The graph of a differentiable function f is shown above on the closed interval 4 3
Let h(x)= Intergal ( upper one is x, bottom one is 1) f(t) dt for 1< or equals X < or equals to 7. the graph of . Find the value of x at which has its maximum on the closed interval [1,7]. 2] ? (B) One the closed interval [0, 21 ? f'(c = fob) - flas The twice-differentiable functions f. (B) On what intervals is g increasing? Justify your answer. Explain your reasoning. docx from HISTORY 141 at Gaylord High School/voc. ≤≤x The second derivative of f has the property that fx′′() 0> for −1. 4 B) 0. f xx 2 4216 4 2 42 6 fb f a f f ba For single-variable functions, the Extreme Value Theorem told us that a continuous function on a closed interval \([a, b]\) always has both an absolute maximum and minimum on that interval, and that these absolute extremes must occur at either an endpoint or at a critical point. collegeboard. Using the subintervals [5, 61, [6, 9], [9,111' and (3. The graph of f', the derivative of f, is shown above. The graph of f has a horizontal tangent line at x = 6. Graph of f' 5. AP® CALCULUS BC 2002 SCORING GUIDELINES (Form B) Question 4 The graph of a differentiable function f on the closed interval [—3,15] is shown in the figure above. 1 5 fcc E. The graph of a function f is shown above. The function f has an absolute maximum at x = 2. The areas of the regions between the graph of f' and the x-axis are labeled in the figure. Verify that f satisfies the hypotheses of Rolle’s Theorem on the interval [0, 8]. (A) Find g(4). If f is inçreasing over the closed interval [0, 3], which of the following could be the value of J 0 f(x)dx'? (A) 50 (C) 77 (D) 100 (E) 154 84. Find h( (4). y = f(x + 1) asked by Anonymous on February 4, 2011; grade 12 AP calculus. fcc 0 C. 6 minutes ago Given a = -3, b = 2, h = 1, and k = -4, what is the equation of the graph if the parent function is y=square root x ? Area = 3 Graph of f' 7. The graph of a differentiable function f on the closed interval [1,7] is shown. Feb 22, 2012 · The graph of a differentiable function must have a non-vertical tangent line at each point in its domain. The graph of f' has horizontal tangent lines at x = l, x = 3, and x = 5. ≤≤x (b) Write an equation of the line 174. The figure above shows the graph of f. When a function is differentiable it is also continuous. Answer to: The graph of a function f(x) is shown below. 8 The table above gives selected values for a continuous function f. —2) Graph of f' The figure above shows the graph of f', the derivative of a twice-differentiable function f, on the closed interval 0 x 8. But a function can be continuous but not differentiable. 5 1. The graph of The graph of f ′ has horizontal tangent lines at x =−3, x =2, and x =5, and a vertical tangent line at x =3. The graph of the function f shown in the Let f be a function defined and continuous on the closed interval [ab,]. Graph of f 76. A function f is continuous on the closed interval and differentiable on the open interval and f has the values given in the table above. The of has a int of inflection at x AP Calculus AB Multiple Choice 2008 Question 92 92. The average rate of change of f is Il. 5 ∫ fx d () x =7 −6 , find the value of −2 which the function fails to be differentiable, identify why. The function f is defined for all real Graph of f 15. The table above gives the values of f and its derivative f ′ for selected points x in the closed interval −1. Let f be the function that is continuous on the closed interval > 2,3@ such that fc(0) the first 4 hours that tickets were on sale. For what values of x, –2 < x< 4, is ƒ not differentiable? A) 0 only •B) 0 and 2 only C) 1 and 3 only D) 0, 1, and 3 only E) 0, 1, 2, and 3 14. 4: The Mean Value Theorem VERY IMPORTANT!!! If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) then there exists a number c in (a, b) such that Example 3: Find a Tangent Line The continuous function f is defined on thc interval —4 < x < 3. Find a function y=f(x) whose derivative is that satisfies the condition tan dy x dx that f(0)=2. 5 15. = The absolute minimum value of f on the closed interval [ ] that this is given by intervals where the graph of f ′ is both decreasing and positive. The graph of a differentiable function f is shown above. _____8. On what intervals, if any, is f increasing? Justify your answer. The above function is constant and equal to 2 if x is greater than -3. 0, 1, and 3 only e. 4. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation f- or with the prefix "arc" (e. 4] with g(1) - 5 and g(4) = 8. On what interval or intervals is the graph of h concave upward? Justify your y f x on the closed interval [2, 7] is shown above. It is known that the point (3, 3 − 5 )is on the graph of . —4 Graph of f 5. The function f is differentiable on the closed interval >−6, 5 @ and satisfies f (−2 ) 7. C) The graph of f', the derivative of the function f, is shown above. f, on the interval [−3 Question: The graph of a differentiable function f on the closed interval {eq}[-3,15] {/eq} is shown in the figure below. Is the converse to this theorem true? Explain why or why not. The table above gives the values of f and its derivative f¢ for selected points along the interval. Justify your answer. (c) On what intervals in the graph of g concave down? Justify your answer. (a) If . The areas of the regions between the graph of and the x-axis are labeled in the figure. none of the above 11. A CALCULATOR MAY NOT BE USED ON THIS PART OF THE REVIEW Directions: After examining the form of the choices, decide which is the best of the choices given and circle your choice. The graph of has horizontal tangent lines at x = 1, x = 3, and x = 5. (a) Find the values of f (−6 ) and f (5). The graph of a piecewise linear function fis shown above. Answer to The graph of a differentiable function f on the closed interval [-4, 4] is shown below. Then estimate the value(s) of c that satisfy the conclusion of Rolle’s Theorem on that interval. (a) Find g (3), g'(3), and (b) Find the average rate of change of g on the interval O < a: < 3. f, consisting of two line segments and a quarter of a circle centered at the point (5, 3). The graph of f' x , the first derivative, is shown below. ) 5 Graph of f = J x f(t) dt, which of the 28. 1 and 3 only d. A. Notice that you are given values of W0(t) and you are asked about W0(t). ( ). Does f satisfy the hypotheses of the Mean Value Theorem on the interval [1, 4]? If (A) (B) (C) 1 (D) 3 (E) 4 Answer 13. The graph of y g x, the derivative of g, consists of a semicircle and three line segments, as shown in the figure above. For which functions do these d two conditions hold? Mike Koehler 3 - 6 Applications of Derivatives (d) According to the model f, given in pan (c), what is the average velocity of the plane, in miles per minute, over the time interval 0 < t < 40 ? Graph of f The graph of a differentiable function f on the closed interval [—3, 151 is shown in the figure above. 0 tan x yfx xdx Enter this equation in y1. 6. ′. The graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and horizontal tangents at the points (1, -1) and (3, 1). 0 and 2 only c. find h' (4) 3. Definition of Concavity Let y = f (x) be a differentiable function on an interval I. The graph of its a twice-differentiable function f is shown in the figure above The point (3, 5) is on the graph Of y = f (x) An equation Of the line tangent to the graph Of f at (3,5) is 26-3) 26+3) (B) (D) (E) The function f is defined on the closed Interval The graph Of its derivative f' is shown above. f x dx. QOD: Differentiability implies continuity, as shown in the theorem above. If f 1998 AP Calculus BC: Section I, Part B The figure above shows the graph of f ′, the derivative of the function f, for −≤ ≤77x. 0 B. The graph of AP Calculus BC Chapter 5 – AP Exam Problems 2 5. For how many values of x in the interval (–5,5) is the function not differentiable? A. AP Calculus 2008 BC Multiple Choice 13 8. On what interval or intervals is the graph of h concave upward? Justify your answer. The function f' is defined for The graph of the function f shown m the figure above has a veltical tangent at the point (2, 0) and horizontal tangents at the points (l, —l) and (3, l) . 2 minutes ago Let C(x, y) mean that student x is enrolled in class y, where the domain for x consists of all students in your school and the domain for y consists o 15. 0 only b. 4) Area = 6 Area = 3 (5. The figure above shows a portion of . Let g be the function given by f(t) dt. b. The graph of the function ƒ shown in the figure above has a vertical tangent at the point (2, 0 ) and horizontal tangents at the points ( 1, –1) and (3, 1). 1 5 fcc D. Time—1 hour Number of questions—4 2017 AP® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part B NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS. Let f be a function defined and continuous on the closed interval [a, b]. The graph of f' has horizontal tangents at x = —1, x = 1, and x = 3. In mathematics, a real-valued function defined on an n-dimensional interval is called convex if the line segment between any two points on the graph of the function lies above or on the graph. 4 —6 (d) Find all values of x on the open interval —6 < x < 3 for which the graph of g has a point of inflection. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4. The graph of the derivative of a function f is shown in the figure below. The graph of f , the derivative of f, consists of a semicircle and three line segments, as shown in the figure above. ' Do not use a calculator on problems 1-27. The function f has first derivative given by f'(x) = x - 6x2 - 8. It can be said that function f is piecewise constant. The graph off has a horizontal tangent lineat x = 6. The function g is twice differentiable except at $\begingroup$ @Todd Trimble: In reference [11] (Zahorski), the corollary implies that all but possibly two of the level sets are either empty or have cardinality continuum. Examples with solutions are included. Complete the following questions from the textbook: - [Voiceover] So we have the function f of x is equal to x to the sixth minus three x to the fifth and we want to know over what intervals is f decreasing and we're going to do it without even having to graph y equals f of x and the way we do that is we look at the derivative of f with respect to x and think about, well, when is that less than zero? Graph of f The graph Of a differentiable function f On the closed interval [—3, 151 is shown in the figure above. and h have second derivatives given above. = The graph of fa, the derivative of f, consists of one line segment and a semicircle, as shown above. (d) Find the value of x at which h has its minimum on the closed interval [1, 7] . Give a reason for your answer. Which of the following could be The function f is continuous for all real numbers, and the average rate of change of f on the closed interval [6,9] is -3/2. (D) The graph of g has at least one horizontal tangent in the open interval (I, 4). (D) d. The graph of f, consisting of four line segments, is shown above. The function f has a point of inflection at x = -1. 7 4 3 f f. Find h(1). The graph of f′, the derivative of f, consists of one line segment and a semicircle, as shown above. 3 E. Examples of the Accumulation Function (ANSWERS) Example 1. For a twice-differentiable function of a single variable, if its second derivative is always nonnegative on its entire For example, the second derivative of f(x) = x4 is f ′′(x) = 12x2, which is zero for x = 0, but x4 is strictly convex. The function f is continuous and differentiable over the interval [-10, 10]. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. The function f is continuous on the closed interval [2, 13] and has values as shown in the table above. Let h(x) = f(t) dt for 1 < x < 7. Let f be a function defined on the closed interval —3 x 4 with of f, consists of one line segment and a semicircle, as shown above. For x ;::: 0, the horizontal line y = 2 is an asymptote for the graph of the function f Which of the following statements must be true? (A) 1(0)=2 Graph off. Let f be a twice differentiable function whose graph is shown in the figure above. In other words, the graph has a tangent somewhere in (a,b) that is parallel to the secant line over [a,b]. Lecture 12. 8 (e) 4. differentiable at x = 3, The graph of the function f shown above has horizontal tangents at , The function f is continuous qn the closed interval 14. The graph of a differentiable function f is shown above on the closed interval [−4, 3]. 2 with f(1)=2. 2 (b) 2. Let g be the function given by g(x) = f (t) dt. The line tangent to the graph of f' at x and f' is not differentiable at x = 2. 2 less than or equal to x less than or equal to 3. 71 is shown above. ue? 0 is vertical, (A) f' does not exist at x = 2. 0 d. What is the value of? f ′()x −7 −5 −3 0 3 5 7 Let f be a function that is differentiable for all real numbers. (E) e. The graph of a differentiable function f on the closed interval [l , 71 is shown above. Suppose that f is a differentiable function such that f (4) = 5. We can use the Fundamental Theorem to write a function whose derivative is tan x: . values of x in the open interval (0, 2) satisfy the conclusion of the Mean Value Theorem for the function f on the closed interval [0. 3 Problem 45E. The graph of y f x on the closed interval ªº¬¼ 3,7 is shown in the figure above. The graph of f has a horizontal tangent line at a: = 6. 1 Definition of the Integral If f is a monotonic function from an interval [a,b] to R≥0, then we have shown that for every sequence {Pn} of partitions on [a,b] such that {µ(Pn)} → 0, 4 of 7 + 100% Graph of f 3. What is the value of AB Calculus - Hardtke AP Review: Avoiding Common Errors SOLUTION KEY Due Date: Thursday, 4/25 1. On the interval 0 x 6, the function f is twice differentiable, with f (x) 0. The graph off is shown Let = . (E) There exists c , where -2 :$; c :$; 1, such that fCc) ~ f(x) for all x on the closed interval-2 :$; x :$; 1 . Graph of f' (a) For —4 < x < 4, find all values of x at which the graph off has a point of inflection. At which of the following values of x does f have a relative maximum? The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b]. The function f is twice differentiable with f(2) = 6. Let f be a twice-differentiable function defined on the interval -1. (a) Find the x-coordinate of each of the points of inflection of the graph of f, Give a reason for your answer. (a) Find all x-coordinates at which f has a relative maximum. The formal definition is as follows: Definition of Increasing and Decreasing Functions A function f is increasing on an interval if for any tow numbers x 1 and x 2 My Chapter 3 Graph Handout 1 1. - 3. Find . this is the Let f be a function defined on the closed interval bb34x with f ()03. How many points of inflection does this graph have on this interval? A) One B) Two C) Three D) Four E) Five 35. 21 Feb 2019 We'll close this section out with a couple of nice facts that can be proved using the Mean Value Theorem. x-3-2-101 fHxL 731 3 7 f¢HxL -5-30 3 5 4. The figure above shows the graph of f', the derivative of the function f, on the closed interval —l x < 5. What is the x-coordinate of a point where the instantaneous rate of change of is the same as the average rate of change of on the interval − < < ? a. Do not spend too much time on any one problem. on the closed interval 2, 4 occurs at x = A) 4 B) 2 C) 1 D) 0 E) 2 2. 3) How Derivatives Affect the Shape of a Graph A function is increasing if its graph moves up as x moves to the right and is decreasing if its graph moves down as x moves to the right. The function f is decreasing on the interval (-∞, -1). Let g(x) — 2x + f (t) (it. A differentiable function f has the property that and . The function f and its derivatives have the properties indicated in the table below. The average value of f is Ill. The domain of f given above is the set of all real numbers except -3: if x = -3 function f is undefined. (d) Does the graph of f have a point of inflection at x 5? Give a reason for your answer. f x d f f f x. = = -. Let f be a function defined on the closed interval -3 ≤ x ≤ 4 with f(0) = 3. g. Examples of derivatives. 1 2 c. values shown for the function f(x), find approximate values for the derivative 7. x A Find g(6), g (6) , (6)g . Limits, Continuity, and Differentiability The graph of a function f is shown above. Textbook solution for Single Variable Calculus 8th Edition James Stewart Chapter 3. 43. o f(x) 1 k 2 2 The ftnction f is continuous on the closed interval [0, 2] and has values that are given in the table above. 0 4 7. The graph of — g'(x), the derivative of g, consists of a semicircle and three line segments, as shown in the figure above. -. On what interval or intervals is the graph of concave upward. Let f be a continuous function defined on the closed interval —l x 4. Also let f have the derivative function f′ that is !continuous and has the graph shown in the figure above. 8. If the velocity at t = 0 is 4 feet per second, the approximation value of the velocity, in feet per second, at t = 8, computed using the Right Riemann Sum with four subdivisions of equal length is The function g is defined and differentiable on the closed interval 7,5 and satisfies g(0) 5. How many values of x in the open interval (-4, 3) satisfy the conclusion of the Mean Value Theorem for f on [-4,3] ? (A) Zero (B) One (C) Two Three values of x in the open interval (0, 2) satisły the conclusion of me Mean Value Theorem for the function f on the closed interval [0. How many values of x in the open interval (−4, 3) satisfy the conclusion of the Mean Value Theorem for f on [−4, 3] ? A zero B one C two D three Graph of f 3. The population density of the city at any point along a strip x miles from the river’s edge is f(x) persons per square mile. Let f be the function that is continuous on the closed interval > @ M 2,3 such that fc (0) does not exist, fc (2) 0 and fxcc ( ) 0 for all x except x = 0. AP Calculus AB semester one review a pam production review packet revised August 2010 AP Calculus AB semester one review Page 3 of 10 15. The graph of f As shown below, the graph on the interval [-2, 3] suggests that f has an absolute maximum of 9 at x = 3 and an absolute minimum of 0 at x = 0. let h(x)= the integral from 1 to x of f(t) dt for 1 is less than or equal to x is less than or equal to 7. The graph of the function f shown in the figure above has a vertical tangent at the point (2, 0) and horizontal tangents at the points (l, —l) and (3,1). (a) Find , and (b) On what intervals is g decreasing? Justify your answer. Find the value of X at which h has its minimum on the closed interval [1, 7]. 2. 1. g. Of the following which must be true? A function fis continuous on the closed interval [5, 121 and differentiable on the open interval (5, 12) and f has the values given in the table above. The function f(x) in the graph is known as a piecewise function, or one that has multiple, well, pieces. The function g is defined and differentiable on the closed p170 Section 3. Therefore, it satisifies the conditions of the Mean Value Theorem. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is A) 0. 6 7. The graph of the continuous function f, consisting of three line segments and a semicircle, is shown above. Let g be defined by g(x) = f xof(t) dt. If f ( 4) 3− = and f (6) 23= then 140 8. An object moves in the xy-plane so that its position at any time t is —t . The derivative of a function f is defined by for —4 < x < O 3 for O < x < 4 The graph of the continuous function f', shown in the figure above, has 2 and x = 3111 2 The graph of g on —4 < x < O x-intercepts at x is a semicircle, and f(O) 5. The graph of a piecewise‐linear function f, for 1 4− ≤≤x, is shown above. The graph has horizontal tangent lines at x = -1, x = 1, and x = 3. (c) Use the Mean Value Theorem on the closed interval >2,4@ to show that f'4 cannot equal 8. (C) g has a maximum value on the closed interval 41. Find the x-coordinate of each of the points of inflection of the graph of f. Let f be a function that is even and continuous on the closed interval [-3, 3]. +. How many points of inflection does f have in the interval shown? (a) None (b) One (c) Two (d) Three (e) Four 16. 16) The figure below shows the graph of f ‘, the derivative of a twice differentiable function f, on the closed interval 0 ≤ x ≤ 8. Which of the following statements is/are true? I. , sin x = arcsin x) 29. If f is continuous on ªº¬¼ 3,7 and differentiable on 3,7 37, then there exists a c, c, such that A. Chapter 4 – AP Exam Problems 7 34. (a) Is fdiffe ? Give a reason for your answer. Let f be a differentiable function such that f (3) = 2 and f '3 5( )= . The figure above shows the graph of f', the derivative of a twice-differentiable function f, on the closed interval 0 < x < 8. (B) b. = ∫. (b) On what intervals is f increasing? Justify your answer. f. What given by the parametric equations — — t4+l and cos rate of change of y with respect to x at (2, 0) ? 12 34 Graph of f 4. 3. On what interval or intervals is the graph of h concave upward? Justify your answers. The function f shown in the graph above has horizontal tangents at (–2,1) and (1,–2) and vertical tangents at (–1,0) and (3,0). The function f is differentiable on the closed interval >−6, 5 @ and satisfies f (− ) 2 7. Area = 3 Graph of f' 4. fc 0 B. 7. d. The graph of f' has horizontal tangent 2: (4. The graph of a differentiable function fon the closed interval [1. Find the x-coordinate of each point of inflection of the graph of f on the open interval -3 The graph of f is shown in the figure above. If 16. A city located beside a river has a rectangular boundary as shown in the figure above. 0) (3. 1. A continuous function f is defined on the closed interval 46 x. 13). Example 3: Functions involving absolute value are Question 4 The figurt above shows the graph of f', the derivative of the function f; on the closed interval —l x 5. √3 3 b. If f is on the interval (d) Find the x-coordinate of each point of inflection of the graph of g on the interval 0 < x < 7. a. Let us give a number of examples that illus-trate di erentiable and non-di erentiable functions. (2013 AB4) (a) Find all values of 8 AP CALCULUS AB PRACTICE EXAM AP CALCULUS AB PRACTICE EXAM 9 21. Continuous. At which value of x is f continuous, but not differentiable? (A) a . Version II AP Calculus AB Calculus Exam 19 (c) Explain why there must be at least one time t, other than t = 10, such that W0(t) = 0:7 GL/day. Let f x be a differentiable function. The graph Off has a horizontal tangent lineat x = 6. I don't know for sure about the two exceptional level sets, but I strongly suspect they are countable. The graph of f ', the derivative of f , consists of one line segment and a semicircle, as shown below. (a) On what intervals, if any, is f increasing? Justify your answer. The graph of a differentiable function on the closed interval [1,7] is shown. III. Di erentiable Functions where AˆR, then we can de ne the di erentiability of f at any interior point c2Asince there is an open interval (a;b) ˆAwith c2(a;b). The graph of f' has horizontal tangent lines at x l, x = 3, and x = 5. f x dx In this problem students were given that a function f is differentiable on the interval [ 6, 5]. Because when a function is differentiable we can use all the power of calculus when working with it. org. For which of the following values of k must there exist two points a and b on the The graph of a differentiable function f on the closed interval [ -3, 15] is shown in the figure to the right. 2 (cont): The Mean Value Theorem (MVT) Theorem 3. the-regions bounded an&lžrespectively. In the xy-plane, the line x – y = k, where k is a constant, is tangent to the graph of y = x2 + 4x + 2. The graph of y f x on the closed interval [2, 7] is shown above. Find a trapezoidal approximation of using six subintervals of length = 3. The graph of f, consisting of three line segments, is shown above. The function f is defined for all real numbers and satisfies f(8) 4. In this section we want to take a look at the Mean Value Theorem. 1 C. function f is also constant and equal to -5 if x is less than -3. 4 f. The function g is defined and differentiable on the closed interval [—7, 5] and satisfies g(0) = 5. (A) –3 (B) –1 (C) 1 (D) 3 (E) 4 Answer 13. The graph of y = g'(x), the derivative of g, consists of a semicircle and three line segments, as shown in the figure above. Let g(x) = for . 5 C) 2. Graph of f' 4. The function f is continuous on the closed interval [2, 8] and has values that are given in the table below. Example 8. The graph of the differentiable function = ( ) is shown to the right. If cos (xy) = x, then 2. (a) Find the x-coordinate of each of the points of inflection of the graph off. ∫. As you can see, the function travels from x=0 to x=3 without interruption, and since the two endpoints are closed (designated by the filled-in black circles), f(x) is continuous on the closed interval [0, 3]. The The graph of the function B shown above consists of six line segments. numbers x for which f(x) is a real number. − and satisfies. 0, 1, 2, and 3 Graph of f Let f be a function whose domain is the closed interval [0, 5]. Example 3: 1995 BC6 4)Let f be a function defined on the closed interval −3≤x≤4 with f(0)=3. For example the absolute value function is actually continuous (though not Area = 3 Graph or f' 4. For what values of x, —2 < x < 4 , is f not 13. 4 E) 5. −≤ ≤x The graph of f consists of a line segment and a curve that is tangent to the x-axis at x = 3, as shown in the figure above. B ) and. The graph of a differentiable function f on the closed interval [-3,15] is shown in the figure above. II. Using the intervals [2, 3], [3, 5], [5, 8], and [8, 13], what is the approximation of 13 2 ∫ f x dx obtained from a left Riemann sum? Ch 5 AP Problems Name_____ NO CALCULATOR 3,4 1. 2] D) (2,-). Find h(1) B. Determine whether a function is increasing or decreasing using information about the derivative. The graph of f consists of a line segment and a curve that is tangent to the x-axis at x = 3, as shown in the figure above. Graphing rational functions as well as a review and a discussion on finding the intercepts, the domain and the asymptotes are presented. Oct 29, 2019 · The area under the continuous graph of over the closed interval is well-defined. $\begingroup$ At the Jahrbuch Database, if you enter "non-differentiable function" into the Title window, then select "Expression" from the drop-down menu, then click the tab labeled "Search", you'll find 3 papers with the title On the zeros of Weierstrass's non-differentiable function. The graph of the function f shown in the figure above has a vertical tangent at the point (2, 0) and The function f 6. The graph of a function f is shown. And graph it in a window: [- AB Calculus Chapter 5 12 Graph of f /1 15 Word Problems The graph of a differentiable function f on the closed interval [—3, 15] is shown in the figure above. f xx 2 is continuous on the interval 2,4 and differentiable on the interval 2,4 . 6 D) 3. ( )2. find h(1) 2. Let for lg. 3 Connecting f ' and f '' with the graph of f Calculus Using the results from the last few statements leads us to the following definition. Which of the f is shown above. = 3. f(2) = Q. In most traditional textbooks this section comes before the sections containing the First and Second Derivative Tests because many of the proofs in those sections need the Mean Value Theorem. A particle moves along the x-axis with its position at time t given by x(t) = (t a)(t constants and a b. (A) g is increasing on the closed interval [1, 41. ~ that f'(c) = 3. If f is not differentiable, even at a single point, the result may not hold. 5 x. If, for all values of x, −3 ≤ f ′(x) ≤ 2, then what range of values can f (10) have? Since −3 ≤ f ′(x) ≤ 2 for all x, by the Mean Value Theorem the average rate of change of f on any interval has to be bounded between −3 and 2 as well. (b) Find the x-coordinate of each point of inflection of the graph of f on the open interval 3 4. The graph of f ′ is shown above. 10 30 40 20 (x) Using the ubinterva [2, 5], [5, 7], and [7, 8], what are the following approximations of the area under the Graph of f' 82. e. . The function f is defined for Let f be a function defined on the closed interval [O, 7]. The graph of a differentiable function f is shown above on the closed interval (-4,3). The graph of f' has horizontal tangent lines at x = 1, x = 3, and x = 5. fcc 5 2. − ≤ ≤ , the derivative of f is specified by a graph consisting of a semicircle and three line Therefore, f is increasing on closed intervals for which ( ) 0. Using the subintervals , , , and , what is the right Riemann sum approximation to ? The figure below shows the graph of f ', the derivative of the function f, on the closed interval from x = -2 to x = 6. c) d) USING DERIVATIVE INFORMATION ASSIGNMENT #7 15 1991 AB 5 5. The graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and horizontal tangents at the points (1, -1) and (3,1). 2. Let f be a function defined on the closed interval [0, 7]. Let . Let f be a function that is continuous on the closed interval [2, 4] with f(2) following is guaranteed by the Intermediate Value Theorem? (A) f(x) = 13 has at least one solution in the open interval (2, 4). 4 22. How many points of inflection does this graph have on this interval? A)One B)Two C)Three D)Four E)Five 35. t (sec) 0 2 4 6 8 a t t() f /sec(2) 2 3 4 3 2 The table for the acceleration of a particle from 0 to 8 is given in the table above. Which of the following is true? (A) f'(-2)<f'(0) The function g is continuous on the closed interval [1. -+. rS7. The graph of a differentiable function f is shown in the figure above. f − = For 6. The graph of f' has horizontal tangent lines at x = I and x = 3. Intuitively, for a real valued function on $\mathbb{R}$ to be differentiable, it means that at each point the graph of the function locally looks like a line. On what intervals is the graph off concave up? (A) (2. Which of the following statements is true? f is continuous at x = a. Let gbe the function defined by g(x) = 5+12 for —l 4. The graph of f', the derivative of f, is shown on the right. The function f is differntiable on the closed interval [1,4 , and it has values as follows: f(1)=3, f(2)=k , and f(4)=5k+2. The volume of the solid formed by revolving the region bounded by the graphs of y As far as differentiable functions on open intervals: If all that is needed is differentiability on the interior of the interval, so much the better. Let fHxL be a twice-differentiable function on the closed interval @-3, 1D. Sample AP Calculus AB Exam Question(s) (taken from the released 2003 MC AP Exam): Graph of f 1. (b) Find /1'(4) . a)On what intervals, if any, is f increasing? Justify your answer. The graph of f consists of a line segment and a curve that is tangent to the x-axis at x 3, as shown in the figure above. If A1 and A2 are positive numbers that represent the areas of the shaded regions, then in terms of A1 and A2, A A1 B A1-A2 C 2A1-A2 D A1+A2 E A1+2A2 19. What is the estimate for using the local linear approximation for f at (a) 2. The figure above shows the graph of f', the derivative of a twice-differentiable function f, on the interval [—3, 4]. ( )8. Analyzing a Function Based on its Derivatives Students need to be able to: Locate critical numbers of the function and its derivatives. View Notes - The function g is defined and differentiable on the closed interval. (c) On what interval or intervals is the graph of h concave upward? Justify your answer. As a result, the graph of a differentiable function must be relatively smooth, and cannot contain any breaks, bends, or cusps, or any points with a vertical tangent. Let 6 5 x gx ftdt for 3 15. At which value(s) of x is f not differentiable? A) intervals is f decreasing? A) only. The continuous function f is defined on the closed interval −6 £ x £ 5. Question 3. Let f(t)dt for 1 K x £7. b)Find the x-coordinate of each point of inflection of the graph of f on the open interval −3<x<4. The average value of f' is (B) (C) (D) None I and Il only 1 and 111 only 11 and 111 only Section 4-7 : The Mean Value Theorem. The function f is twice differentiable Wilh . A continuous function f is defined on the closed interval 4 x 6. f(x) 9 The table above gives values of a function f at selected values of x. In the examples above, in order to ensure that the graph of a function has a tangent line that is parallel to the secant line, the function must be continuous on [a, b] andifferentiable on ( )ab, . On what interval or intervals is the graph of h concave upward? Mar 12, 2008 · The graph of a differentiable function f on the closed interval [1, 7] is shown below. A continuous function f is defined on the closed interval —4 x 6. For c [6,9] there is no such value that f'(c) = -3/2. For the horizontal line is an asymptote for the graph of the function. The graph of a function f is shown in the figure above. -2) lines at x = I and x = 3. Which of the following statements must be true? A). How much oil leaks out of the tanker from time Chapter 8 Integrable Functions 8. NO CALCULATOR IS ALLOWED FOR THIS QUESTION 2) The graph of the continuous function g is shown above for d d35x. 5. For what values of x, —2 < x < 4, is f not The graph of a differentiable function f is shown above on the closed interval [−4, 3]. The function f is defined for The graph of the function f shown in the figure above has a vertical tangent at the point (2,0) and horizontal tangents at the points () 1, 1− and ( 3,1 ) . The main thing that I ultimately want to emphasize in this blog post is the relationship between the Infinity Principle and the idea of a differentiable function. ( lim f 3. ) only Graph off. If f has a relative on this interval. Let g(x) = 5 + f(t)dt for f', the derivative of a twice-differentiable function f, on the closed interval 08x. On the interval 06,<<x the function f is twice differentiable, with fx′′()> 0. The graph of f the derivative (a) (b) (c) (d) On what intervals, if any, is f increasing? Justify your answer. The graph of the function f shown above consists of two line segments and a semicircle. (a) Find g(3) and g( 2). of f has a horizontal tangent line at x = 6. ~. The maximum acceleration attained on the interval 03 t by the particle whose velocity is given by v t t t t( ) 3 12 4 32 is A) 9 B) 12 C) 14 D) 21 E) 40 3. Determine the graph of the function given the graph of its derivative and vice versa. Which of the following statements is true . B) for all. The graph of a differentiable function f on the closed interval [-3, 15] is shown in the figure above. Graph the secant and tangent lines to verify your answer. AP Calculus 2008 BC Multiple Choice x 2 3 5 8 13 f x() 6 −2 −1 3 9 8. Which of the 0--2. Graph of g' 2. = ′. The graph of a differentiable function f on the closed interval [1, 7] is shown above (the points i provided). 13. 77. [No Calculator Allowed] Let f be a function defined on the closed interval –3 < x < 4 with f (0) = 3. Let G(x) = integrate limit -4 to —3 -2 -1 1 2 SAMPLE EXAMINATION 11 57 3 4 ILK. The gra h of the function fis shown above. Find h' (4) C. B On what intervals is g decreasing? Justify your answer. On the interval 0 < x < 6, the function f is twice differentiable, with f "(x) > 0. Can the graph of a bounded function ever have an unbounded derivative? function ever have an unbounded derivative? function on a closed interval is bounded (d) Find the value Of x at which h has its minimum on the closed interval [l, 7]. Thank you so much for the help. The equation f (x) must have at least two solutions in the Interval [0, 2] if k An important point about Rolle’s theorem is that the differentiability of the function \(f\) is critical. (B) is differentiable on the opcn intcrval (l , 4). The graph of f' Iras horizontal tangent at x = I and x = 3. c. (a) Find . on the closed interval [0, 8]. Sketch the graph of the equation. For what values of x , −< <24 x , is f not Graph of f' The figure above shows the graph of f', (he derivative of the function f, on the closed interval —l < x < 5. Area = 3 (5. Bldg. Using the subintervals , , , 3. The function f is twice differentiable with (a) Find the x-coordinate of each of the points of inflection of the graph of f. √3 12. (a) (b) Find . graph of f' x (a) Use the graph of to locate the x value(s) of all relative extrema of f [Justify completely] (b) Use the graph of to locate the x value(s) of all points of inflection of f [Justify completely] The graph of y=f(x) is shown in the figure above. ex. In fact, we can use methods of calculus to show that this area equals 18. Does f satisfy the hypotheses of the Mean Value Theorem on the interval [1, 4]? If Answer to: The graph of a function f(x) is shown below. 2 D. -2) Graph of f' The figure above shows the graph of f', the derivative of a twice-differentiable function f, on the closed interval 0 x 8. f ′, the derivative of a twice-differentiable function . Of the following. Which of the following statements is u. The graph . Let ( )= + − . On what intervals is the graph off concave up? The figure above shows the graph of ,f ′ the derivative of a twice-differentiable function f, on the closed interval. <<x AP® CALCULUS AB 2015 SCORING GUIDELINES The figure above shows the graph of . The graph of a piecewise-linear function f, for , is shown above. The figure above shows the graph of the derivative of a function f. For what values of x, -2<x<4, is f not differentiable? a. 1995 AB6 The graph of a differentiable function f on the closed interval [l , 7] is shown above. For f(x)=√x over the interval [0,9], show that f satisfies the hypothesis of the Mean Value Theorem, and 3. The graph 4. 8 (c) 3. Nov 05, 2006 · The graph of a function f with domain [0, 4] is shown in the figure. 4 (d) 3. A continuous function f is defined on the closed interval 4 6. Differentiable ⇒ Continuous. For which of the following values of t is the particle at rest? (A) t = ab b), where a and b are (C) Graph of f 17. The amount At of a certain item produced in a factory is given by A(t) = 4000 + 48(t – 3) – 4(t – 3)3 Oct 23, 2016 · 4 minutes ago Estimate the value of the function at x = 4 given the following graph. Sketch a possible graph of a continuous function f that has domain [0,3], where f(0)=-3 and the graph of y=f'(x) is shown. No partial credit will be given. Which of the following must be tille for the function f on the closed interval [—2, 8] I. If f is a linear function and 0 0 < a < b, then 13. Note that in both of these facts we are assuming the functions are continuous and differentiable on the interval [ 18 Oct 2018 Let f be a continuous function over the closed interval [a,b] and differentiable over the open interval (a,b) such that Case 3: The case when there exists a point x∈ (a,b) such that f(x)<k is analogous to case 2, with maximum Both points are in the interval [−2,2], and, therefore, both points satisfy the conclusion of Rolle's theorem as shown in the following graph. The two examples above show that the existence of absolute maxima and minima depends on the domain of the function. Find the value of x at which h has its minimum on the closed interval [1,7]. Oil is leaking from a tanker at the rate of gallons per hour, where t is measured in hours. (c) For how many values c, where O < c < 3, is g'(c) equal to the Note: This the graph of the derivative of f, NOT the graph of f! Let f be a function that has domain closed interval [−1,4] and range the closed interval [−1,2]!Let f(−1)=−1, f(0)=0, and f(4)=1. The graph of f consists of two quarter circles and one line segment, as shown in the figure above. Find g'(x) and evaluate (b) Determine the a-coordinate of the point at which g has an absolute maximum on the interval —4 x 3. 1 3 e. (B is decreasing on the interval (2, 4). These statements are not, in general, true if the function is defined on an open interval (a,b) (or any set that is not both closed and bounded), as, for example, the continuous function f(x) = 1/x, defined on the open interval (0,1), does not attain a maximum, being unbounded above. (C) c. The graph of f ‘ has horizontal tangent lines at x = 1, x = 3, and x = 5, and the function f is defined for all real numbers. How many values of x in the open interval (−4, 3) satisfy the conclusion of the Mean Value Theorem for f on [−4, 3] ? A zero B one C two D three AP Calculus AB Multiple Choice 1998 Question 11 - 15 11. We have step-by-step solutions for your textbooks written by Bartleby experts! The graph of a function f is shown. -OV -4 -3 -2 1 2 3 Graph of f The graph of a differentiable function fis shown above on the closed interval (-4,3]. Jan 21, 2011 · Determine whether Rolle's Theorem can be applied to f on the closed interval [a,b]. How many values of x in the open interval (-4,3) satisfy the conclusion of the Mean Value Theorem for fon (-4,3] ? May 17, 2011 · okey dokey, the question is. (a) ( ). Visit the College Board on the Web: www. (a) Find g(—3). the graph of a differentiable function f is shown above on the closed interval 4 3
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