Midterm 3 Review Questions

Transcription

1 Midterm 3 Review Questions Matthew Weaver Ma These problems should give an accurate depiction of the problems on the midterm exam and the techniques and material you should know. Some problems will be similar to old homework and quiz questions while some may come from past exams. Problem 1: Solve the following equation for the exact value(s) of x. lnpx 5q ` lnpx ` 13q lnpx 2 q Problem 2: Solve the following equation for the exact value(s) of x. lnpxq ` lnpx 3q lnp4x 12q Problem 3: Solve the following equation for the exact value(s) of x. 2 3x 4 3 x`1 Problem 4: Solve the following equation for the exact value(s) of x. e 2x 4e x 12 0 Problem 5: Cobalt-60 has a half-life of 7 years. Assuming that it decays according to the formula Aptq A 0 e kt find the following: 1. the decay constant k 2. If there are initially 200 grams present, how long will it take to decay to 115 grams? 3. How long will it take for 30% of the material to decay? Problem 6: Solve for t: A P e rt 1

2 Problem 7: solve for t: c ap3 e bt q Problem 8: List some angles that are coterminal to 40. Problem 9: List some angles that are coterminal to π 9. Problem 10: Convert to degrees, minutes, and seconds. Round to the nearest second. Problem 11: Convert 64, 23 1, 19 2 to decimal degrees. Round to three decimal places. Problem 12: Convert θ 2 to degrees, minutes, and seconds. Round to three decimal places. Problem 13: 1. Find the complement and supplement of Find the complement and supplement of 2π Find the complement and supplement of 57, 46 1, Problem 14: Find the acute reference angles of 6π 5, , and 97, 35 2, 2 2. Problem 15: Determine the values of all six trig functions of θ given the triangle 11 θ 8 Problem 16: Find the exact value of the remaining trigonometric functions for an acute angle θ in the given quadrant. cos θ 2 5 in Quadrant III 2

3 Problem 17: Find the exact value of the remaining trigonometric functions for an acute angle θ in the given quadrant. cot θ 11 in Quadrant II 6 Problem 18: Determine the quadrant which would contain the angle θ with the following conditions. sec θ ą 0 and tan θ ă 0 Problem 19: Find the exact value of the remaining trigonometric functions for an acute angle θ cot θ 5 and sec θ ą 0 Problem 20: Find the exact values of all six trigonometric functions for the following angles (without looking at a unit circle) 1. π π π 3 Problem 21: Find the amplitude, the period, the domain, and the range of 5 sinp7xq Problem 22: Find the amplitude, the period, the domain, and the range of ˆ 2x 3 cos 5 Problem 23: Find the period, the domain, and the range of 2 tanpπxq Problem 24: Using the function in problem 21, determine when it is positive, negative, increasing, decreasing on the interval r0, 10πs 3

4 Problem 25: Using the function in problem 22, determine when it is positive, negative, increasing, decreasing on the interval r1, 3s Problem 26: Approximate, to the nearest 0.1, all angles r0, 360 q that satisfy csc θ 3.2 Problem 27: Find the general solution to cospxq 1 2 Problem 28: Find the general solution to cscpxq 2 Problem 29:Find the general solution to tanpxq? 3 Problem 30: Find the general solution to sinp3xq 1 Problem 31: Find the general solution to the equation 24 cos θ ` 12? 3 0 Problem 32: Find the general solution to the equation 6 sec 2 θ 6 0 Problem 33: Find all solutions in r0, 2πq to the equation 2 sinpxq cospxq sinpxq 0 Problem 34: Find all solutions in r0, 2πq to the equation 7 cotpxq cospxq ` 7 cotpxq 0 Problem 35: Find all solutions in r0, 2πq to the equation 2 sin 2 pxq 3 sinpxq ` 1 0 4

5 Problem 36: Find the general solution to the equation tan 2 θ 1 0 Problem 37: Find the general solution to the equation csc 2 θ 4 0 Problem 38: Find the general solution to the equation 2 sin 2 θ ` cos θ 1 Problem 39: Find the general solution to the equation 2 cosp2xq?2 0 Problem 40: The headlights of an automobile are set such that the beam drops 1 in for each 30 ft in front of the car. What is the angle between the beam and the road? Round your answer to 3 decimal places. Problem 41: When the elevation of the sun is 0.7 rad a tree has a shadow 21m long. How tall is the tree? Problem 42: A driver in a car traveling along a straight level road at 40 miles per hour sees a mountain in the distance. The driver determines that the angle of elevation to the top of mountain to be 12. After 45 minutes (driving closer), the driver determines the angle of elevation to the top of the mountain to be 41. Calculate the height of the mountain to the nearest foot. Problem 43: Solve the system of equations " 3x ` y 9 3x ` 2y 6 Problem 44:Solve the system of equations " x2 ` y 2 20 x ` y 2 Problem 45: A boat sets out for a cruise on a river, sailing upstream 10 miles and then turning around and returning to port. The trip upstream 5

6 requires 75 minutes and the trip downstream requires 30 minutes. Find the speed of the boat in still water and the speed of the current in miles per hour. Round to the nearest whole number. Problem 46: What quantities of silver 70% pure and 95% pure must be mixed together to give 12 grams of silver 80% pure? Give your answer to one decimal place. 6