AP CALCULUS BC - College Board
[Pages:8]AP? CALCULUS BC 2007 SCORING GUIDELINES
Question 4
Let f be the function defined for x > 0, with f (e) = 2 and f , the first derivative of f, given by f ( x) = x2 ln x. (a) Write an equation for the line tangent to the graph of f at the point (e, 2).
(b) Is the graph of f concave up or concave down on the interval 1 < x < 3 ? Give a reason for your answer.
(c) Use antidifferentiation to find f ( x).
(a) f (e) = e2
An equation for the line tangent to the graph of f at the
point (e, 2) is y - 2 = e2 ( x - e).
2
:
1 1
: :
f (e)
equation
of
tangent
line
(b) f ( x) = x + 2x ln x.
For 1 < x < 3, x > 0 and ln x > 0, so f ( x) > 0. Thus, the graph of f is concave up on (1, 3).
3
:
2 1
: :
f ( x)
answer
with
reason
( ) (c) Since f ( x) = x2 ln x dx, we consider integration by
parts.
u = ln x
dv = x2 dx
( ) du
=
1 x
dx
v = x2 dx = 1 x3 3
Therefore,
f ( x) = ( x2 ln x) dx
( ) =
1 x3 ln 3
x
-
1 x3 1 3x
dx
= 1 x3 ln x - 1 x3 + C.
3
9
Since f (e) = 2, 2 = e3 - e3 + C and C = 2 - 2 e3.
39
9
Thus, f ( x) = x3 ln x - 1 x3 + 2 - 2 e3.
3
9
9
4
:
2 1
: :
antiderivative
uses f (e) = 2
1 : answer
? 2007 The College Board. All rights reserved. Visit apcentral. (for AP professionals) and apstudents (for students and parents).
?2007 The College Board. All rights reserved. Visit apcentral. (for AP professionals) and apstudents (for students and parents).
?2007 The College Board. All rights reserved. Visit apcentral. (for AP professionals) and apstudents (for students and parents).
?2007 The College Board. All rights reserved. Visit apcentral. (for AP professionals) and apstudents (for students and parents).
?2007 The College Board. All rights reserved. Visit apcentral. (for AP professionals) and apstudents (for students and parents).
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