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-bc_course-and-exam-description 24 -bc_sample-questions 18
-bc_sample-questions

18 maths questions

QFR1 Connected Rates of Change Volume/Height Related Rates for Containers and Solids View
The height of the water in a conical storage tank is modeled by a differentiable function $h$, where $h ( t )$ is measured in meters and $t$ is measured in hours. At time $t = 0$, the height of the water in the tank is 25 meters. The height is changing at the rate $h ^ { \prime } ( t ) = 2 - \frac { 24 e ^ { - 0.025 t } } { t + 4 }$ meters per hour for $0 \leq t \leq 24$.
(a) When the height of the water in the tank is $h$ meters, the volume of water is $V = \frac { 1 } { 3 } \pi h ^ { 3 }$. At what rate is the volume of water changing at time $t = 0$ ? Indicate units of measure.
(b) What is the minimum height of the water during the time period $0 \leq t \leq 24$ ? Justify your answer.
(c) The line tangent to the graph of $h$ at $t = 16$ is used to approximate the height of the water in the tank. Using the tangent line approximation, at what time $t$ does the height of the water return to 25 meters?
QFR2 Indefinite & Definite Integrals Accumulation Function Analysis View
The graph of a differentiable function $f$ is shown above for $- 3 \leq x \leq 3$. The graph of $f$ has horizontal tangent lines at $x = - 1 , x = 1$, and $x = 2$. The areas of regions $A , B , C$, and $D$ are 5, 4, 5, and 3, respectively. Let $g$ be the antiderivative of $f$ such that $g ( 3 ) = 7$.
(a) Find all values of $x$ on the open interval $- 3 < x < 3$ for which the function $g$ has a relative maximum. Justify your answer.
(b) On what open intervals contained in $- 3 < x < 3$ is the graph of $g$ concave up? Give a reason for your answer.
(c) Find the value of $\lim _ { x \rightarrow 0 } \frac { g ( x ) + 1 } { 2 x }$, or state that it does not exist. Show the work that leads to your answer.
(d) Let $h$ be the function defined by $h ( x ) = 3 f ( 2 x + 1 ) + 4$. Find the value of $\int _ { - 2 } ^ { 1 } h ( x ) d x$.
Q1 Chain Rule Limit Involving Derivative Definition of Composed Functions View
$\lim _ { x \rightarrow \pi } \frac { \cos x + \sin ( 2 x ) + 1 } { x ^ { 2 } - \pi ^ { 2 } }$ is
(A) $\frac { 1 } { 2 \pi }$
(B) $\frac { 1 } { \pi }$
(C) 1
(D) nonexistent
Q2 Curve Sketching Limit Computation from Algebraic Expressions View
$\lim _ { x \rightarrow \infty } \frac { \sqrt { 9 x ^ { 4 } + 1 } } { x ^ { 2 } - 3 x + 5 }$ is
(A) 1
(B) 3
(C) 9
(D) nonexistent
Q3 Curve Sketching Continuity and Differentiability of Special Functions View
The graph of the piecewise-defined function $f$ is shown in the figure above. The graph has a vertical tangent line at $x = - 2$ and horizontal tangent lines at $x = - 3$ and $x = - 1$. What are all values of $x , - 4 < x < 3$, at which $f$ is continuous but not differentiable?
(A) $x = 1$
(B) $x = - 2$ and $x = 0$
(C) $x = - 2$ and $x = 1$
(D) $x = 0$ and $x = 1$
Q4 Connected Rates of Change Reverse-Engineering a Geometric Quantity from Given Rates View
An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical shape. The volume of the sphere is decreasing at a constant rate of $2 \pi$ cubic meters per hour. At what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment when the radius is 5 meters? (Note: For a sphere of radius $r$, the surface area is $4 \pi r ^ { 2 }$ and the volume is $\frac { 4 } { 3 } \pi r ^ { 3 }$.)
(A) $\frac { 4 \pi } { 5 }$
(B) $40 \pi$
(C) $80 \pi ^ { 2 }$
(D) $100 \pi$
Q5 Differential equations Slope Field Identification View
Shown above is a slope field for which of the following differential equations?
(A) $\frac { d y } { d x } = x y + x$
(B) $\frac { d y } { d x } = x y + y$
(C) $\frac { d y } { d x } = y + 1$
(D) $\frac { d y } { d x } = ( x + 1 ) ^ { 2 }$
Q6 Differentiation from First Principles View
Let $f$ be the piecewise-linear function defined by $$f ( x ) = \begin{cases} 2 x - 2 & \text { for } x < 3 \\ 2 x - 4 & \text { for } x \geq 3 \end{cases}$$ Which of the following statements are true? I. $\lim _ { h \rightarrow 0 ^ { - } } \frac { f ( 3 + h ) - f ( 3 ) } { h } = 2$ II. $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( 3 + h ) - f ( 3 ) } { h } = 2$ III. $f ^ { \prime } ( 3 ) = 2$
(A) None
(B) II only
(C) I and II only
(D) I, II, and III
Q7 Chain Rule Chain Rule Combined with Fundamental Theorem of Calculus View
If $f ( x ) = \int _ { 1 } ^ { x ^ { 3 } } \frac { 1 } { 1 + \ln t } d t$ for $x \geq 1$, then $f ^ { \prime } ( 2 ) =$
(A) $\frac { 1 } { 1 + \ln 2 }$
(B) $\frac { 12 } { 1 + \ln 2 }$
(C) $\frac { 1 } { 1 + \ln 8 }$
(D) $\frac { 12 } { 1 + \ln 8 }$
Q8 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
Which of the following limits is equal to $\int _ { 3 } ^ { 5 } x ^ { 4 } d x$ ?
(A) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { k } { n } \right) ^ { 4 } \frac { 1 } { n }$
(B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { k } { n } \right) ^ { 4 } \frac { 2 } { n }$
(C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { 2 k } { n } \right) ^ { 4 } \frac { 1 } { n }$
(D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( 3 + \frac { 2 k } { n } \right) ^ { 4 } \frac { 2 } { n }$
Q9 Indefinite & Definite Integrals Average Value of a Function View
The function $f$ is continuous for $- 4 \leq x \leq 4$. The graph of $f$ shown above consists of five line segments. What is the average value of $f$ on the interval $- 4 \leq x \leq 4$ ?
(A) $\frac { 1 } { 8 }$
(B) $\frac { 3 } { 16 }$
(C) $\frac { 15 } { 16 }$
(D) $\frac { 3 } { 2 }$
Q10 Exponential Equations & Modelling Exponential Growth/Decay Modelling with Contextual Interpretation View
Let $y = f ( t )$ be a solution to the differential equation $\frac { d y } { d t } = k y$, where $k$ is a constant. Values of $f$ for selected values of $t$ are given in the table below:
$t$02
$f ( t )$412

Which of the following is an expression for $f ( t )$ ?
(A) $4 e ^ { \frac { t } { 2 } \ln 3 }$
(B) $e ^ { \frac { t } { 2 } \ln 9 } + 3$
(C) $2 t ^ { 2 } + 4$
(D) $4 t + 4$
Q11 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 could be the graph of $f$?
(A), (B), (C), (D) [graphs as shown in the exam]
Q12 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = e ^ { \sin x } - \cos x - 1$ for $0 < x < 9$. On what intervals is $f$ decreasing?
(A) $0 < x < 0.633$ and $4.115 < x < 6.916$
(B) $0 < x < 1.947$ and $5.744 < x < 8.230$
(C) $0.633 < x < 4.115$ and $6.916 < x < 9$
(D) $1.947 < x < 5.744$ and $8.230 < x < 9$
Q13 Connected Rates of Change Table-Based Estimation with Rate of Change Interpretation View
The temperature of a room, in degrees Fahrenheit, is modeled by $H$, a differentiable function of the number of minutes after the thermostat is adjusted. Of the following, which is the best interpretation of $H ^ { \prime } ( 5 ) = 2$ ?
(A) The temperature of the room is 2 degrees Fahrenheit, 5 minutes after the thermostat is adjusted.
(B) The temperature of the room increases by 2 degrees Fahrenheit during the first 5 minutes after the thermostat is adjusted.
(C) The temperature of the room is increasing at a constant rate of $\frac { 2 } { 5 }$ degree Fahrenheit per minute.
(D) The temperature of the room is increasing at a rate of 2 degrees Fahrenheit per minute, 5 minutes after the thermostat is adjusted.
Q14 Proof True/False Justification View
A function $f$ is continuous on the closed interval $[ 2,5 ]$ with $f ( 2 ) = 17$ and $f ( 5 ) = 17$. Which of the following additional conditions guarantees that there is a number $c$ in the open interval $( 2,5 )$ such that $f ^ { \prime } ( c ) = 0$ ?
(A) No additional conditions are necessary.
(B) $f$ has a relative extremum on the open interval $( 2,5 )$.
(C) $f$ is differentiable on the open interval $( 2,5 )$.
(D) $\int _ { 2 } ^ { 5 } f ( x ) d x$ exists.
Q15 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View
A rain barrel collects water off the roof of a house during three hours of heavy rainfall. The height of the water in the barrel increases at the rate of $r ( t ) = 4 t ^ { 3 } e ^ { - 1.5 t }$ feet per hour, where $t$ is the time in hours since the rain began. At time $t = 1$ hour, the height of the water is 0.75 foot. What is the height of the water in the barrel at time $t = 2$ hours?
(A) 1.361 ft
(B) 1.500 ft
(C) 1.672 ft
(D) 2.111 ft
Q16 Variable acceleration (1D) Compute total distance traveled over an interval View
A race car is traveling on a straight track at a velocity of 80 meters per second when the brakes are applied at time $t = 0$ seconds. From time $t = 0$ to the moment the race car stops, the acceleration of the race car is given by $a ( t ) = - 6 t ^ { 2 } - t$ meters per second per second. During this time period, how far does the race car travel?
(A) 188.229 m
(B) 198.766 m
(C) 260.042 m
(D) 267.089 m