1998 AP Calculus AB Scoring Guidelines
[Pages:1]1998 AP Calculus AB Scoring Guidelines
6. Consider the curve defined by 2y3 + 6x2y - 12x2 + 6y = 1. dy 4x - 2xy
(a) Show that dx = x2 + y2 + 1 . (b) Write an equation of each horizontal tangent line to the curve.
(c) The line through the origin with slope -1 is tangent to the curve at point P . Find the x? and y?coordinates of point P .
(a)
6y2 dy
+ 6x2 dy
+ 12xy
dy - 24x + 6
=
0
dx
dx
dx
dy (6y2 + 6x2 + 6) = 24x - 12xy dx
dy dx
=
24x - 12xy 6x2 + 6y2 + 6
=
4x - 2xy x2 + y2 + 1
1:
2
1:
implicit differentiation dy
verifies expression for dx
dy (b) = 0
dx 4x - 2xy = 2x(2 - y) = 0 x = 0 or y = 2 When x = 0, 2y3 + 6y = 1 ; y = 0.165 There is no point on the curve with y coordinate of 2. y = 0.165 is the equation of the only horizontal tangent line.
(c) y = -x is equation of the line. 2(-x)3 + 6x2(-x) - 12x2 + 6(-x) = 1 -8x3 - 12x2 - 6x - 1 = 0 x = -1/2 , y = 1/2 ?or? dy dx = -1 4x - 2xy = -x2 - y2 - 1 4x + 2x2 = -x2 - x2 - 1 4x2 + 4x + 1 = 0 x = -1/2 , y = 1/2
1:
1:
4
1:
1:
dy sets = 0
dx dy
solves = 0 dx
uses solutions for x to find equations of horizontal tangent lines
verifies which solutions for y yield equations of horizontal tangent lines
Note: max 1/4 [1-0-0-0] if dy/dx = 0 is not of the form g(x, y)/h(x, y) = 0 with solutions for both x and y
1:
3
1:
1:
y = -x substitutes y = -x into equation of curve
solves for x and y
?or?
1:
3 1: 1:
dy sets = -1
dx dy
substitutes y = -x into dx solves for x and y
Note: max 2/3 [1-1-0] if importing incorrect derivative from part (a)
? Copyright 1998 College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- score sjsu
- math 111 quiz 6 solutions 1 consider the equation 1 which
- homework 3 department of physics
- lagrange multipliers illinois institute of technology
- equations with regular singular points sect 5 5 equations with
- unit 5 implicit di erentiation related rates
- ch 14 6 directional derivatives and gradient vector
- name solutions date 10 06 2016
- unit 22 unconstrained optimization local extrema queen s u
- 1998 ap calculus ab scoring guidelines
Related searches
- calculus ab cheat sheet
- ap calculus derivatives test pdf
- ap calculus ab textbook pdf
- ap calculus book pdf
- ap calculus textbook finney pdf
- finney ap calculus 5th ed
- ap calculus problems and solutions
- ap calculus textbook larson pdf
- larson calculus ap edition pdf
- ap calculus graphical numerical algebraic
- ap calculus derivative problems
- ap calculus finney pdf