ap-calculus-ab

QFR1 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 $t = 25$ ?

QFR2 Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View

Let $R$ be the region in the first quadrant bounded by the $x$-axis and the graphs of $y = \ln x$ and $y = 5 - x$, as shown in the figure above.
(a) Find the area of $R$.
(b) Region $R$ is the base of a solid. For the solid, each cross section perpendicular to the $x$-axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.
(c) The horizontal line $y = k$ divides $R$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.

QFR3 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.

QFR4 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View

The function $f$ is defined by $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ for $- 5 \leq x \leq 5$.
(a) Find $f ^ { \prime } ( x )$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = - 3$.
(c) Let $g$ be the function defined by $g ( x ) = \left\{ \begin{array} { l } f ( x ) \text { for } - 5 \leq x \leq - 3 \\ x + 7 \text { for } - 3 < x \leq 5 . \end{array} \right.$
Is $g$ continuous at $x = - 3$ ? Use the definition of continuity to explain your answer.
(d) Find the value of $\int _ { 0 } ^ { 5 } x \sqrt { 25 - x ^ { 2 } } d x$.

QFR5 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 $B ( 0 ) = 20$.

QFR6 Variable acceleration (1D) Multi-part particle motion analysis (formula-based velocity) View

For $0 \leq t \leq 12$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t ) = \cos \left( \frac { \pi } { 6 } t \right)$. The particle is at position $x = - 2$ at time $t = 0$.
(a) For $0 \leq t \leq 12$, when is the particle moving to the left?
(b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time $t = 0$ to time $t = 6$.
(c) Find the acceleration of the particle at time $t$. Is the speed of the particle increasing, decreasing, or neither at time $t = 4$ ? Explain your reasoning.
(d) Find the position of the particle at time $t = 4$.

Q1 Chain Rule Straightforward Polynomial or Basic Differentiation View

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

Q2 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $f$ be the function given by $f ( x ) = 300 x - x ^ { 3 }$. On which of the following intervals is the function $f$ increasing?
(A) $( - \infty , - 10 ]$ and $[ 10 , \infty )$
(B) $[ - 10,10 ]$
(C) $[ 0,10 ]$ only
(D) $[ 0,10 \sqrt { 3 } ]$ only
(E) $[ 0 , \infty )$

Q3 Standard Integrals and Reverse Chain Rule Standard Antiderivative Identification (MCQ) View

$\quad \int \sec x \tan x \, d x =$
(A) $\sec x + C$
(B) $\tan x + C$
(C) $\frac { \sec ^ { 2 } x } { 2 } + C$
(D) $\frac { \tan ^ { 2 } x } { 2 } + C$
(E) $\frac { \sec ^ { 2 } x \tan ^ { 2 } x } { 2 } + C$

Q4 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View

If $f ( x ) = 7 x - 3 + \ln x$, then $f ^ { \prime } ( 1 ) =$
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Q6 Variable acceleration (1D) Compute total distance traveled over an interval View

A particle moves along the $x$-axis. The velocity of the particle at time $t$ is $6 t - t ^ { 2 }$. What is the total distance traveled by the particle from time $t = 0$ to $t = 3$ ?
(A) 3
(B) 6
(C) 9
(D) 18
(E) 27

Q7 Chain Rule Chain Rule with Composition of Explicit Functions View

If $y = \left( x ^ { 3 } - \cos x \right) ^ { 5 }$, then $y ^ { \prime } =$
(A) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 }$
(B) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 }$
(C) $5 \left( 3 x ^ { 2 } + \sin x \right)$
(D) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 } \cdot ( 6 x + \cos x )$
(E) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 } \cdot \left( 3 x ^ { 2 } + \sin x \right)$

Q8 Numerical integration Tabular Data Multi-Part (Derivative, Integral Approximation, and Interpretation) 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.

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

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

Q9 Composite & Inverse Functions Find or Apply an Inverse Function Formula View

Let $f$ be the function defined above. $$f ( x ) = \begin{cases} \frac { ( 2 x + 1 ) ( x - 2 ) } { x - 2 } & \text { for } x \neq 2 \\ k & \text { for } x = 2 \end{cases}$$ For what value of $k$ is $f$ continuous at $x = 2$ ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5

Q10 Areas by integration View

What is the area of the region in the first quadrant bounded by the graph of $y = e ^ { x / 2 }$ and the line $x = 2$ ?
(A) $2 e - 2$
(B) $2 e$
(C) $\frac { e } { 2 } - 1$
(D) $\frac { e - 1 } { 2 }$
(E) $e - 1$

Q11 Curve Sketching Continuity and Differentiability of Special Functions 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 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View

Using the substitution $u = \sqrt { x }$, $\int _ { 1 } ^ { 4 } \frac { e ^ { \sqrt { x } } } { \sqrt { x } } d x$ is equal to which of the following?
(A) $2 \int _ { 1 } ^ { 16 } e ^ { u } d u$
(B) $2 \int _ { 1 } ^ { 4 } e ^ { u } d u$
(C) $2 \int _ { 1 } ^ { 2 } e ^ { u } d u$
(D) $\frac { 1 } { 2 } \int _ { 1 } ^ { 2 } e ^ { u } d u$
(E) $\int _ { 1 } ^ { 4 } e ^ { u } d u$

Q13 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

The function $f$ is defined by $f ( x ) = \left\{ \begin{array} { l l } 2 & \text { for } x < 3 \\ x - 1 & \text { for } x \geq 3 \end{array} \right.$. What is the value of $\int _ { 1 } ^ { 5 } f ( x ) d x$ ?
(A) 2
(B) 6
(C) 8
(D) 10
(E) 12

Q14 Chain Rule Chain Rule with Composition of Explicit Functions View

If $f ( x ) = \sqrt { x ^ { 2 } - 4 }$ and $g ( x ) = 3 x - 2$, then the derivative of $f ( g ( x ) )$ at $x = 3$ is
(A) $\frac { 7 } { \sqrt { 5 } }$
(B) $\frac { 14 } { \sqrt { 5 } }$
(C) $\frac { 18 } { \sqrt { 5 } }$
(D) $\frac { 15 } { \sqrt { 21 } }$
(E) $\frac { 30 } { \sqrt { 21 } }$

Q15 Stationary points and optimisation Analyze function behavior from graph or table of derivative 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 Applied differentiation Kinematics via differentiation View

A particle moves along the $x$-axis with its position at time $t$ given by $x ( t ) = ( t - a ) ( t - b )$, where $a$ and $b$ are constants and $a \neq b$. For which of the following values of $t$ is the particle at rest?
(A) $t = a b$
(B) $t = \frac { a + b } { 2 }$
(C) $t = a + b$
(D) $t = 2 ( a + b )$
(E) $t = a$ and $t = b$

Q17 Curve Sketching Sketching a Curve from Analytical Properties View

The figure above shows the graph of $f$. If $f ( x ) = \int _ { 2 } ^ { x } g ( t ) d t$, which of the following could be the graph of $y = g ( x )$ ?
(A) [graph A]
(B) [graph B]
(C) [graph C]
(D) [graph D]
(E) [graph E]

Q18 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

$\lim _ { h \rightarrow 0 } \frac { \ln ( 4 + h ) - \ln ( 4 ) } { h }$ is
(A) 0
(B) $\frac { 1 } { 4 }$
(C) 1
(D) $e$
(E) nonexistent

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 Composite & Inverse Functions Derivative of an Inverse Function View

Let $f ( x ) = ( 2 x + 1 ) ^ { 3 }$ and let $g$ be the inverse function of $f$. Given that $f ( 0 ) = 1$, what is the value of $g ^ { \prime } ( 1 )$ ?
(A) $- \frac { 2 } { 27 }$
(B) $\frac { 1 } { 54 }$
(C) $\frac { 1 } { 27 }$
(D) $\frac { 1 } { 6 }$
(E) 6

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 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

Let $f$ be the function defined by $f ( x ) = \frac { \ln x } { x }$. What is the absolute maximum value of $f$ ?
(A) 1
(B) $\frac { 1 } { e }$
(C) 0
(D) $- e$
(E) $f$ does not have an absolute maximum value.

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 Stationary points and optimisation Determine parameters from given extremum conditions View

Let $g$ be the function given by $g ( x ) = x ^ { 2 } e ^ { k x }$, where $k$ is a constant. For what value of $k$ does $g$ have a critical point at $x = \frac { 2 } { 3 }$ ?
(A) $-3$
(B) $- \frac { 3 } { 2 }$
(C) $- \frac { 1 } { 3 }$
(D) 0
(E) There is no such $k$.

Q25 Differential equations Solving Separable DEs with Initial Conditions View

Which of the following is the solution to the differential equation $\frac { d y } { d x } = 2 \sin x$ with the initial condition $y ( \pi ) = 1$ ?
(A) $y = 2 \cos x + 3$
(B) $y = 2 \cos x - 1$
(C) $y = - 2 \cos x + 3$
(D) $y = - 2 \cos x + 1$
(E) $y = - 2 \cos x - 1$

Q26 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

Let $g$ be a function with first derivative given by $g ^ { \prime } ( x ) = \int _ { 0 } ^ { x } e ^ { - t ^ { 2 } } d t$. Which of the following must be true on the interval $0 < x < 2$ ?
(A) $g$ is increasing, and the graph of $g$ is concave up.
(B) $g$ is increasing, and the graph of $g$ is concave down.
(C) $g$ is decreasing, and the graph of $g$ is concave up.
(D) $g$ is decreasing, and the graph of $g$ is concave down.
(E) $g$ is decreasing, and the graph of $g$ has a point of inflection on $0 < x < 2$.

Q27 Connected Rates of Change Parametric or Curve-Based Particle Motion Rates View

If $( x + 2 y ) \cdot \frac { d y } { d x } = 2 x - y$, what is the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 3,0 )$ ?
(A) $- \frac { 10 } { 3 }$
(B) 0
(C) 2
(D) $\frac { 10 } { 3 }$
(E) Undefined

Q28 Applied differentiation Kinematics via differentiation View

For $t \geq 0$, the position of a particle moving along the $x$-axis is given by $x ( t ) = \sin t - \cos t$. What is the acceleration of the particle at the point where the velocity is first equal to 0 ?
(A) $- \sqrt { 2 }$
(B) $-1$
(C) 0
(D) 1
(E) $\sqrt { 2 }$

Q76 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

The graph of the function $f$ is shown in the figure above. For which of the following values of $x$ is $f ^ { \prime } ( x )$ positive and increasing?
(A) $a$
(B) $b$
(C) $c$
(D) $d$
(E) $e$

Q77 Curve Sketching Number of Solutions / Roots via Curve Analysis View

Let $f$ be a function that is continuous on the closed interval $[ 2,4 ]$ with $f ( 2 ) = 10$ and $f ( 4 ) = 20$. Which of the following is guaranteed by the Intermediate Value Theorem?
(A) $f ( x ) = 13$ has at least one solution in the open interval $( 2,4 )$.
(B) $f ( 3 ) = 15$
(C) $f$ attains a maximum on the open interval $( 2,4 )$.
(D) $f ^ { \prime } ( x ) = 5$ has at least one solution in the open interval $( 2,4 )$.
(E) $f ^ { \prime } ( x ) > 0$ for all $x$ in the open interval $( 2,4 )$.

Q78 Tangents, normals and gradients Find tangent line equation at a given point View

The graph of $y = e ^ { \tan x } - 2$ crosses the $x$-axis at one point in the interval $[ 0,1 ]$. What is the slope of the graph at this point?
(A) 0.606
(B) 2
(C) 2.242
(D) 2.961
(E) 3.747

Q79 Variable acceleration (1D) Find displacement/position by integrating velocity View

A particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t )$, and the acceleration of the particle at time $t$ is given by $a ( t )$. Which of the following gives the average velocity of the particle from time $t = 0$ to time $t = 8$ ?
(A) $\frac { a ( 8 ) - a ( 0 ) } { 8 }$
(B) $\frac { 1 } { 8 } \int _ { 0 } ^ { 8 } v ( t ) d t$
(C) $\frac { 1 } { 8 } \int _ { 0 } ^ { 8 } | v ( t ) | d t$
(D) $\frac { 1 } { 2 } \int _ { 0 } ^ { 8 } v ( t ) d t$
(E) $\frac { v ( 0 ) + v ( 8 ) } { 2 }$

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

Water is pumped into a tank at a rate of $r ( t ) = 30 \left( 1 - e ^ { - 0.16 t } \right)$ gallons per minute, where $t$ is the number of minutes since the pump was turned on. If the tank contained 800 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 20 minutes?
(A) 380 gallons
(B) 420 gallons
(C) 829 gallons
(D) 1220 gallons
(E) 1376 gallons

Q82 Stationary points and optimisation Find critical points and classify extrema of a given function View

If $f ^ { \prime } ( x ) = \sqrt { x ^ { 4 } + 1 } + x ^ { 3 } - 3 x$, then $f$ has a local maximum at $x =$
(A) $-2.314$
(B) $-1.332$
(C) $0.350$
(D) $0.829$
(E) $1.234$

Q83 Variable acceleration (1D) Compute total distance traveled over an interval View

The graph above gives the velocity, $v$, in ft/sec, of a car for $0 \leq t \leq 8$, where $t$ is the time in seconds. Of the following, which is the best estimate of the distance traveled by the car from $t = 0$ until the car comes to a complete stop?
(A) 21 ft
(B) 26 ft
(C) 180 ft
(D) 210 ft
(E) 260 ft

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

The graph of $f ^ { \prime }$, the derivative of $f$, is shown in the figure above. The function $f$ has a local maximum at $x =$
(A) $-3$
(B) $-1$
(C) 1
(D) 3
(E) 4

Q86 Exponential Functions True/False or Multiple-Statement Verification 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 Stationary points and optimisation Find concavity, inflection points, or second derivative properties View

The graph of $f ^ { \prime \prime }$, the second derivative of $f$, is shown above for $- 2 \leq x \leq 4$. What are all intervals on which the graph of the function $f$ is concave down?
(A) $-1 < x < 1$
(B) $0 < x < 2$
(C) $1 < x < 3$ only
(D) $-2 < x < -1$ only
(E) $-2 < x < -1$ and $1 < x < 3$

Q88 Connected Rates of Change Shadow Rate of Change Problem View

A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person's shadow is lengthening?
(A) $1.5 \text{ ft/sec}$
(B) $2.667 \text{ ft/sec}$
(C) $3.75 \text{ ft/sec}$
(D) $6 \text{ ft/sec}$
(E) $10 \text{ ft/sec}$

Q89 Applied differentiation Kinematics via differentiation 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 Integration by Substitution Substitution to Prove an Integral Identity or Equality View

Let $f$ be a function such that $\int _ { 6 } ^ { 12 } f ( 2 x ) d x = 10$. Which of the following must be true?
(A) $\int _ { 12 } ^ { 24 } f ( t ) d t = 5$
(B) $\int _ { 12 } ^ { 24 } f ( t ) d t = 20$
(C) $\int _ { 6 } ^ { 12 } f ( t ) d t = 5$
(D) $\int _ { 6 } ^ { 12 } f ( t ) d t = 20$
(E) $\int _ { 3 } ^ { 6 } f ( t ) d t = 5$

Q91 Stationary points and optimisation Analyze function behavior from graph or table of derivative View

Let $f$ be a polynomial function with values of $f ^ { \prime } ( x )$ at selected values of $x$ given in the table above.

$x$	$-2$	0	3	5	6
$f ^ { \prime } ( x )$	3	1	4	7	5

Which of the following must be true for $-2 < x < 6$ ?
(A) The graph of $f$ is concave up.
(B) The graph of $f$ has at least two points of inflection.
(C) $f$ is increasing.
(D) $f$ has no critical points.
(E) $f$ has at least two relative extrema.

Q92 Volumes of Revolution Volume by Cross Sections with Known Geometry View

Let $R$ be the region in the first quadrant bounded below by the graph of $y = x ^ { 2 }$ and above by the graph of $y = \sqrt { x }$. $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. What is the volume of the solid?
(A) 0.129
(B) 0.300
(C) 0.333
(D) 0.700
(E) 1.271

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