ap-calculus-bc

Papers (38)
2025
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2024
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2023
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2011
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2008
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2007
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2006
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2005
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2004
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2003
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2002
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2001
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1999
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1998
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2003 free-response

3 maths questions

Q4 Stationary points and optimisation Analyze function behavior from graph or table of derivative View
4. Let $f$ be a function defined on the closed interval $- 3 \leq x \leq 4$ with $f ( 0 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.
(a) On what intervals, if any, is $f$ increasing? Justify your answer.
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $- 3 < x < 4$. Justify your answer.
(c) Find an equation for the line tangent to the graph of $f$ at the point ( 0,3 ).
(d) Find $f ( - 3 )$ and $f ( 4 )$. Show the work that leads to your answers. [Figure]
Q5 Connected Rates of Change Applied Modeling with Differential Equations View
5. A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let $h$ be the depth of the coffee in the pot, measured in inches, where $h$ is a function of time $t$, measured in seconds. The volume $V$ of coffee in the pot is changing at the rate of $- 5 \pi \sqrt { h }$ cubic inches per second. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(a) Show that $\frac { d h } { d t } = - \frac { \sqrt { h } } { 5 }$.
(b) Given that $h = 17$ at time $t = 0$, solve the differential equation $\frac { d h } { d t } = - \frac { \sqrt { h } } { 5 }$ for $h$ as a function of $t$.
(c) At what time $t$ is the coffeepot empty?
Q6 Taylor series Use series to analyze function properties (extrema, monotonicity, concavity) View
6. The function $f$ is defined by the power series
$$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } = 1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$$
for all real numbers $x$.
(a) Find $f ^ { \prime } ( 0 )$ and $f ^ { \prime \prime } ( 0 )$. Determine whether $f$ has a local maximum, a local minimum, or neither at $x = 0$. Give a reason for your answer.
(b) Show that $1 - \frac { 1 } { 3 ! }$ approximates $f ( 1 )$ with error less than $\frac { 1 } { 100 }$.
(c) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } + y = \cos x$.
END OF EXAMINATION