1. ____________ Evaluate $f\left(x\right)=5^{2x-4}$ for x = 3.
2. ____________ Calculate: 12 log₈(4).
3. Rewrite 343 = 7^x in logarithmic form and then solve for x.
4. log₁₉(x) = 2 in exponential form and then solve for x.
5. Solve for x in the following equations:
   
   ____________ $4.90625=2{\left(\frac{5}{4}\right)}^x+1$
6. ____________ 6 ln(6x) = 32.2517
7. ____________ $e^{\left(\frac{x}{7}\right)}=1096.6332$
8. The soil on Bikini atoll in the Marshall Islands is contaminated by the radioactive element Cesium-137. Like all radioactive elements, Cesium-137 will eventually decay into non-radioactive elements. The rate of decay is an exponential decay given by the formula
   $N=P{e}^{\left(-0.023t\right)}$ where P is the starting amount of Cesium in the soil in kilograms, t is the number of years, and N is the remaining amount of radioactive Cesium after that number of years.
   
   
   1. __________ If there were 100 kilograms of Cesium-137 in the soil in 1954 as a result of the Castle Bravo thermonuclear hydrogen bomb test, how many kilograms will be in the soil now in 2017?
   2. __________ How many years after 1954 until there is only 1 kilogram of the original 100 kilograms of Cesium-137 left in the soil?
9. A ball was dropped and the bounce heights for consecutive bounces were measured.
   

   x1	y1
   0	100
   1	80
   2	64
   3	51.2
   4	40.96

   
   where x₁ is the number of bounces and y₁ is the height of that bounce in centimeters. Make a table and use the regression y₁~ab^x₁.
   1. a = ____________
   2. b = ____________
   3. _____________ Use the equation to predict the height of bounce number five.
10. Design specifications for the 2018 world cup soccer ball have yet to leak, and only a few photos of the ball have appeared. The ball is thought to behave like the 2014 world cup soccer ball, the Brazuca. Soccer balls have a coefficient of drag that changes with the speed, Note that the actual function for the coefficient of drag is far more complex than a logarithmic equation can model.

    x2	y2
    23	0.49
    40	0.40
    47	0.20
    62	0.18
    123	0.20

    
    where x₂ is the speed of the ball in kilometers per hour and y₂ is the coefficient of drag.
    Use the regression y₂~cln(x₂)+d
    1. c = ____________
    2. d = ____________
    3. _____________ Use the equation to predict the coefficient of drag at 100 kilometers per hour.

Test one
1	25
2	8
3	log_7(343), 3
4	19^2, 361
5	3
6	36
7	49
8a	23.48
8b	200.225
9a	100
9b	0.8
9c	32.768
10a	-0.1846
10b	1.0180
10c	0.1680
 

1. _________ Find the arc length in the diagram on the right.
2. _________ Find the area of the sector in the diagram on the right.
3. _________ Convert $\frac{\pi }{12}$ to an angle in degrees.
4. _________ Convert 225° to radians.
5. For the following exercises, use the figure to evaluate each trigonometric function of angle A.
   
   1. sin A = _________
   2. cos A = _________
   3. tan A = _________
6. For the following exercises, use the figure to answer the questions.
   
   • b = _________
   • c = _________
7. (_________, _________) Find the coordinates of the point on a circle with radius 20 corresponding to an angle of 220°
8. Nine points are spaced equally around a 360° unit circle.
   
   1. _________ What is the angle of the shaded area?
   2. ( _________ , _________) What are (x,y) coordinates of the point A?

Solutions presented in class. No key for two.

1. With Desmos set to radians, use Desmos to graph two full periods of $f\left(x\right)=5sin\left(\frac{\mathrm{\pi }\mathrm{x}}{3}\right)$
   1. _________ State the amplitude.
   2. _________ Calculate the period.
   3. _________ State the midline.
2. For the following graph:
   
   1. _________ State the amplitude.
   2. _________ Determine the period.
   3. _________ State the midline.
   4. Write the equation the function:
3. For f(x) = sin(x) on the interval from zero to 2π, find the x values where $f\left(x\right)=\frac{\sqrt{3}}{2}$. Desmos should be set in radians.
4. The graph depicts three trigonometric functions. One as a solid line, one as a dashed line, and one as a dotted line. Desmos is set in radians. Match the function to the line on the graph.
   
   1. f(x)=tan(x):
   2. f(x)=sec(x):
   3. f(x)=-cos(x):
5. Calculate the values of θ = ______________° and φ = ______________° in degrees based on the following diagrams.
   
6. θ = __________° In the movie Blade Runner 2049 a guy-wire holds up a dead tree. Use the information in the diagram to calculate the angle θ in degrees. Note that the horizontal value is 18, the vertical value is 12 in the diagram.
   

Solutions presented in class. No key for three.

1. Determine the solutions for $sin\left(x\right)+\frac{sin\left(3x\right)}{3}=\frac{2}{3}$ for the interval {0 ≤ x ≤ 2π}.
   
   
2. Determine the solutions for $cos\left(x\right)=4$ for the interval {0 ≤ x ≤ 2π}.
   
   
3. Determine the solutions for ${tan}^2\left(x\right)+4tan\left(x\right)=-4$ for the interval {0 ≤ x ≤ 2π}.
   
   
4. Determine the solutions for $sin\left(x\right)+1=-sin\left(x\right)-1$ for the interval {0 ≤ x ≤ 2π}.
   
   
5. Toughie: Determine the solutions for x for the following equations:
   
   $x^2+y^2={\pi }^2$
   and
   $y=sin\left(x\right)$
   
   with the sine function limited to the interval {-π ≤ x ≤ π}.
   
   
   

1. 1
2. 33.9481958283
3. π/3, 2π/3 [1.0471975512, 2.09439510239]
4. 0.3747, π/2, 2.7669
5. No solutions
6. 2.034, 5.176
7. 3π/2
8. -π and π

1. Convert the following from polar coordinates (r,θ) to Cartesian coordinates (x,y) with r>0 and 0 ≤ θ ≤ 2π radians.
   1. (7.0711, $\frac{\mathrm{\pi }}{4}$) = (__________ , __________)
   2. (536, $\frac{5\mathrm{\pi }}{6}$) = (__________ , __________)
2. Convert the following the given Cartesian coordinates (x,y) to polar coordinates (r,θ) with r>0 and 0 ≤ θ ≤ 2π radians.
   1. (40,41) = (__________ , __________)
   2. (275,252) = (__________ , __________)
3. Rewrite x² + y² = 7y as a polar equation.
4. Rewrite r = 4 sin θ as a Cartesian equation.
5. Set Desmos to polar grid with the mode in radians. The following shape is the result of using the polar variables r, θ, and the sine function. Write the function that generated the following graph:
   
6. For the parametric equations:
   x(t)= a sin(bt)
   y(t)= c sin(dt)
   
   Find the values of a, b, c, and d for the following graph in the parametric form (a cos(bt),c sin(dt)).
   1. = ____
   2. = ____
   3. = ____
   4. = ____
7. A ball is thrown at 609 cm/s perpendicular to the direction of RipStik traveling at 346 cm/s as seen in the diagram. The velocity vector for the ball over the ground is given by the sum of the velocity of the RipStik plus the launch velocity for the ball: v_ground = v_ripstik + v_ball
   
   1. __________ Calculate the magnitude (length) for the velocity vector of the ball over the ground.
   2. __________ Calculate the direction angle θ for the velocity vector of the ball in degrees.
8. __________ A tractor tire is pulled with a force F of 900 Newtons at an angle of 20° for a distance d of 10 meters. Use the dot product Fd_distancecos(θ) to calculate the work done (energy expended). Round answer to nearest whole number.
   
9. __________ A tractor tire is pulled with a force of 900 Newtons at an angle of 20°. The tire pivots about a point that is 1.5 meters behind the attachment point. Use 1.5 meters as the lever arm. Use the cross product Fd_{lever arm}sin(θ) to calculate the torque exerted on the tire.

1a. (5,5)
1b. (-464.1896, 268)
2a. (57.28, 0.7977)
2b. (373, 0.74178)
3. r = 7 sin θ
4. x²+y²=4y
5. r=4 sin(4θ)
6. 4,1,6,3 or 4,k,6,3k
7a. 700.426
7b. 60.4°
8. 8457
9. 461.7