ap-calculus-bc

Q1 Chain Rule Chain Rule with Composition of Explicit Functions View

If $y = \sin ^ { 3 } x$, then $\frac { d y } { d x } =$
(A) $\cos ^ { 3 } x$
(B) $3 \cos ^ { 2 } x$
(C) $3 \sin ^ { 2 } x$
(D) $- 3 \sin ^ { 2 } x \cos x$
(E) $3 \sin ^ { 2 } x \cos x$

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

The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W ( t )$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is $55 ^ { \circ } \mathrm { F }$. The water is heated for 30 minutes, beginning at time $t = 0$. Values of $W ( t )$ at selected times $t$ for the first 20 minutes are given in the table above.

$t$ (minutes)	0	4	9	15	20
$W ( t )$ (degrees Fahrenheit)	55.0	57.1	61.8	67.9	71.0

(a) Use the data in the table to estimate $W ^ { \prime } ( 12 )$. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$. Using correct units, interpret the meaning of $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$ in the context of this problem.
(c) For $0 \leq t \leq 20$, the average temperature of the water in the tub is $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water temperature has first derivative given by $W ^ { \prime } ( t ) = 0.4 \sqrt { t } \cos ( 0.06 t )$. Based on the model, what is the temperature of the water at time $t = 25$ ?

Q2 Parametric differentiation View

The position of a particle moving in the $x y$-plane is given by the parametric equations $x ( t ) = t ^ { 3 } - 3 t ^ { 2 }$ and $y ( t ) = 12 t - 3 t ^ { 2 }$. At which of the following points $( x , y )$ is the particle at rest?
(A) $( - 4,12 )$
(B) $( - 3,6 )$
(C) $( - 2,9 )$
(D) $( 0,0 )$
(E) $( 3,4 )$

Q2 (Free Response) Variable acceleration (vectors) View

For $t \geq 0$, a particle is moving along a curve so that its position at time $t$ is $( x ( t ) , y ( t ) )$. At time $t = 2$, the particle is at position $( 1,5 )$. It is known that $\frac { d x } { d t } = \frac { \sqrt { t + 2 } } { e ^ { t } }$ and $\frac { d y } { d t } = \sin ^ { 2 } t$.
(a) Is the horizontal movement of the particle to the left or to the right at time $t = 2$ ? Explain your answer. Find the slope of the path of the particle at time $t = 2$.
(b) Find the $x$-coordinate of the particle's position at time $t = 4$.
(c) Find the speed of the particle at time $t = 4$. Find the acceleration vector of the particle at time $t = 4$.
(d) Find the distance traveled by the particle from time $t = 2$ to $t = 4$.

Q3 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

The graph of $f$ is shown above for $0 \leq x \leq 4$. What is the value of $\int _ { 0 } ^ { 4 } f ( x ) d x$ ?
(A) - 1
(B) 0
(C) 2
(D) 6
(E) 12

Q3 (Free Response) Indefinite & Definite Integrals Accumulation Function Analysis View

Let $f$ be the continuous function defined on $[ - 4,3 ]$ whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let $g$ be the function given by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$.
(a) Find the values of $g ( 2 )$ and $g ( - 2 )$.
(b) For each of $g ^ { \prime } ( - 3 )$ and $g ^ { \prime \prime } ( - 3 )$, find the value or state that it does not exist.
(c) Find the $x$-coordinate of each point at which the graph of $g$ has a horizontal tangent line. For each of these points, determine whether $g$ has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers.
(d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.

Q4 Areas by integration View

Which of the following integrals gives the length of the curve $y = \ln x$ from $x = 1$ to $x = 2$ ?
(A) $\int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 1 } { x ^ { 2 } } } d x$
(B) $\int _ { 1 } ^ { 2 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) d x$
(C) $\int _ { 1 } ^ { 2 } \sqrt { 1 + e ^ { 2 x } } d x$
(D) $\int _ { 1 } ^ { 2 } \sqrt { 1 + ( \ln x ) ^ { 2 } } d x$
(E) $\int _ { 1 } ^ { 2 } \left( 1 + ( \ln x ) ^ { 2 } \right) d x$

Q4 (Free Response) Integration by Parts Definite Integral Evaluation by Parts View

The function $f$ is twice differentiable for $x > 0$ with $f ( 1 ) = 15$ and $f ^ { \prime \prime } ( 1 ) = 20$. Values of $f ^ { \prime }$, the derivative of $f$, are given for selected values of $x$ in the table above.

$x$	1	1.1	1.2	1.3	1.4
$f ^ { \prime } ( x )$	8	10	12	13	14.5

(a) Write an equation for the line tangent to the graph of $f$ at $x = 1$. Use this line to approximate $f ( 1.4 )$.
(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$. Use the approximation for $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$ to estimate the value of $f ( 1.4 )$. Show the computations that lead to your answer.
(c) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the computations that lead to your answer.
(d) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 1.4 )$.

Q5 Taylor series Identify a closed-form function from its Taylor series View

The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( - \frac { x } { 4 } \right) ^ { n }$. What is the value of $f ( 3 )$ ?
(A) - 3
(B) $- \frac { 3 } { 7 }$
(C) $\frac { 4 } { 7 }$
(D) $\frac { 13 } { 16 }$
(E) 4

Q5 (Free Response) Differential equations Multi-Part DE Problem (Slope Field + Solve + Analyze) View

The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time $t = 0$, when the bird is first weighed, its weight is 20 grams. If $B ( t )$ is the weight of the bird, in grams, at time $t$ days after it is first weighed, then
$$\frac { d B } { d t } = \frac { 1 } { 5 } ( 100 - B ) .$$
Let $y = B ( t )$ be the solution to the differential equation above with initial condition $B ( 0 ) = 20$.
(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning.
(b) Find $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ in terms of $B$. Use $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ to explain why the graph of $B$ cannot resemble the following graph.
(c) Use separation of variables to find $y = B ( t )$, the particular solution to the differential equation with initial condition $B ( 0 ) = 20$.

Q6 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View

Using the substitution $u = x ^ { 2 } - 3 , \int _ { - 1 } ^ { 4 } x \left( x ^ { 2 } - 3 \right) ^ { 5 } d x$ is equal to which of the following?
(A) $2 \int _ { - 2 } ^ { 13 } u ^ { 5 } d u$
(B) $\int _ { - 2 } ^ { 13 } u ^ { 5 } d u$
(C) $\frac { 1 } { 2 } \int _ { - 2 } ^ { 13 } u ^ { 5 } d u$
(D) $\int _ { - 1 } ^ { 4 } u ^ { 5 } d u$
(E) $\frac { 1 } { 2 } \int _ { - 1 } ^ { 4 } u ^ { 5 } d u$

Q6 (Free Response) Taylor series Determine radius or interval of convergence View

The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is
$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 3 } = \frac { x } { 3 } - \frac { x ^ { 3 } } { 5 } + \frac { x ^ { 5 } } { 7 } - \cdots$$
(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for $g$.
(b) The Maclaurin series for $g$ evaluated at $x = \frac { 1 } { 2 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $g \left( \frac { 1 } { 2 } \right)$ using the first two nonzero terms of this series is $\frac { 17 } { 120 }$. Show that this approximation differs from $g \left( \frac { 1 } { 2 } \right)$ by less than $\frac { 1 } { 200 }$.
(c) Write the first three nonzero terms and the general term of the Maclaurin series for $g ^ { \prime } ( x )$.

Q7 Laws of Logarithms Solve a Logarithmic Equation View

If $\arcsin x = \ln y$, then $\frac { d y } { d x } =$
(A) $\frac { y } { \sqrt { 1 - x ^ { 2 } } }$
(B) $\frac { x y } { \sqrt { 1 - x ^ { 2 } } }$
(C) $\frac { y } { 1 + x ^ { 2 } }$
(D) $e ^ { \arcsin x }$
(E) $\frac { e ^ { \arcsin x } } { 1 + x ^ { 2 } }$

Q8 Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View

A tank contains 50 liters of oil at time $t = 4$ hours. Oil is being pumped into the tank at a rate $R ( t )$, where $R ( t )$ is measured in liters per hour, and $t$ is measured in hours. Selected values of $R ( t )$ are given in the table above. Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time $t = 15$ hours?

$t$ (hours)	4	7	12	15
$R ( t )$ (liters/hour)	6.5	6.2	5.9	5.6

(A) 64.9
(B) 68.2
(C) 114.9
(D) 116.6
(E) 118.2

Q9 Sequences and Series Convergence/Divergence Determination of Numerical Series View

Which of the following series converge?
I. $\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } } { n ! }$
II. $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { n ^ { 100 } }$
III. $\sum _ { n = 1 } ^ { \infty } \frac { n + 1 } { ( n ) ( n + 2 ) ( n + 3 ) }$
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

Q10 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

$\int _ { 1 } ^ { 4 } t ^ { - 3 / 2 } d t =$
(A) - 1
(B) $- \frac { 7 } { 8 }$
(C) $- \frac { 1 } { 2 }$
(D) $\frac { 1 } { 2 }$
(E) 1

Q11 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View

Let $f$ be the function defined by $f ( x ) = \sqrt { | x - 2 | }$ for all $x$. Which of the following statements is true?
(A) $f$ is continuous but not differentiable at $x = 2$.
(B) $f$ is differentiable at $x = 2$.
(C) $f$ is not continuous at $x = 2$.
(D) $\lim _ { x \rightarrow 2 } f ( x ) \neq 0$
(E) $x = 2$ is a vertical asymptote of the graph of $f$.

Q12 Differential equations Qualitative Analysis of DE Solutions View

The points $( - 1 , - 1 )$ and $( 1 , - 5 )$ are on the graph of a function $y = f ( x )$ that satisfies the differential equation $\frac { d y } { d x } = x ^ { 2 } + y$. Which of the following must be true?
(A) $( 1 , - 5 )$ is a local maximum of $f$.
(B) $( 1 , - 5 )$ is a point of inflection of the graph of $f$.
(C) $( - 1 , - 1 )$ is a local maximum of $f$.
(D) $( - 1 , - 1 )$ is a local minimum of $f$.
(E) $( - 1 , - 1 )$ is a point of inflection of the graph of $f$.

Q13 Sequences and Series Power Series Expansion and Radius of Convergence View

What is the radius of convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 4 ) ^ { 2 n } } { 3 ^ { n } }$ ?
(A) $2 \sqrt { 3 }$
(B) 3
(C) $\sqrt { 3 }$
(D) $\frac { \sqrt { 3 } } { 2 }$
(E) 0

Q14 Differential equations Applied Modeling with Differential Equations View

Let $k$ be a positive constant. Which of the following is a logistic differential equation?
(A) $\frac { d y } { d t } = k t$
(B) $\frac { d y } { d t } = k y$
(C) $\frac { d y } { d t } = k t ( 1 - t )$
(D) $\frac { d y } { d t } = k y ( 1 - t )$
(E) $\frac { d y } { d t } = k y ( 1 - y )$

Q15 Indefinite & Definite Integrals Accumulation Function Analysis View

The graph of a differentiable function $f$ is shown above. If $h ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, which of the following is true?
(A) $h ( 6 ) < h ^ { \prime } ( 6 ) < h ^ { \prime \prime } ( 6 )$
(B) $h ( 6 ) < h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 )$
(C) $h ^ { \prime } ( 6 ) < h ( 6 ) < h ^ { \prime \prime } ( 6 )$
(D) $h ^ { \prime \prime } ( 6 ) < h ( 6 ) < h ^ { \prime } ( 6 )$
(E) $h ^ { \prime \prime } ( 6 ) < h ^ { \prime } ( 6 ) < h ( 6 )$

Q16 Differential equations Euler's Method Approximation View

Let $y = f ( x )$ be the solution to the differential equation $\frac { d y } { d x } = x - y$ with initial condition $f ( 1 ) = 3$. What is the approximation for $f ( 2 )$ obtained by using Euler's method with two steps of equal length starting at $x = 1$ ?
(A) $- \frac { 5 } { 4 }$
(B) 1
(C) $\frac { 7 } { 4 }$
(D) 2
(E) $\frac { 21 } { 4 }$

Q17 Taylor series Identify a closed-form function from its Taylor series View

For $x > 0$, the power series $1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$ converges to which of the following?
(A) $\cos x$
(B) $\sin x$
(C) $\frac { \sin x } { x }$
(D) $e ^ { x } - e ^ { x ^ { 2 } }$
(E) $1 + e ^ { x } - e ^ { x ^ { 2 } }$

Q18 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

The graph of $f ^ { \prime }$, the derivative of a function $f$, consists of two line segments and a semicircle, as shown in the figure above. If $f ( 2 ) = 1$, then $f ( - 5 ) =$
(A) $2 \pi - 2$
(B) $2 \pi - 3$
(C) $2 \pi - 5$
(D) $6 - 2 \pi$
(E) $4 - 2 \pi$

Q19 Tangents, normals and gradients Find tangent line with a specified slope or from an external point View

The function $f$ is defined by $f ( x ) = \frac { x } { x + 2 }$. What points $( x , y )$ on the graph of $f$ have the property that the line tangent to $f$ at $( x , y )$ has slope $\frac { 1 } { 2 }$ ?
(A) $( 0,0 )$ only
(B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 5 } \right)$ only
(C) $( 0,0 )$ and $( - 4,2 )$
(D) $( 0,0 )$ and $\left( 4 , \frac { 2 } { 3 } \right)$
(E) There are no such points.

Q20 Partial Fractions View

$\int _ { 0 } ^ { 1 } \frac { 5 x + 8 } { x ^ { 2 } + 3 x + 2 } d x$ is
(A) $\ln ( 8 )$
(B) $\ln \left( \frac { 27 } { 2 } \right)$
(C) $\ln ( 18 )$
(D) $\ln ( 288 )$
(E) divergent

Q21 Curve Sketching Asymptote Determination View

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

Q22 Sequences and Series Power Series Expansion and Radius of Convergence View

The power series $\sum _ { n = 0 } ^ { \infty } a _ { n } ( x - 3 ) ^ { n }$ converges at $x = 5$. Which of the following must be true?
(A) The series diverges at $x = 0$.
(B) The series diverges at $x = 1$.
(C) The series converges at $x = 1$.
(D) The series converges at $x = 2$.
(E) The series converges at $x = 6$.

Q23 Differential equations Applied Modeling with Differential Equations View

If $P ( t )$ is the size of a population at time $t$, which of the following differential equations describes linear growth in the size of the population?
(A) $\frac { d P } { d t } = 200$
(B) $\frac { d P } { d t } = 200 t$
(C) $\frac { d P } { d t } = 100 t ^ { 2 }$
(D) $\frac { d P } { d t } = 200 P$
(E) $\frac { d P } { d t } = 100 P ^ { 2 }$

Q24 Integration by Parts Multiple-Choice Primitive Identification View

Let $f$ be a differentiable function such that $\int f ( x ) \sin x \, d x = - f ( x ) \cos x + \int 4 x ^ { 3 } \cos x \, d x$. Which of the following could be $f ( x )$ ?
(A) $\cos x$
(B) $\sin x$
(C) $4 x ^ { 3 }$
(D) $- x ^ { 4 }$
(E) $x ^ { 4 }$

Q25 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

$\int _ { 1 } ^ { \infty } x e ^ { - x ^ { 2 } } d x$ is
(A) $- \frac { 1 } { e }$
(B) $\frac { 1 } { 2 e }$
(C) $\frac { 1 } { e }$
(D) $\frac { 2 } { e }$
(E) divergent

Q26 Polar coordinates View

What is the slope of the line tangent to the polar curve $r = 1 + 2 \sin \theta$ at $\theta = 0$ ?
(A) 2
(B) $\frac { 1 } { 2 }$
(C) 0
(D) $- \frac { 1 } { 2 }$
(E) - 2

Q27 Sequences and Series Convergence/Divergence Determination of Numerical Series View

For what values of $p$ will both series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 p } }$ and $\sum _ { n = 1 } ^ { \infty } \left( \frac { p } { 2 } \right) ^ { n }$ converge?
(A) $- 2 < p < 2$ only
(B) $- \frac { 1 } { 2 } < p < \frac { 1 } { 2 }$ only
(C) $\frac { 1 } { 2 } < p < 2$ only
(D) $p < \frac { 1 } { 2 }$ and $p > 2$
(E) There are no such values of $p$.

Q28 Indefinite & Definite Integrals Finding a Function from an Integral Equation View

Let $g$ be a continuously differentiable function with $g ( 1 ) = 6$ and $g ^ { \prime } ( 1 ) = 3$. What is $\lim _ { x \rightarrow 1 } \frac { \int _ { 1 } ^ { x } g ( t ) d t } { g ( x ) - 6 }$ ?
(A) 0
(B) $\frac { 1 } { 2 }$
(C) 1
(D) 2
(E) The limit does not exist.

Q76 Stationary points and optimisation Find critical points and classify extrema of a given function View

The function $f$, whose graph is shown above, is defined on the interval $- 2 \leq x \leq 2$. Which of the following statements about $f$ is false?
(A) $f$ is continuous at $x = 0$.
(B) $f$ is differentiable at $x = 0$.
(C) $f$ has a critical point at $x = 0$.
(D) $f$ has an absolute minimum at $x = 0$.
(E) The concavity of the graph of $f$ changes at $x = 0$.

Q77 Chain Rule Straightforward Polynomial or Basic Differentiation View

Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = x ^ { 4 }$. On what intervals is the rate of change of $f ( x )$ greater than the rate of change of $g ( x )$ ?
(A) $( 0.831, 7.384 )$ only
(B) $( - \infty , 0.831 )$ and $( 7.384 , \infty )$
(C) $( - \infty , - 0.816 )$ and $( 1.430, 8.613 )$
(D) $( - 0.816, 1.430 )$ and $( 8.613 , \infty )$
(E) $( - \infty , \infty )$

Q78 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

The graph of the piecewise linear function $f$ is shown above. What is the value of $\int _ { - 1 } ^ { 9 } ( 3 f ( x ) + 2 ) d x$ ?
(A) 7.5
(B) 9.5
(C) 27.5
(D) 47
(E) 48.5

Q79 Taylor series Construct Taylor/Maclaurin polynomial from derivative values View

Let $f$ be a function having derivatives of all orders for $x > 0$ such that $f ( 3 ) = 2 , f ^ { \prime } ( 3 ) = - 1 , f ^ { \prime \prime } ( 3 ) = 6$, and $f ^ { \prime \prime \prime } ( 3 ) = 12$. Which of the following is the third-degree Taylor polynomial for $f$ about $x = 3$ ?
(A) $2 - x + 6 x ^ { 2 } + 12 x ^ { 3 }$
(B) $2 - x + 3 x ^ { 2 } + 2 x ^ { 3 }$
(C) $2 - ( x - 3 ) + 6 ( x - 3 ) ^ { 2 } + 12 ( x - 3 ) ^ { 3 }$
(D) $2 - ( x - 3 ) + 3 ( x - 3 ) ^ { 2 } + 4 ( x - 3 ) ^ { 3 }$
(E) $2 - ( x - 3 ) + 3 ( x - 3 ) ^ { 2 } + 2 ( x - 3 ) ^ { 3 }$

Q80 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following statements must be true?
I. $f$ has a relative minimum at $x = - 3$.
II. The graph of $f$ has a point of inflection at $x = - 2$.
III. The graph of $f$ is concave down for $0 < x < 4$.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Q81 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

Let $f$ be a function that is twice differentiable on $- 2 < x < 2$ and satisfies the conditions in the table above. If $f ( x ) = f ( - x )$, what are the $x$-coordinates of the points of inflection of the graph of $f$ on $- 2 < x < 2$ ?

\cline{2-3} \multicolumn{1}{c\|}{}	$0 < x < 1$	$1 < x < 2$
$f ( x )$	Positive	Negative
$f ^ { \prime } ( x )$	Negative	Negative
$f ^ { \prime \prime } ( x )$	Negative	Positive

(A) $x = 0$ only
(B) $x = 1$ only
(C) $x = 0$ and $x = 1$
(D) $x = - 1$ and $x = 1$
(E) There are no points of inflection on $- 2 < x < 2$.

Q82 Indefinite & Definite Integrals Average Value of a Function View

What is the average value of $y = \sqrt { \cos x }$ on the interval $0 \leq x \leq \frac { \pi } { 2 }$ ?
(A) - 0.637
(B) 0.500
(C) 0.763
(D) 1.198
(E) 1.882

Q83 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View

If the function $f$ is continuous at $x = 3$, which of the following must be true?
(A) $f ( 3 ) < \lim _ { x \rightarrow 3 } f ( x )$
(B) $\lim _ { x \rightarrow 3 ^ { - } } f ( x ) \neq \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(C) $f ( 3 ) = \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = \lim _ { x \rightarrow 3 ^ { + } } f ( x )$
(D) The derivative of $f$ at $x = 3$ exists.
(E) The derivative of $f$ is positive for $x < 3$ and negative for $x > 3$.

Q84 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

For $- 1.5 < x < 1.5$, let $f$ be a function with first derivative given by $f ^ { \prime } ( x ) = e ^ { \left( x ^ { 4 } - 2 x ^ { 2 } + 1 \right) } - 2$. Which of the following are all intervals on which the graph of $f$ is concave down?
(A) (-0.418, 0.418) only
(B) $( - 1, 1 )$
(C) $( - 1.354 , - 0.409 )$ and $( 0.409, 1.354 )$
(D) $( - 1.5 , - 1 )$ and $( 0, 1 )$
(E) $( - 1.5 , - 1.354 ) , ( - 0.409, 0 )$, and $( 1.354, 1.5 )$

Q85 Connected Rates of Change Parametric or Curve-Based Particle Motion Rates View

The fuel consumption of a car, in miles per gallon (mpg), is modeled by $F ( s ) = 6 e ^ { \left( \frac { s } { 20 } - \frac { s ^ { 2 } } { 2400 } \right) }$, where $s$ is the speed of the car, in miles per hour. If the car is traveling at 50 miles per hour and its speed is changing at the rate of 20 miles/hour$^{2}$, what is the rate at which its fuel consumption is changing?
(A) 0.215 mpg per hour
(B) 4.299 mpg per hour
(C) 19.793 mpg per hour
(D) 25.793 mpg per hour
(E) 515.855 mpg per hour

Q86 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

If $f ^ { \prime } ( x ) > 0$ for all real numbers $x$ and $\int _ { 4 } ^ { 7 } f ( t ) d t = 0$, which of the following could be a table of values for the function $f$ ?
(A)

$x$	$f ( x )$
4	- 4
5	- 3
7	0

(B)

$x$	$f ( x )$
4	- 4
5	- 2
7	5

(C)

$x$	$f ( x )$
4	- 4
5	6
7	3

(D)

$x$	$f ( x )$
4	0
5	0
7	0

(E)

$x$	$f ( x )$
4	0
5	4
7	6

Q87 Volumes of Revolution Volume by Cross Sections with Known Geometry View

Let $R$ be the region in the first quadrant bounded above by the graph of $y = \ln ( 3 - x )$, for $0 \leq x \leq 2$. $R$ is the base of a solid for which each cross section perpendicular to the $x$-axis is a square. What is the volume of the solid?
(A) 0.442
(B) 1.029
(C) 1.296
(D) 3.233
(E) 4.071

Q88 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

The derivative of a function $f$ is increasing for $x < 0$ and decreasing for $x > 0$. Which of the following could be the graph of $f$ ?
(A), (B), (C), (D), (E) [graphs shown in figures above]

Q89 Variable acceleration (1D) Find displacement/position by integrating velocity View

A particle moves along a line so that its acceleration for $t \geq 0$ is given by $a ( t ) = \frac { t + 3 } { \sqrt { t ^ { 3 } + 1 } }$. If the particle's velocity at $t = 0$ is 5, what is the velocity of the particle at $t = 3$ ?
(A) 0.713
(B) 1.134
(C) 6.134
(D) 6.710
(E) 11.710

Q90 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View

If the series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges and $a _ { n } > 0$ for all $n$, which of the following must be true?
(A) $\lim _ { n \rightarrow \infty } \left| \frac { a _ { n + 1 } } { a _ { n } } \right| = 0$
(B) $\left| a _ { n } \right| < 1$ for all $n$
(C) $\sum _ { n = 1 } ^ { \infty } a _ { n } = 0$
(D) $\sum _ { n = 1 } ^ { \infty } n a _ { n }$ diverges.
(E) $\sum _ { n = 1 } ^ { \infty } \frac { a _ { n } } { n }$ converges.

Q91 Polar coordinates View

The figure above shows the graphs of the polar curves $r = 2 \cos ( 3 \theta )$ and $r = 2$. What is the sum of the areas of the shaded regions?
(A) 0.858
(B) 3.142
(C) 8.566
(D) 9.425
(E) 15.708

Q92 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

The function $h$ is differentiable, and for all values of $x$, $h ( x ) = h ( 2 - x )$. Which of the following statements must be true?
I. $\int _ { 0 } ^ { 2 } h ( x ) d x > 0$
II. $h ^ { \prime } ( 1 ) = 0$
III. $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 2 ) = 1$
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

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