----- o Implicit Differentiation ID: A. dy r.---; d 2 Y 2. If- = '" 1-y- then - = dx 'dx 2. a c. -1 d. -2 e.

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1 Name: Class: Date: ID: A Implicit Differentiation Multiple Choice Identify the choice that best completes the statement or answers the question.. The slope of the line tangent to the curve y + (xy + ) = 0 at (, -) is a. b. c. d. e. o dy r.---; d Y. If- = '" -y- then - = dx 'dx a. -y b. -y -y C. ~_y d. y e. ~ dy.:>. If y = xy + x +, then when x = -, dx is a. - b. c. - d. - e. nonexistent

2 008 AB 6 Form B Consider the closed curve in the xy-plane l'x-~ ~~f~~?l --( ~ given by X + x + y + y = 5. (a) Show that dy = -(x+l) dx (y+)' (b) Write an equation for the line tangent to the curve at the point (-, ). (c) Find the coordinates of the two points on the curve where the line tangent to the curve is vertical. (d) Is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis? Explain your reasoning. 005 AB 5 Form B Consider the curve given by y = + xy. (a) Show that dy = -y-. dx y-x (b) Find all points (x,y) on the curve where the line tangent to the curve has slope~. (c) Show that there are no points (x, y) on the curve where the line tangent to the curve is horizontal. (d) Let x and y be functions of time t that are related by the equation y = + xy. At time t = 5, the value of y is and dy = 6. Find the value of dx at time t = 5. dt dt 00AB The function/is differentiable for all real numbers. The point (,±) is on the graph of y = I(x), and the slope at each point (x, y) on the graph is given by dy = y(6 - x). dx d (a) Find:' and evaluate it at the point (, '). (b) Find y = I(x) by solving the differential equation : = y(6_ x) with the initial condition () =..

3 Ap CALCULUS AB 008 SCORING GUIDELINES (Form B) > Question 6 Consider the closed curve in the xy-plane given by x + x + y + y = 5. d -(x+l) (a) Show that dx y - (i + ) (b) Write an equation for the line tangent to the curve at the point (-,). (c) Find the coordinates of the two points on the curve where the line tangent to the curve is vertical. (d) Is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis? Explain your reasoning. (a) x + + i : + : = 0 (i+): =-x- dy -(x + ) -(x + ) dx = (i +) = (y+) : : implicit differentiation : verification (b) dyl = -(-+) =! dx (-,) (+ ) Tangent line: y = + ±(x + ) I: slope : : tangent line equation (c) Vertical tangent lines occur at points on the curve where i + = 0 (or y = - ) and x *- -. On the curve, y = - implies that x + x + I - = 5, so x = - or x =. Vertical tangent lines occur at the points (-, -) and (, -). I: y =- : I: substitutes y = - into the equation of the curve : answer (d) Horizontal tangents occur at points on the curve where x = - and y *- -. The curve crosses the x-axis where y = o. (_) + (-) *- 5 I: works with x = -lor y = 0 '. I: answer with reason No, the curve cannot have a horizontal tangent where it crosses the x-axis. 008 The College Board. Allrights reserved. Visit the College Board on the Web:

4 - Consider the curve given by i = + ~. (a) Show that dy = _y_. dx y - x Ap CALCULUS AB 005 SCORING GUIDELINES (Form B) Question 5 (b) Find all points (x, y) on the curve where the line tangent to the curve has slope ~. (c) Show that there are no points (x, y) on the curve where the line tangent to the curve is horizontal. (d) Let x and y be functions of time t that are related by the equation y = +~. At time t = 5, the value of y is and ~ = 6. Find the value of;; at time t = 5. (a) yy' = y + xy' (y - x)y' = y, y y=-- y-x : I : implicit diff~rentiation : solves for y (b) y _ y - x -" y =y-x x=o y=±j (0, J), (0, -J) l.-y =. :. y - x : answer (c) -y_=o y-x y=o The curve has no horizontal tangent since 0 * + x-o for any x. l:y=o. : explanation (d) When y =, = + x so x =. dy_dy dx_ y dx dt - dx. dt - y - x. dt At t=5 ;;Ls = dx 9 dx 6 =-_._=_.-, 6 _ 7... dt dt I: solves for x : : chain rule : answer Copyright 005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 6

5 Ap CALCULUS AB 00 SCORING GUIDELINES 5 Question 6 The function f is differentiable for all real numbers. The point (, ~) is on the graph of y = f(x), and the slope at each point (x,y) on the graph is given by dy = y (6 - x). dx (a) Find d; and evaluate it at the point (,.!.). dx (b) Find y = f(x) by solving the differential equation ~~ = y (6 - x) with the initial condition f() = ~. (a) d y dy -. = y-(6 - x) - y dx dx = y(6 - x? - y d y I () dx ( ~ \ = 0 - " ' 8 : d y : - dx < - > product rule or chain rule error : value at (,~) (b) dy = (6 - x)dx y - - = 6x -x + C y - = C = 9 + C C = - 6: : separates variables : antiderivative of dy term rantiderivative of dx term : constant of integration : uses initial condition f() = ~ : solves for y y = x - 6x + Note: max /6 [ ] if no constant of integration Note: 0/6 if no separation of variables Copyright 00 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 7