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face-to-face student instruction in the AP Calculus  \nclassroom  \n1  \n\n|  |\n| --- |\n| |\n| 2 |\n\nAP Calculus Free-Response Questions  \n1969 AB 1  \nConsider the following functions defined for all x: f1 (x) 􀀠 x  \nf 2 (x) 􀀠 xcos x  \nf 3 (x) 􀀠 3e2 x  \nf 4 (x) 􀀠 x 􀀐 x   \nAnswer the following questions (a, b, c, and d) about each of these functions. Indicate your answer by writing either yes or no in the appropriate space in the given rectangular grid. No justification is required but each blank space will be scored as an incorrect answer.  \n\n| Questions |  |  | Functions |  |\n| --- | --- | --- | --- | --- |\n|  | f1 | f 2 | f3 | f4 |\n| (a) Does f (􀀐x) 􀀠 􀀐f (x) |  |  |  |  |\n| (b) Does the inverse function exist for all x? |  |  |  |  |\n| (c) Is the function periodic? |  |  |  |  |\n| (d) Is the function continuous at x = 0? |  |  |  |  |\n\n1969 AB 2  \nA particle moves along the x-axis in such a way that its position at time t is given by x 􀀠 3t4 􀀐16t3 􀀎 24t2 for 􀀐5 􀁤 t 􀁤 5.  \na. Determine the velocity and acceleration of the particle at time t .  \nb. At what values of t is the particle at rest?  \nc. At what values of t does the particle change direction?  \nd. What is the velocity when the acceleration is first zero?  \n1969 AB 3  \nGiven f (x) 􀀠 1x 􀀎 ln x, defined only on the closed interval 1e 􀁤 x 􀀟 e.  \na. Showing your reasoning, determine the value of x at which f has its  \n(i) absolute maximum  \n(ii) absolute minimum  \nb. For what values of x is the curve concave up?  \nc. On the coordinate axis provided, sketch the graph of f over the  \n3  \ninterval 1e 􀁤 x 􀀟 e.  \nd. Given that the mean value (average ordinate) of f over the interval is  \n 2   \n, state in words a geometrical interpretation of this number relative e􀀐1  \nto the graph.  \n1969 AB 4 BC 4  \nThe number of bacteria in a culture at time t is given approximately by  \n􀀐t  \ny 􀀠 1000(25 􀀎 te20 ) for 0 􀁤 t 􀁤100.  \na. Find the largest number and the smallest number of bacteria in the culture during the interval.  \nb. At what time during the interval is the rate of change in the number of bacteria a minimum?  \n1969 AB 5  \nLet R denote the region enclosed between the graph of y 􀀠 x2 and the graph of y 􀀠 2x.  \na. Find the area of region R.  \nb. Find the volume of the solid obtained by revolving the region R about the y-axis.  \n1969 AB 6  \nAn arched window with base width 2b and height h is set into a wall. The archis to be either an arc of a parabola or a half-cycle of a cosine curve.  \n4  \na. If the arch is an arc of a parabola, write an equation for the parabola relative to the coordinate system shown in the figure. (x-intercepts are (􀀐b,0) and (b,0). y-intercept is (0, h). )  \nb. If the arch is a half-cycle of a cosine curve, write an equation for the cosine curve relative to the coordinate system shown in the figure.  \nc. Of these two window designs, which has the greater area? Justify your answer.  \n1969 AB 7  \na. On the coordinate axes provided, sketch the graph of y 􀀠 ex~~ ~~􀀎2e􀀐~~ ~~x~~ ~~ .  \nb. Let R be a point on the curve and let the x-coordinate of R be r (r 􀁺 0). The tangent line to the curve at R crosses the x-axis at a point Q. Find the coordinates of Q.  \nc. If P is the point (r,0), find the length of PQ as a function of r and the limiting value of this length as r increases without bound.  \n1970 AB 1 BC 1  \nGiven the parabola y 􀀠 x2 􀀐 2x􀀎3:  \na. Find an equation for the line L , which contains the point (2, 3) and is perpendicular to the line tangent to the parabola at (2, 3) .  \nb. Find the area of that part of the first quadrant which lies below both the line L and the parabola.  \n1970 AB 2  \nA function f is defined on the closed interval from-3 to 3 and has the graph shown below.  \na. Sketch the entire graph of y 􀀠 f (x) .  \nb. Sketch the entire graph of y 􀀠 f (x) .  \nc. Sketch the entire graph of y 􀀠 f (􀀐x) .  \n5  \n􀂧 1 􀂷  \nd. Sketch the entire graph of y 􀀠 f 􀂨 ~~ ~~ x 􀂸 .  \n􀂩2 􀂹  \ne. Sketch the enti","cbCaioUCFzHMUSWR","https://ap.wps.com/l/cbCaioUCFzHMUSWR","pdf",1978693,"English","# AP Calculus Free-Response Questions 1969-2010\n## 1969 AB 1\n## 1969 AB 2\n## 1969 AB 3\n## 1969 AB 4\n## 1969 AB 5\n## 1969 AB 6\n## 1969 AB 7\n## 1970 AB 1\n## 1970 AB 2\n## 1970 AB 3\n## 1970 AB 4\n## 1970 AB 5\n## 1970 AB 6\n## 1970 AB 7\n## 1971 AB 1","[{\"question\":\"What types of skills are tested across these AP Calculus free-response questions?\",\"answer\":\"The set includes conceptual checks (like periodicity, continuity, and inverse-function existence) and computation tasks involving derivatives, tangents, concavity, graph sketching, and definite integrals for area and volume.\"},{\"question\":\"How are some answers expected to be recorded in 1969 AB 1?\",\"answer\":\"For the listed questions (a–d), answers are indicated as either yes or no in a rectangular grid, with each blank space scored as incorrect if left wrong.\"},{\"question\":\"What mathematical modeling themes appear in the later problems?\",\"answer\":\"Problems model changing quantities over time (e.g., particle motion, bacteria growth, and geometric solids with rates of change), and then ask for velocities/accelerations, extrema, or minimum/maximum rates using calculus.\"}]","AP Calculus Free-Response Questions - 1969-2010 | PDF",1789900512,265]