Online Calculus 1 MAC 2311 Quiz #1 For all questions, you must show your work. Correct answers without supporting work will not receive credit. 1) Find all values of x in the interval [0,2𝜋] that satisfy the equation. Show algebraic work and express all angles in RADIANS. a) 2 cos 2 𝑥 = 1 b) 2 + cos 2𝑥 = 3 cos 𝑥 4𝑥−9 2) Solve the inequality ≥ 3. Use the test point method described in your notes/book and 𝑥−3 show all work for credit (number line, intervals, test values, resulting signs, etc.). 3) Rewrite |2𝑥 − 1| without using the absolute value symbol. 4) Solve the equations. Leave the solution(s) in exact form (no decimals). a) ln √𝑥 + 1 = 2 b) 𝑒 −2𝑥 = 7 c) cos (sin−1 5 )=𝑥 √41 5) The graph of f is given (the graph does not extend beyond what you see here) a) What are the domain and range of 𝑓 −1? b) What is the value of 𝑓 −1 (2)? 6) A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. t (min) 36 38 40 42 44 Heartbeats 2530 2661 2806 2948 3080 The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient’s heart rate after 42 minutes using the secant line between the points with the given values of t. a) 𝑡 = 36 and 𝑡 = 42 b) 𝑡 = 38 and 𝑡 = 42 c) 𝑡 = 40 and 𝑡 = 42 d) 𝑡 = 42 and 𝑡 = 44 e) What do these values reveal about the patient’s heart rate after 42 minutes? 7) Sketch a graph of a function that has the following properties: lim 𝑓(𝑥) = −2 𝑥→3− lim 𝑓(𝑥) = 2 𝑥→3+ 𝑓(3) = 1 lim 𝑓(𝑥) = 1 𝑥→0 𝑓(0) = −1 8) Use a table of values to estimate the limit. Include in your table all of the values of t that you use and the results, but feel free to use a calculator for the arithmetic. Make sure to state your conclusion. a) lim 𝑡→0 5𝑡 −1 𝑡 𝑥+1 b) lim+ 𝑥−5 𝑥→5 9) Evaluate the following limits using algebra/limit laws. Show all work for credit but you do not need to state which limit laws you are using. No credit will be given for limits found numerically, graphically, etc. Please note that notation will be a part of the grade. √4𝑥+1−3 𝑥−2 𝑥→2 a) lim b) lim 2𝑥 2 +3𝑥+1 𝑡→−1 𝑥 2 −2𝑥−3 c) lim ℎ→0 1 1 − (𝑥+ℎ)2 𝑥2 ℎ