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Practice Test 11 - AP Precalculus

20 questions. Answer them in any order, then submit to see your score and every explanation.

  1. A launcher uses r(t)=⟨(vcos⁡a)t, (vsin⁡a)t−4.9t2⟩\mathbf{r}(t)=\langle (v\cos a)t,\,(v\sin a)t-4.9t^2\rangle; which vector is the velocity v(t)\mathbf{v}(t)?
  2. Let f(x)=2x2+1f(x) = 2x^2 + 1. The rate of change at x=1x=1 is to be approximated by the average rate of change on the interval [1,1.1][1, 1.1]. What is this approximation?
  3. An Earth-observing satellite's distance from a tracking station is approximated in polar coordinates by r(θ)=20+5sin⁡(2θ)r(\theta)=20+5\sin(2\theta), where rr (thousands of km) is distance from the station and θ\theta (radians) is the satellite's angular position. The derivative drdθ\dfrac{dr}{d\theta} measures how the satellite's distance changes per radian of orbital angle. Because the sine input is 2θ2\theta, the chain rule is required: ddθ[sin⁡(2θ)]=2cos⁡(2θ)\dfrac{d}{d\theta}[\sin(2\theta)]=2\cos(2\theta). Interpret the meaning of the rate of change of r(θ)=20+5sin⁡(2θ)r(\theta)=20+5\sin(2\theta) at θ=0\theta=0 in this satellite-tracking context.​
  4. A Ferris wheel seat height follows h(t)=18cos⁡(π20t)+20h(t)=18\cos\left(\frac{\pi}{20}t\right)+20, with tt in minutes. The wheel rotates steadily, starting at maximum height at t=0t=0. The amplitude is the radius, and the vertical shift is the axle height. Based on the scenario, what is the period of the sinusoidal function described?​
  5. A Ferris wheel is modeled by x(t)=10cos⁡(π8t)x(t)=10\cos\left(\frac{\pi}{8}t\right), y(t)=10+10sin⁡(π8t)y(t)=10+10\sin\left(\frac{\pi}{8}t\right) (meters). Based on these, calculate the speed at t=4t=4 s. Give your answer in simplest form.
  6. Let MM be a 2×22 \times 2 matrix. Which of the following statements is NOT always true?
  7. A surveyor records a point at (x,y)=(−3, 33)(x,y)=(-3,\,3\sqrt{3}) meters relative to a marker. Convert this location to polar form (r,θ)(r,\theta) with θ\theta in radians.
  8. An account follows A(t)=2000(1.05)tA(t)=2000(1.05)^t. Determine the least integer tt such that A(t)≥3000A(t)\ge 3000 using logarithms.​
  9. In Phoenix, Arizona, average monthly temperature varies seasonally and is well-approximated by a sinusoid. The average temperatures (F) for selected months are: Jan 57, Mar 65, May 84, Jul 95, Sep 88, Nov 67. The annual mean is about 76F (vertical shift), and the typical peak-to-trough change is about 38F, giving amplitude about 19F. The period is 12 months, and the hottest month is July, so the phase shift should place a maximum near month 7. Using the information given in the scenario, what period should be used to fit the data accurately?
  10. Which of the following describes the radian measure of an angle in standard position?
  11. If v⃗=[3−2]\vec{v} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} and the linear transformation matrix is A=[21−13]A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}, what is Av⃗A\vec{v}?
  12. If a 2×22 \times 2 matrix AA has a determinant of 4, which of the following statements must be true?
  13. Using the polynomial provided, consider the quadratic r(x)=x2+8x+20r(x)=x^2+8x+20, a degree 22 polynomial. Compute b2−4ac=82−4(1)(20)=64−80=−16b^2-4ac=8^2-4(1)(20)=64-80=-16, so x=−8±−162=−8±4i2=−4±2i.x=\frac{-8\pm\sqrt{-16}}{2}=\frac{-8\pm 4i}{2}=-4\pm 2i. These nonreal zeros do not correspond to real xx-intercepts, yet they satisfy r(x)=0r(x)=0 in the complex system. What are the complex zeros of the polynomial function r(x)r(x)?
  14. Money grows continuously: A(t)=1200e0.045tA(t)=1200e^{0.045t}. How long until the balance reaches 20002000 dollars?
  15. A rational function is a quotient of two polynomials. For f(x)=x2−4x−2f(x)=\dfrac{x^2-4}{x-2}, zeros satisfy f(x)=0f(x)=0, and vertical asymptotes typically occur where the denominator is zero. Since x2−4=(x−2)(x+2)x^2-4=(x-2)(x+2), the factor (x−2)(x-2) cancels, leaving a removable discontinuity at x=2x=2 (a hole), not a vertical asymptote. Based on the function described, determine the vertical asymptotes of f(x)=x2−4x−2f(x)=\dfrac{x^2-4}{x-2}.
  16. Consider the function s(x)=x2+x−6x−2s(x)=\dfrac{x^2+x-6}{x-2} for a demand curve; at which xx-value is ss undefined due to a removable discontinuity?
  17. Using the matrix provided, what does the matrix A=(0−1−10)\mathbf{A}=\begin{pmatrix}0&-1\\-1&0\end{pmatrix} represent in the context of the transformation?
  18. A city's population is modeled linearly by P(t)=120000+2500tP(t)=120000+2500t, where t is years since 2025, used for housing projections. Based on the scenario, how does the rate of change in population compare at the beginning and end of the time period?​
  19. An air-traffic radar observes a plane whose distance from the radar varies with bearing according to r(θ)=25+8sin⁡(θ)r(\theta)=25+8\sin(\theta), where rr (km) is distance and θ\theta (radians) is the bearing angle. In polar modeling, drdθ\dfrac{dr}{d\theta} is the instantaneous rate at which the plane's distance changes per radian as the bearing increases. Using ddθ[sin⁡θ]=cos⁡θ\dfrac{d}{d\theta}[\sin\theta]=\cos\theta, compute the derivative and evaluate it at the specified angle. Given r(θ)=25+8sin⁡(θ)r(\theta)=25+8\sin(\theta), what is the rate of change of rr with respect to θ\theta at θ=π4\theta=\dfrac{\pi}{4}?​
  20. Which of the following statements correctly describes a characteristic of the function f(x)=−3sin⁡(12x+π2)−1f(x) = -3\sin(\frac{1}{2}x + \frac{\pi}{2}) - 1?

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