ap-calculus-ab

Papers (46)
2025
free-response 6
2024
free-response 6
2023
free-response 6
2022
free-response 6
2021
free-response 6
2019
free-response 6
2018
free-response 6
2017
free-response 6
2016
free-response 6
2015
free-response 6
2014
free-response 6
2013
free-response 6
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
1999 free-response_formB

7 maths questions

Q1 Tangents, normals and gradients Chain Rule with Composition of Explicit Functions View
Let $f$ be the real-valued function defined by $f ( x ) = \sqrt { 1 + 6 x }$. (a) Give the domain and range of $f$. (b) Determine the slope of the line tangent to the graph of $f$ at $x = 4$. (c) Determine the y -intercept of the line tangent to the graph of f at $\mathrm { x } = 4$. (d) Give the coordinates of the point on the graph of $f$ where the tangent line is parallel to $y = x + 12$.
Q2 Factor & Remainder Theorem Custom Operation or Property Verification View
Given the two functions $f$ and $h$ such that $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 4 x + 12$ and $h ( x ) = \left\{ \begin{array} { l } \frac { f ( x ) } { x - 3 } , \text { for } x \neq 3 \\ p , \text { for } x = 3 . \end{array} \right.$ (a) Find all zeros of the function $f$. (b) Find the value of $p$ so that the function $h$ is continuous at $x = 3$. Justify your answer. (c) Using the value of $p$ found in (b), determine whether $h$ is an even function. Justify your answer.
Q3 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
Let $R$ be the region bounded by the curves $f ( x ) = \frac { 4 } { x }$ and $g ( x ) = ( x - 3 ) ^ { 2 }$. (a) Find the area of R . (b) Find the volume of the solid generated by revolving R about the X -axis.
Q4 Connected Rates of Change Parametric or Curve-Based Particle Motion Rates View
(a) A point moves on the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ so that its $y$-coordinate is increasing at a constant rate of 4 units per second. How fast is the $x$-coordinate changing when $x = 4$ ? (b) For what values of $k$ will the line $2 x + 9 y + k = 0$ be normal to the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ ?
Q5 Stationary points and optimisation Find critical points and classify extrema of a given function View
Given the function defined by $\mathrm { y } = \mathrm { e } ^ { \sin \mathrm { x } }$ for all x such that $- \pi \leqq \mathrm { x } \leqq 2 \pi$. (a) Find the x - and y -coordinates of all maximum and minimum points on the given interval. Justify your answe (b) On the axes provided, sketch the graph of the function. (c) Write an equation for the axis of symmetry of the graph.
Q6 Indefinite & Definite Integrals Finding a Function from an Integral Equation View
(a) Given $5 x ^ { 3 } + 40 = \int _ { C } ^ { X } f ( t ) d t$. (i) Find $f ( x )$. (ii) Find the value of $c$. (b) If $F ( x ) = \int _ { x } ^ { 3 } \sqrt { 1 + t ^ { 16 } } d t$, find $F ^ { \prime } ( x )$.
Q7 Differentiation from First Principles Limit Involving Derivative Definition of Composed Functions View
For a differentiable function $f$, let $f ^ { * }$ be the function defined by $f ^ { * } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x - h ) } { h }$. (a) Determine $f ^ { * } ( x )$ for $f ( x ) = x ^ { 2 } + x$. (b) Determine $f ^ { * } ( x )$ for $f ( x ) = \cos x$. (c) Write an equation that expresses the relationship between the functions $f ^ { * }$ and $f ^ { \prime }$, where $f ^ { \prime }$ denotes the usual derivative of $f$.