MATH 170

Final - Practice 1 | General

1. At 9:00 pm Bob leaves his home and starts walking to the bus stop at a constant speed of $9.3$ feet per second. Bob's house is $4320$ feet from the bus stop.
a. Define variables for the quantities that are changing (BE SPECIFIC).

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b. What does it mean to say Bob walks at a constant speed of $9.3$ feet per second?

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c. Consider the formula $d = 4320 - 9.3t$.
i. What does $t$ represent?

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ii. What does $9.3t$ represent?

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iii. What does $d$ represent?

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2. A rock is dropped into the middle of a lake creating a circular ripple. The radius of the circle increases at a constant rate of $5$ cm each second. Define a function $g$ that determines the radius of the circular ripple after $t$ seconds have elapsed since the rock hit the water.

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Find $(g \circ f)(x)$, where $f(x) = e^{2x} - 1$, and $g(x) = \ln (x + 1)$.

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3. Let $d$ be the distance of a car, measured in feet, from mile marker $120$, and let $t$ be the number of seconds that have elapsed since the vehicle has passed mile marker $120$. The distance $d$ is a function of the time $t$ and is represented by the function $f$, defined $d = f(t)$ where $f(t) = t^2 + t$.
a. As the number of seconds elapsed increases from $2$ to $3.5$ seconds, what is the change in the car's distance from mile marker $120$?

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b. Determine the average speed of the car over the time interval of $1$ second to $4$ seconds.

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c. Explain the meaning of average speed in this context.

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4) $f(x) = x^3 - 10$

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4. Consider the function that defines the revenue in dollars obtained by selling $x$ units, $R(x)$, and another function that defines the cost in dollars, $C(x)$, obtained by producing $x$ units.
a. Explain the meaning of $R(120)$.

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b. Explain the meaning of $R(120 + h) - R(120)$

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c. Explain the meaning of $R(120) - C(120)$.

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d. Explain the meaning of $R^{-1}(60000)$

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5. Consider the function $f(x) = \frac{x^2 - 4}{3x^2 - 10x - 13}$.
a. What is the domain of this function?

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b. What is the vertical intercept?

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c. What are the zeros of function $f$?

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d. What is the horizontal asymptote?

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e. What are the vertical asymptote(s)?

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f. Sketch a graph of this function.

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6. It takes $67$ maple trees to make $30$ gallons of maple syrup each year.
a. How many maple trees will it take to make $217$ gallons of maple syrup? Explain your reasoning.

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b. How much maple syrup can you get from $8$ maple trees? Explain your reasoning.

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3) $e^{2x} - 7e^x - 18 = 0$

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7. Determine the domain of the following functions.
a. $f(t) = \sqrt{t + 6}$

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b. $g(n) = \frac{1}{n^2 - 9}$

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c. $h(p) = \frac{\sqrt{p + 9}}{p - 7}$

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8. Consider the graph of the function $g$.


a. What are the roots of $g$?

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b. Determine the $y$-intercept.

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c. On what interval(s) of $x$ is the rate of change of $y$ with respect to $x$ decreasing?

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d. On what interval(s) of $x$ is the rate of change of $y$ with respect to $x$ increasing?

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e. On what interval(s) of $x$ is the function decreasing?

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f. On what interval(s) of $x$ is the fucntion increasing?

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g. $\lim_{x \to \infty} g(x)$

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$\lim_{x \to -\infty} g(x)$

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9. Suppose the following graph of the function $f$ represents the number of dollars in Meg's bank account as a function of the number of days, $t$, since January 1, 2012.


a. Evaluate $f(30)$. What does this value represent in the context of the problem?

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b. What is the meaning of $f(60) = 1400$ in this context?

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c. Determine whether the following statements are true or false.
i. As the number of days since January 1st increases from $60$ to $70$ days, for equal changes in the number of days elapsed the change in the number of dollars in Meg's bank account is decreasing.

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ii. $80$ days after January 1st, the amount of money in Meg's bank account increases at a constant rate of change.

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iii. As the number of days since January 1st increases from $0$ to $20$ days, the amount of money in Meg's bank account is increasing and the rate of change of the amount of money with respect to the number of days elapsed decreases.

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10. Every two hours, the amount of caffeine in the body decreases by $18%$. Suppose you start with $180$ mg of caffeine.
a. Define an exponential function $C = f(t)$ to determine the amount of caffeine in the body after $t$ hours.

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b. After $3.8$ hours, how much caffeine remains in the body? Show your work.

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c. By what percentage does the amount of caffeine decrease every half-hour? Explain your reasoning.

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11. The number of asthma sufferers in the world was about $62$ million in $1975$ and $82$ million in $1995$. Let $N$ represent the number of asthma sufferers (in millions) worldwide, $t$ years after $1975$.
a. What is the $20$-year growth factor?

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b. By what percentage did the number of asthma sufferers increase over the $20$-year period of time?

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c. What is the annual growth factor?

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d. Define an exponential function that determines the number of asthma sufferers $N$ in terms of the number of years $t$ that have passes since $1975$.

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e. Use the function you created in part (d) to determine the number of asthma sufferers in the year $2010$.

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12. Solve each of the following for $x$.
a. $\log_5(30x^2) = 3$

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b. $\log(1.5x) = 0$

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c. $\log_3(x) + \log_3(2x) = 3$

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4) Suppose $y = -3\cos(2x + \pi)$; find the amplitude, period and phase shift.

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13. Find the formula for a parabola (a quadratic function) with horizontal intercepts (roots) at $x = -6$ and $x = 2$ and passes through the point $(0, 5)$.

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2) $\sec(\alpha)$

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3) $\tan(\alpha)$

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4) $\cot(\alpha)$

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5) $\csc(\alpha)$

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6) $\cos(2\alpha)$

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7) $\sin(2\alpha)$

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8) $\tan(2\alpha)$

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9) $\cos \left( \frac{\alpha}{2} \right)$

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10) $\sin \left( \frac{\alpha}{2} \right)$

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11) $\sin(\alpha - \beta)$

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12) $\cos(\alpha + \beta)$

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14. A quadratic function has a vertex of $(1, 3)$ and a root at $x = 4.5$. What is the other root of this quadratic function?

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2) Find an algebraic expression for the $\cos(\tan^{-1}(x))$, where $x$ is positive and in the domian of the given inverse function.

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15. A race-car on is driving counter-clockwise on circular track in which the starting position is the $3$ o'clock position from your view. The radisu of the circular track is $1.3$ miles.
a. How much distance (in miles) has a race-casr travelled if the angle that has been swept out is #3.4$ radisn?

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b. What is the angle that has been swept out if the car has travelled $6.3$ miles?

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c. Define a function that determines the vertical distance of the race-car above the center of the track (in radii) in terms of the distance the race-car has traveled on the track.

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d. How many miles above the horizontal diameter is the car when the angle swept out is $2.1$ radians?

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e. How many miles to the right of the center of the track is the car when the car has travelled $2$ miles?

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Michael is sitting on a Ferris wheel. He is exactly $35$ feet from the center and is at the $3$ o'clock position when the Ferris wheel begins moving. The bottom of the Ferris wheel is $5$ feet above the ground.
16. If Michael swept out $150$ degrees as the Ferris wheel rotated, how many feet did he travel along the arc?

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2) Solve $2\sin^2x - 5\sin x + 2 = 0$ in the interval $[0, 2\pi)$.

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17. Suppose Michael traveled $22$ feet along the arc.
a. What angle measure (in radians) has Michael swept out?

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b. What angle measure (in degrees) has Michael swept out?

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c. Determine how many feet above or below the horizontal diameter of the Ferris wheel Michael is after traveling $22$ feet along the arc.

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18. Michael's sister is on a Ferris wheel with an arm (radius) that is twice as long as the arm of the Ferris Wheel Michael is on.
a. If they both open the same angle measure how do the distances they have traveled along the arc compare?

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b. If they both travel the same distance along the arc how do the angle measures they have opened compare?

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19. The Ferris wheel broke down when Michael was at the point $(-33.807, -9.059)$.
a. What is the measure of the angle (in radians) that has a vertex $(0,0)$ and rays through the point $(35, 0)$ and $(-33.807, -9.059)$?

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b. How many feet did Michael travel along the arc before stopping?

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20. Define a function $f$ that determines mIchael's vertical distance above the ground (measured in feet) as a function of the measure of the angle (in radians) swept out by the Ferris wheel as it moves counter-clockwise from the $3$ o'clock position. Be sure to define your variables.

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2) $v = 6i + j, w = 2i - 3j$
a) $v \cdot w =$
b) $||v|| =$
c) $|| 2w || =$
d) What is the angle between $v$ and $w$?

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21. Suppose the Ferris wheel completes $3$ revolutions in $55$ minutes.
a. How many radians does the Ferris wheel sweep out per minute?

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b. Define a formula that relates the measure odf the angle (in radians) swept out by the Ferris wheel in terms of the number of minutes since the Ferris wheel began to move.

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c. Define a functino $g$ to relate Michael's vertical distance above the horizontal diameter of the Ferris wheel (measured in feet) as a function of the number of minutes since the ball began to move. Be sure to define your variables.

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22. Use the diagram below to answer the following questions. Assume the measure of the angle is given in radians.


a. Determine the coordinates of the point $(x, y)$.

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b. What is the slope of the terminal ray?

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23. Complete the following statements.
a. As $\theta$ increases from $0$ to $\pi/2$ radians, $\sin(\theta)$ varies from ____ to ____ radii.

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b. As $\theta$ increases from $\pi$ to $3\pi/2$ radians $\cos(\theta)$ varies from ____ to ____ radii.

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c. As $\theta$ increases from $3\pi/2$ to $2\pi$ radians $\cos(\theta)$ varies from ____ to ____ radii.

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24. Suppose $g(t) = 6\cos(4t)$.
a. What is the amplitude of function $g$?

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b. What is the period of function $g$?

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c. What quantity does $4t$ represent?

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25. A "Hokie" is a new unit of measure invented by a Virginia Tech Alumna. An angle corresponding to one full rotation around a circle measures exactly $234$ Hokies.
a. What fraction of a circle's circumference is cut-off by an angle measuring $123$ Hokies?

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b. An angle whose openness is $45%$ of one full rotation has a measure of how many Hokies?

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c. Convert $101$ degrees to Hokies.

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d. Convert $101$ Hokies to radians.

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26. Use the given table to evaluate the following.

a. $g^{-1}(2)$

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b. $f(g(-1))$

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c. $f^{-1}(6)$

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d. $g^{-1}(f^{-1}(4))$

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27. Determine the rule of the inverse relationship of the following functions.
$n(p) = 3p + 4$

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b. $k(f) = 3(f - 2)^4$

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c. $c(s) = e^{s + 4}$

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28. The graphs of two periodic functions, $f$ and $g$, are given below.


a. What is the amplitude of function $f$?

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b. What is the period of function $f$?

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c. What is the amplitude of fucntion $g$?

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d. What is the period of function $g$?

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e. Express function $f$ in terms of function $g$.

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29. For the following situations, define a function that describes the number of pennies on a given square of a chessboard. Let $n$ represent the number of the square and $P = f(n)$ represent the number of pennies on the given square.
a. Suppose you start with $4$ pennies on a table. On the 1st square you double the initial amount of pennies. The number of pennies on any given square is double the number of pennies on the previous square.

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b. Suppose you start with $4$ pennies on the first square of the chess board. On the 2nd square you triple the initial amount of pennies. The number of pennies on any given square is triple the number of pennies on the previous square.

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30. Suppose the number of bees in a hive can be modeled by the exponential function $P(n) = 54321(0.876)^n$ where $n$ represents the number of days since November 1, 2014.
a. What does $54321$ represent?

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b. What does $0.876$ represent?

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c. What would $0.876$ represent?

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d. What would $0.876^{365}$ represent?

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31. The point $(3, -3.65)$ line on a line with a constant rate of change of $y$ with respect to $x$ of $-1.75$.
a. If the change in $x$ is $4.2$, what is the corresponding change in $y$?

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b. When $x = 1.8$, what is the corresponding value of $y$?

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c. Define a function $j$ that expresses the value for $y$ in terms of the value for $x$.

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32. For the tables below, determine if the relationship is linear. If the table represents a linear relationship, write a formula to represent how the quantities change together.


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