ap-calculus-ab

Papers (46)
2025
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2024
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2023
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2022
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2019
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2018
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2017
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2016
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2015
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2014
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2013
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2012
free-response 6 practice-exam 50
2011
free-response 6 free-response_formB 6
2010
free-response 6 free-response_formB 3
2009
free-response 6 free-response_formB 6
2008
free-response 6 free-response_formB 6
2007
free-response 6 free-response_formB 6
2006
free-response 6 free-response_formB 6
2005
free-response 6 free-response_formB 6
2004
free-response 3 free-response_formB 6
2003
free-response 6
2002
free-response 5 free-response_formB 6
2001
free-response 6 free-response_formB 6
2000
free-response 6 free-response_formB 6
1999
free-response 6 free-response_formB 7
1998
free-response 6 free-response_formB 5 full-exam 18
Unknown
-bc_1969-1998_multiple-choice-collection 29 -bc_course-and-exam-description 26 -bc_sample-questions 18 1988_full-exam 3
2004 free-response

3 maths questions

Q3 Areas by integration Definite Integral Evaluation (Computational) View
3. Let $f$ be the function given by $f ( x ) = \sqrt { x - 3 } . \wedge B$
(a) On the axes provided below, sketch the graph of $f$ and shade the region $R$ enclosed by the graph of $f$, the $x$-axis, and the vertical line $x = 6$.
Note: The axes for this graph are provided in the pink test booklet only.
(b) Find the area of the region $R$ described in part (a).
(c) Rather than using the line $x = 6$ as in part (a), consider the line $x = w$, where $w$ can be any number greater than 3 . Let $A ( w )$ be the area of the region enclosed by the graph of $f$, the $x$-axis, and the vertical line $x = w$. Write an integral expression for $A ( w )$.
(d) Let $A ( w )$ be as described in part (c). Find the rate of change of $A$ with respect to $w$ when $w = 6$.
Q4 Stationary points and optimisation Determine parameters from given extremum conditions View
4. Let $f$ be the function given by $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + p$, where $p$ is an arbitrary constant.
(a) Write an expression for $f ^ { \prime } ( x )$ and use it to find the relative maximum and minimum values of $f$ in terms of $p$. Show the analysis that leads to your conclusion.
(b) For what values of the constant $p$ does $f$ have 3 distinct real roots?
(c) Find the value of $p$ such that the average value of $f$ over the closed interval $[ - 1,2 ]$ is 1 . [Figure]
Q5 Stationary points and optimisation Accumulation Function Analysis View
5. The graph of a function $f$ consists of a semicircle and two line segments as shown above. Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) Find $g ( 3 )$.
(b) Find all values of $x$ on the open interval $( - 2,5 )$ at which $g$ has a relative maximum. Justify your answer.
(c) Write an equation for the line tangent to the graph of $g$ at $x = 3$.
(d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $( - 2,5 )$. Justify your answer.