of the plot of largest possible area that can be enclosed with $750 worth of fence.. Three rectangular gardens must be constructed as shown on the right. The three pens must enclose a total area of 800 square feet. What are the dimensions of the gardens that can be constructed with the least amount of fencing material? x y 4. Find the dimensions of the rectangle of greatest area that can be inscribed in a semicircle of radius 4. 5. A rectangular plot of land is to be fenced off using two kinds of fencing. Two opposite sides will use heavyduty fencing selling for $ per foot, while the remaining two sides will use standard fencing selling for $ per foot. What he the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000? 6. A rectangular area of 00 square feet is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $ per foot. Find the dimensions of the rectangle of least cost. 7. A closed rectangular container with a square base is to have a volume of 50 in. The material for the top and bottom of the container will cost $ per in, and the material for the sixes will cost $ per in. Find the dimensions of the container of least cost. Mike Koehler - 5 Applications of Derivatives
8. A page of a book is to have an area of 90 in, with a 1 in margins at the bottom and sides and a.5 in margin at the top. Find the dimensions of the page that will allow the largest printed area. 9. An offshore oil well is located at point W 5 miles from the closest point A on a straight shoreline. Oil is to be pumped from the well to a point B that is 8 miles from A by piping it on a straight line under water from W to some point P on shore between A and B, and then on to B via a pipeline on shore. If the cost of laying pipe is 1 million dollars per mile under water and one-half million dollars per mile on land, where should the point P be located to minimize the cost of laying the pipe? 10. Orange trees grown in California produce 600 oranges per year if no more than 0 trees are planted per acre. For each additional tree planted per acre, the yield per tree decreases by 15 oranges. How many trees per acre should be planted to obtain the greatest number of oranges? Answers 1. 50 ft by 500 ft. 7.5 ft by 5.571 ft. 0 ft by 40 ft 4..884... by 5.6568... 5. 500 ft by 750 ft 6. 80 ft ($1 fencing) by 40 ft ($ fencing) 7. 15 in by 15 in by 10 in 8. 10.954 in by 8.158 in 9. Approximately.89 miles from A 10. 0 trees Mike Koehler - 6 Applications of Derivatives
AP Calculus Chapter 4 Section 6 Related Rates 1. Sand is being dropped at the rate of 10 ft /min onto a conical pile. If the height of the pile is always twice the base radius, at what rate is the height increasing when the pile is 8 ft high?. A light is hung 0 ft above a straight horizontal path. A man 6 ft tall is walking toward the light at the rate of 4 ft/sec. (a) How fast is his shadow shortening? (b) At what rate is the tip of the man s shadow moving?. A man 6 ft tall is walking toward a building at the rate of 5 ft/sec. If there is a light on the ground 50 ft from the building, how fast is the man s shadow on the building growing shorter when he is 0 ft from the building? 4. A water tank in the form of an inverted cone is being emptied at the rate of 6 ft /min. The altitude of the cone is 4 ft, and the base radius is 1 ft. Find how fast the water level is lowering when the water is 10 ft deep. 5. A trough is 1 ft long and its ends are in the form of inverted isosceles triangles having an altitude of ft and a base of ft. Water is flowing into the trough at the rate of ft /min. How fast is the water level rising when the water is 1 ft deep? 6. An automobile traveling at a rate of 0 ft/sec is approaching an intersection. When the automobile is 10 ft from the intersection, a truck traveling at the rate of 40 ft/sec crosses the intersection. The automobile and the truck are on roads that are at right angles to each other. How fast are the automobile and the truck separating sec after the truck leaves the intersection? 7. A ladder 5 ft long is leaning against a house. If the base of the ladder is pulled away from the house wall at a rate of ft/sec, how fast is the top moving down the wall when the base of the ladder is 7 ft from the wall? 8. A stone is dropped into a lake, causing circular waves whose radii increase at a constant rate of.5 m/sec. a) At what rate is the circumference of a wave changing when its radius is 4 m? b) At what rate is the area of the wave changing when its radius is 4 m? 9. A spherical balloon is inflated with gas at the rate of 0 ft /min. How fast is the radius of the balloon increasing at the instant the radius is: (a) 1 ft? (b) ft? 10. A revolving beacon in a lighthouse makes one revolution every 15 sec. The beacon is 00 ft from the nearest point P on a straight shoreline. Find the rate at which a ray from the light moves along the shore at a point 400 ft from P. 11. A horizontal trough is 16 ft long, and its ends are isosceles trapezoids with an altitude of 4 ft, a lower base of 4 ft, and an upper base of 6 ft. Water is being poured into the trough at the rate of 10 ft per min. How fast is the water level rising when the water is ft deep? 1. A tanker accident has spilled oil in a bay. Oil-eating bacteria are gobbling at the rate of 5 ft /hr. The oil slick has the form of a circular cylinder. When the radius of the cylinder is 500 ft, the thickness of the slick is.01 ft and decreasing at a rate of.001 ft/hr. At what rate is the area of the slick changing at this time? Is the area of the slick increasing or decreasing? V da Hint: V = Ah A = =? h dt Mike Koehler - 7 Applications of Derivatives
AP Calculus Chapter 4 Section 6 Related Rates Answers 1. 10 ft min 16π.19894. a) -1/7 ft/sec b) -40/7 ft/sec. Decreasing at the rate of.75 ft/sec 4. 4 ft min 100π 5. 1/6 ft/min 6. 14 ft/sec 7. -7/1 ft/sec 8. a) π m b) sec 4π m sec 9. a) 5/π = 1.5915 ft/min b) 5/(4π ) =.978 ft/min 1 10. 8000 π ft or 1 π ft min sec 11. 1/8 ft/min 1. 7809.816 ft /hr decreasing Mike Koehler - 8 Applications of Derivatives