ap-calculus-ab

Q1 Small angle approximation View

$\lim _ { x \rightarrow 0 } \frac { 1 - \cos ^ { 2 } ( 2 x ) } { ( 2 x ) ^ { 2 } } =$
(A) 0
(B) $\frac { 1 } { 4 }$
(C) $\frac { 1 } { 2 }$
(D) 1

Q1 (Free-Response) Indefinite & Definite Integrals Multi-Part Applied Integration with Context (Trapezoidal/Numerical Estimation) View

$t$ (hours)	0	2	4	6	8	10	12
$R ( t )$ (vehicles per hour)	2935	3653	3442	3010	3604	1986	2201

On a certain weekday, the rate at which vehicles cross a bridge is modeled by the differentiable function $R$ for $0 \leq t \leq 12$, where $R ( t )$ is measured in vehicles per hour and $t$ is the number of hours since 7:00 A.M. $(t = 0)$. Values of $R ( t )$ for selected values of $t$ are given in the table above.
(a) Use the data in the table to approximate $R ^ { \prime } ( 5 )$. Show the computations that lead to your answer. Using correct units, explain the meaning of $R ^ { \prime } ( 5 )$ in the context of the problem.
(b) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of $\int _ { 0 } ^ { 12 } R ( t ) \, d t$. Indicate units of measure.
(c) On a certain weekend day, the rate at which vehicles cross the bridge is modeled by the function $H$ defined by $H ( t ) = - t ^ { 3 } - 3 t ^ { 2 } + 288 t + 1300$ for $0 \leq t \leq 17$, where $H ( t )$ is measured in vehicles per hour and $t$ is the number of hours since 7:00 A.M. $(t = 0)$. According to this model, what is the average number of vehicles crossing the bridge per hour on the weekend day for $0 \leq t \leq 12$?
(d) For $12 < t < 17$, $L ( t )$, the local linear approximation to the function $H$ given in part (c) at $t = 12$, is a better model for the rate at which vehicles cross the bridge on the weekend day. Use $L ( t )$ to find the time $t$, for $12 < t < 17$, at which the rate of vehicles crossing the bridge is 2000 vehicles per hour. Show the work that leads to your answer.

Q2 Tangents, normals and gradients Find tangent line equation at a given point View

$$f ( x ) = \begin{cases} \frac { 2 } { x } & \text { for } x < - 1 \\ x ^ { 2 } - 3 & \text { for } - 1 \leq x \leq 2 \\ 4 x - 3 & \text { for } x > 2 \end{cases}$$
Let $f$ be the function defined above. At what values of $x$, if any, is $f$ not differentiable?
(A) $x = - 1$ only
(B) $x = 2$ only
(C) $x = - 1$ and $x = - 2$
(D) $f$ is differentiable for all values of $x$.

Q2 (Free-Response) Stationary points and optimisation Analyze function behavior from graph or table of derivative View

The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $[ 0,4 ]$. The areas of the regions bounded by the graph of $f ^ { \prime }$ and the $x$-axis on the intervals $[ 0,1 ] , [ 1,2 ] , [ 2,3 ]$, and $[ 3,4 ]$ are $2, 6, 10$, and $14$, respectively. The graph of $f ^ { \prime }$ has horizontal tangents at $x = 0.6 , x = 1.6$, $x = 2.5$, and $x = 3.5$. It is known that $f ( 2 ) = 5$.
(a) On what open intervals contained in $( 0,4 )$ is the graph of $f$ both decreasing and concave down? Give a reason for your answer.
(b) Find the absolute minimum value of $f$ on the interval $[ 0,4 ]$. Justify your answer.
(c) Evaluate $\int _ { 0 } ^ { 4 } f ( x ) f ^ { \prime } ( x ) \, d x$.
(d) The function $g$ is defined by $g ( x ) = x ^ { 3 } f ( x )$. Find $g ^ { \prime } ( 2 )$. Show the work that leads to your answer.

Q3 Product & Quotient Rules View

$x$	$f ( x )$	$f ^ { \prime } ( x )$	$g ( x )$	$g ^ { \prime } ( x )$
1	2	- 4	- 5	3
2	- 3	1	8	4

The table above gives values of the differentiable functions $f$ and $g$ and their derivatives at selected values of $x$. If $h$ is the function defined by $h ( x ) = f ( x ) g ( x ) + 2 g ( x )$, then $h ^ { \prime } ( 1 ) =$
(A) 32
(B) 30
(C) - 6
(D) - 16

Q3 (Free-Response) Variable acceleration (vectors) View

For $0 \leq t \leq 5$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 1$, the particle is at position $( 2 , - 7 )$. It is known that $\frac { d x } { d t } = \sin \left( \frac { t } { t + 3 } \right)$ and $\frac { d y } { d t } = e ^ { \cos t }$.
(a) Write an equation for the line tangent to the curve at the point $( 2 , - 7 )$.
(b) Find the $y$-coordinate of the position of the particle at time $t = 4$.
(c) Find the total distance traveled by the particle from time $t = 1$ to time $t = 4$.
(d) Find the time at which the speed of the particle is 2.5. Find the acceleration vector of the particle at this time.

Q4 Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View

If $x ^ { 3 } - 2 x y + 3 y ^ { 2 } = 7$, then $\frac { d y } { d x } =$
(A) $\frac { 3 x ^ { 2 } + 4 y } { 2 x }$
(B) $\frac { 3 x ^ { 2 } - 2 y } { 2 x - 6 y }$
(C) $\frac { 3 x ^ { 2 } } { 2 x - 6 y }$
(D) $\frac { 3 x ^ { 2 } } { 2 - 6 y }$

Q4 (Free-Response) Taylor series Determine radius or interval of convergence View

The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k + 1 } x ^ { k } } { k ^ { 2 } } = x - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 3 } } { 9 } - \cdots$ on its interval of convergence.
(a) Use the ratio test to determine the interval of convergence of the Maclaurin series for $f$. Show the work that leads to your answer.
(b) The Maclaurin series for $f$ evaluated at $x = \frac { 1 } { 4 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $f \left( \frac { 1 } { 4 } \right)$ using the first two nonzero terms of this series is $\frac { 15 } { 64 }$. Show that this approximation differs from $f \left( \frac { 1 } { 4 } \right)$ by less than $\frac { 1 } { 500 }$.
(c) Let $h$ be the function defined by $h ( x ) = \int _ { 0 } ^ { x } f ( t ) \, d t$. Write the first three nonzero terms and the general term of the Maclaurin series for $h$.

Q5 Connected Rates of Change Volume/Height Related Rates for Containers and Solids View

The radius of a right circular cylinder is increasing at a rate of 2 units per second. The height of the cylinder is decreasing at a rate of 5 units per second. Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius $r$ and height $h$ of the cylinder? (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(A) $- 20 \pi r$
(B) $- 2 \pi r h$
(C) $4 \pi r h - 5 \pi r ^ { 2 }$
(D) $4 \pi r h + 5 \pi r ^ { 2 }$

Q6 Differentiation from First Principles View

Which of the following is equivalent to the definite integral $\int _ { 2 } ^ { 6 } \sqrt { x } \, d x$ ?
(A) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 4 } { n } \sqrt { \frac { 4 k } { n } }$
(B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 6 } { n } \sqrt { \frac { 6 k } { n } }$
(C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 4 } { n } \sqrt { 2 + \frac { 4 k } { n } }$
(D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 6 } { n } \sqrt { 2 + \frac { 6 k } { n } }$

Q7 Indefinite & Definite Integrals Accumulation Function Analysis View

The figure above shows the graph of the continuous function $g$ on the interval $[ 0,8 ]$. Let $h$ be the function defined by $h ( x ) = \int _ { 3 } ^ { x } g ( t ) \, d t$. On what intervals is $h$ increasing?
(A) $[ 2,5 ]$ only
(B) $[ 1,7 ]$
(C) $[ 0,1 ]$ and $[ 3,7 ]$
(D) $[ 1,3 ]$ and $[ 7,8 ]$

Q8 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View

$\int \frac { x } { \sqrt { 1 - 9 x ^ { 2 } } } d x =$
(A) $- \frac { 1 } { 9 } \sqrt { 1 - 9 x ^ { 2 } } + C$
(B) $- \frac { 1 } { 18 } \ln \sqrt { 1 - 9 x ^ { 2 } } + C$
(C) $\frac { 1 } { 3 } \arcsin ( 3 x ) + C$
(D) $\frac { x } { 3 } \arcsin ( 3 x ) + C$

Q9 Differential equations Slope Field Identification View

Shown above is a slope field for which of the following differential equations?
(A) $\frac { d y } { d x } = \frac { y - 2 } { 2 }$
(B) $\frac { d y } { d x } = \frac { y ^ { 2 } - 4 } { 4 }$
(C) $\frac { d y } { d x } = \frac { x - 2 } { 2 }$
(D) $\frac { d y } { d x } = \frac { x ^ { 2 } - 4 } { 4 }$

Q10 Areas by integration View

Let $R$ be the region bounded by the graph of $x = e ^ { y }$, the vertical line $x = 10$, and the horizontal lines $y = 1$ and $y = 2$. Which of the following gives the area of $R$?
(A) $\int _ { 1 } ^ { 2 } e ^ { y } \, d y$
(B) $\int _ { e } ^ { e ^ { 2 } } \ln x \, d x$
(C) $\int _ { 1 } ^ { 2 } \left( 10 - e ^ { y } \right) d y$
(D) $\int _ { e } ^ { 10 } ( \ln x - 1 ) d x$

Q11 Curve Sketching Limit Reading from Graph View

The graph of the function $f$ is shown in the figure above. The value of $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ is
(A) - 2
(B) - 1
(C) 2
(D) nonexistent

Q12 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View

The velocity of a particle moving along a straight line is given by $v ( t ) = 1.3 t \ln ( 0.2 t + 0.4 )$ for time $t \geq 0$. What is the acceleration of the particle at time $t = 1.2$?
(A) - 0.580
(B) - 0.548
(C) - 0.093
(D) 0.660

Q13 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

$x$	- 1	0	2	4	5
$f ^ { \prime } ( x )$	11	9	8	5	2

Let $f$ be a twice-differentiable function. Values of $f ^ { \prime }$, the derivative of $f$, at selected values of $x$ are given in the table above. Which of the following statements must be true?
(A) $f$ is increasing for $- 1 \leq x \leq 5$.
(B) The graph of $f$ is concave down for $- 1 < x < 5$.
(C) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime } ( c ) = - \frac { 3 } { 2 }$.
(D) There exists $c$, where $- 1 < c < 5$, such that $f ^ { \prime \prime } ( c ) = - \frac { 3 } { 2 }$.

Q14 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

Let $f$ be the function with derivative defined by $f ^ { \prime } ( x ) = 2 + ( 2 x - 8 ) \sin ( x + 3 )$.
How many points of inflection does the graph of $f$ have on the interval $0 < x < 9$?
(A) One
(B) Two
(C) Three
(D) Four

Q15 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View

Honey is poured through a funnel at a rate of $r ( t ) = 4 e ^ { - 0.35 t }$ ounces per minute, where $t$ is measured in minutes. How many ounces of honey are poured through the funnel from time $t = 0$ to time $t = 3$?
(A) 0.910
(B) 1.400
(C) 2.600
(D) 7.429

Q16 Integration by Parts Definite Integral Evaluation by Parts View

$x$	2	5
$f ( x )$	4	7
$f ^ { \prime } ( x )$	2	3

The table above gives values of the differentiable function $f$ and its derivative $f ^ { \prime }$ at selected values of $x$. If $\int _ { 2 } ^ { 5 } f ( x ) \, d x = 14$, what is the value of $\int _ { 2 } ^ { 5 } x \cdot f ^ { \prime } ( x ) \, d x$?
(A) 13
(B) 27
(C) $\frac { 63 } { 2 }$
(D) 41

Q17 First order differential equations (integrating factor) View

The number of fish in a lake is modeled by the function $F$ that satisfies the logistic differential equation $\frac { d F } { d t } = 0.04 F \left( 1 - \frac { F } { 5000 } \right)$, where $t$ is the time in months and $F ( 0 ) = 2000$. What is $\lim _ { t \rightarrow \infty } F ( t )$?
(A) 10,000
(B) 5000
(C) 2500
(D) 2000

Q18 Parametric differentiation View

A curve is defined by the parametric equations $x ( t ) = t ^ { 2 } + 3$ and $y ( t ) = \sin \left( t ^ { 2 } \right)$. Which of the following is an expression for $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $t$?
(A) $- \sin \left( t ^ { 2 } \right)$
(B) $- 2 t \sin \left( t ^ { 2 } \right)$
(C) $\cos \left( t ^ { 2 } \right) - 2 t ^ { 2 } \sin \left( t ^ { 2 } \right)$
(D) $2 \cos \left( t ^ { 2 } \right) - 4 t ^ { 2 } \sin \left( t ^ { 2 } \right)$

Q20 Taylor series Derive series via differentiation or integration of a known series View

Let $f$ be the function defined by $f ( x ) = e ^ { 2 x }$. Which of the following is the Maclaurin series for $f ^ { \prime }$, the derivative of $f$?
(A) $1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } + \cdots$
(B) $2 + 2 x + \frac { 2 x ^ { 2 } } { 2 ! } + \frac { 2 x ^ { 3 } } { 3 ! } + \cdots + \frac { 2 x ^ { n } } { n ! } + \cdots$
(C) $1 + 2 x + \frac { ( 2 x ) ^ { 2 } } { 2 ! } + \frac { ( 2 x ) ^ { 3 } } { 3 ! } + \cdots + \frac { ( 2 x ) ^ { n } } { n ! } + \cdots$
(D) $2 + 2 ( 2 x ) + \frac { 2 ( 2 x ) ^ { 2 } } { 2 ! } + \frac { 2 ( 2 x ) ^ { 3 } } { 3 ! } + \cdots + \frac { 2 ( 2 x ) ^ { n } } { n ! } + \cdots$

Q22 Taylor series Lagrange error bound application View

The function $f$ has derivatives of all orders for all real numbers. It is known that $\left| f ^ { ( 4 ) } ( x ) \right| \leq \frac { 12 } { 5 }$ and $\left| f ^ { ( 5 ) } ( x ) \right| \leq \frac { 3 } { 2 }$ for $0 \leq x \leq 2$. Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. The Taylor series for $f$ about $x = 0$ converges at $x = 2$. Of the following, which is the smallest value of $k$ for which the Lagrange error bound guarantees that $\left| f ( 2 ) - P _ { 4 } ( 2 ) \right| \leq k$?
(A) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 3 } { 2 }$
(B) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 12 } { 5 }$
(C) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 3 } { 2 }$
(D) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 12 } { 5 }$

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