# Ap calculus ab 2015 scoring guidelines

2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

Let f and also g be the functions defined by ( )2 21 x xf x x e = + + and

( ) 4 26.5 6 2.g x x x x= + + Let R and also S be the two regions enclosed by the graphs of f and also g shown in the figure over.

(a) Find the amount of the locations of regions R and also S.

(b) Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid.

(c) Let h be the vertical distance between the graphs of f and g in region S. Find the price at which h transforms through respect to x once

1.8.x =

(a) The graphs of ( )y f x= and also ( )y g x= intersect in the initially

( ) ( )< > ( ) ( )< >2

00.997427 1.006919 2.00

e

4

Ar aA

Af xg x g x dxf x dx +

= +

=

=

1 : limits4 : 2 : integrands

(b) ( ) ( )< >2 2 1.28Volum 3eA

f x g x dx == { 2 : integrand3 : 1 : answer

(c) ( ) ( ) ( )h x f x g x= ( ) ( ) ( )h x f x g x = ( ) ( ) ( )1.8 1.8 1.8 3.812 (or 3.811)h f g = =

{ 1 : considers 2 : 1 : answer h

2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.

You watching: Ap calculus ab 2015 scoring guidelines

2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.

2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.

2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.

2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.

2015 The College Board.Visit the College Board on the Web: www.collegeboard.org.

AP CALCULUS AB 2015 SCORING COMMENTARY

2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

Question 2

Overview

In this problem students were provided a graph of the boundary curves of two planar areas R and also S in the first

quadrant. One boundary curve is identified by ( )2 21 ,x xf x x e = + + and also the various other boundary is identified by

( ) 4 26.5 6 2.g x x x x= + + In component (a) students were asked to uncover the sum of the locations of areas R and S. Two interarea points of the boundary curves, ( )0, 2 and also ( )2, 4 , are offered, and also students were intended to uncover the other suggest of interarea by using the calculator. The interarea point is ( ) ( ), 1.032832, 2.401108 .A B =

The sum of the locations of R and S is ( ) ( )( ) ( ) ( )( )2

0.

A

Ag x f x dx f x g x dx + Students were expected to use

the calculator to evaluate the integrals. In component (b) students were asked to find the volume of a solid with S as its base. Students had to analyze the area of the cross sections as ( ) ( )< >2 ,f x g x and usage the calculator to evaluate

the volume as ( ) ( )< >2 2 .A

f x g x dx In component (c) students had actually to discover the price of change of the vertical distance, h, in between the graphs of f and also g at 1.8.x = Students were expected to identify and also communicate ( ) ( ) ( ) ,h x f x g x = then evaluate ( )1.8h using the numerical derivative at a suggest capcapability of the calculator.

Sample: 2A Score: 9

The response earned all 9 points.

Sample: 2B Score: 6

The response earned 6 points: 3 points in part (a), 3 points in component (b), and no points in part (c). In component (a) the student presents correct integrals for the areas of the 2 areas and earned the first 3 points. The student evaluates the locations of the two areas properly. The student does not discover the amount and did not earn the answer point. In part (b) the students work-related is correct. In component (c) the student presents an incorrect expression for .h

Sample: 2C Score: 3

The response earned 3 points: 2 points in component (a), no points in part (b), and 1 point in component (c). In part (a) the student supplies 1x = as the x-coordinate of the suggest of intersection. The student did not earn the first allude. For each of the regions, the student presents the correct integrand also, so the second and also 3rd points were earned. The student is not eligible for the answer allude. In part (b) the student presents an incorrect integrand also. In part (c) the student considers h f g = and earned the first allude. The testimonial of ( )1.8h is incorrect.