Calc BC 2018 Free-Response Questions

AP Calculus BC 2018 Free-Response Questions

Section II, Part A

• Time allocated: 30 minutes

• Number of questions: 2

• Calculator: A graphing calculator is required for these questions.

Question 1: Escalator Line Model

• Function Modeling Rate of People Entry:

  • The rate function is defined as:

  
  r(t) = \begin{cases}
  \frac{t}{3} +4 \quad & \text{for } 0 \leq t \leq 300\ \
  \frac{100 - t}{300} & \text{for } t > 300 \ \
  0 & \text{otherwise}
  \end{cases}
  

  • Variables:

  • $r(t)$ is measured in people per second.

  • $t$ is measured in seconds.

• Constant Exit Rate: People exit the line at a constant rate of 0.7 person per second.

• Initial Condition: There are 20 people in line at time $t = 0$.

Parts of Question 1:

1. (a) Total People Entering Line (0 ≤ t ≤ 300):

   • To find the number of people entering the line over this interval, integrate the rate function:

     $<br>\text{Total Entry} = \int_0^{300} r(t) \ dt<br>$

2. (b) People in Line at t = 300:

   • At time $t = 300$, calculate the number of people still in line:

     $<br>N(t) = N(0) + \int_0^{300} r(t) \ dt - 0.7 \cdot 300<br>$

   • Where $N(0) = 20$ is the initial count.

3. (c) When is the Line Empty (t > 300):

   • Set the equation for the line to zero, solve for $t$:

     $<br>N(t) = N(300) - 0.7(t - 300) = 0<br>$

4. (d) Minimum Line Count:

   • Find the minimum number of people in line while $0 ≤ t ≤ 300$:

   • Justification involves evaluating critical points where $N'(t) = 0$.

   • Minimize how many people are in line by considering both entry and exit rates.

Question 2: Plankton Cell Investigation

• Context: Researchers are studying plankton cells at various underwater depths.

• Density Function:

  • Density of plankton cells given by:

  
  p(h) = \begin{cases}
  0.2h e^{-0.0025h} \quad & \text{for } 0 \leq h \leq 30 \ \
  f(h) \quad & \text{for } h \geq 30
  \end{cases}
  

Parts of Question 2:

1. (a) Rate of Change at Depth (h = 25):

   • Compute the derivative $p'(25)$:

   • Interpret the result in terms of plankton density and depth.

2. (b) Total Cells in Column (h = 0 to 30):

   • Volume of column: 3 square meters constant area.

   • Total number of plankton cells:

     $<br>\text{Total Cells} = \int_0^{30} p(h) \cdot 3 \, dh<br>$

   • Round to the nearest million cells.

3. (c) Expression for Cells in Column (h ≥ 30):

   • Expression to calculate total cells in depth $K$, where K > 30:

     $<br>N = \int<em>{30}^{K} f(h) \, dh + \int</em>0^{30} p(h) \, dh<br>$

   • Justification: Explain why this value is ≤ 2000 million due to density function constraints.

4. (d) Boat Movement:

   • Position of the boat given by:

     $<br>(x(t), y(t)) \text{ where } x'(t) = 662 \sin(5t), \ y'(t) = 880 \cos(6t)<br>$

   • Calculate total distance traveled in the interval $0 \leq t \leq 1$:

     $<br>D = \int_0^{1} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt<br>$

Section II, Part B

• Time allocated: 1 hour

• Number of questions: 4

• Calculator: No calculator allowed for these questions.

Question 3: Function g and its Derivative

• Continuous function: g defined as piecewise linear for -5 \leq x < 3 and $g(x) = -4x^2 + 2$ for $3 \leq x \leq 6$.

Parts of Question 3:

1. (a) Value of f(-5):

   • Given that $f(1)=3$, find $f(-5)$ by integrating $g(x)$.

2. (b) Evaluate Integral from 1 to 6:

   • Compute:

     $<br>\int_1^{6} g(x) \, dx<br>$

3. (c) Increase and Concavity:

   • Identify intervals where $f$ is both increasing and concave up.

   • Reasoning must include the behavior of $g$ and its derivative $f'$.

4. (d) Points of Inflection:

   • To find points of inflection, set the second derivative of $f$ to zero and identify intervals where concavity changes.

Question 4: Tree Height Model

• Function H(t): Height of a tree modeled as a twice-differentiable function.

Parts of Question 4:

1. (a) Estimate H'(6):

   • Use data from a table to estimate and interpret $H'(6)$ with appropriate units.

2. (b) Intermediate Value Theorem:

   • Justify the existence of a time $t$ such that $H'(t) = 2$ for 2 < t < 10 using continuity.

3. (c) Average Height:

   • Approximate the average height using the trapezoidal sum based on table values:

     $<br>\text{Average Height} = \frac{1}{b-a} \int_a^{b} H(t) \, dt<br>$

4. (d) Model G: Height $G(x)$ related to diameter:

   • Given by $G(x) = \frac{100x}{1 + x}$ with $x$ measured in meters.

   • Find the rate of change of height with respect to time when $H = 50$ meters.

Question 5: Polar Curves

• Polar Curves: Given curves $r = 4$ and $r = 2 + 3 ext{cos}(\theta)$ intersecting points $\theta = \frac{\pi}{3}$ and $\theta = \frac{5\pi}{3}$.

Parts of Question 5:

1. (a) Area of the Region R:

   • Write area expression for region R:

     $<br>A = \int<em>{\theta</em>1}^{\theta<em>2} \left(\frac{1}{2}(r</em>1^2 - r_2^2)\right) d\theta<br>$

2. (b) Slope of Tangent Line:

   • Find the slope of the tangent line for $r(\theta)$ at intersection points.

3. (c) Particle Movement:

   • Derive the rate of change of angle $\theta$ with respect to time at a specific point on the curve, given the constant increase in distance from the origin:

     $\frac{d\theta}{dt}$ depending on angular position and radial distance rate.

Conclusion

• This study guide encapsulates key concepts and detailed calculations from the 2018 AP Calculus BC Free-Response Questions. Each section builds a comprehensive understanding necessary for successfully tackling similar problems in exams.