advertisement

Welcome to Jeopardy! AP Calculus Contestants! Call 911! We Need a Parametric. Too hip to be squared. Can You Function in the Morning? Opposites Attract I Saw the Sine. 100 100 100 100 100 200 200 200 200 200 300 300 300 300 300 400 400 400 400 400 500 500 500 500 500 Graph the curve and determine the initial and terminal points, if any. 𝒙 = 𝟒 𝐬𝐢𝐧 𝒕 𝒚 = 𝟓 𝐜𝐨𝐬 𝒕 𝟎 ≤ 𝒕 ≤ 𝟐𝝅 And the Answer Is: 𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒂𝒏𝒅 𝑻𝒆𝒓𝒎𝒊𝒏𝒂𝒍 𝑷𝒐𝒊𝒏𝒕𝒔: (𝟎, 𝟓) Back to the Board. Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? 𝟐 𝒙 = 𝒔𝒆𝒄 𝒕 − 𝟏 𝒚 = 𝐭𝐚𝐧 𝒕 𝝅 𝝅 − <𝒕< 𝟐 𝟐 And the Answer Is: 𝒙= 𝟐 𝒚 All Back to the Board. Find the values of t that produce the graph in Quadrant IV. 𝒙=𝟑− 𝒕 𝒚=𝒕−𝟏 −𝟓 ≤ 𝒕 ≤ 𝟓 And the Answer Is: −𝟑 < 𝒕 < 𝟏 Back to the Board. Find a parametrization for the part of the graph that lies in Quadrant I. 𝒚= 𝒙+𝟑 Possible Answers Include: 𝒙=𝒕 𝒚= 𝒕+𝟑 𝒕<𝟎 Back to the Board. Find a parametrization for the left half of the parabola 𝟐 𝒚 = 𝒙 + 𝟐𝒙 Possible Answers Include: 𝒙=𝒕 𝟐 𝒚 = 𝒕 + 𝟐𝒕 𝒕 ≤ −𝟏 Back to the Board. Rewrite the following expression to have base 3: 𝟏 𝟐𝟕 𝒙 And the Answer Is: 𝟑 Back to the Board. −𝟑𝒙 Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded daily. And the Answer Is: ≈ 19.106 𝑦𝑒𝑎𝑟𝑠 Back to the Board. If John invests $2300 in a savings account with a 6% interest rate compounded annually, how long will it take until John’s account has a balance of $4150? And the Answer Is: ≈ 𝟏𝟎. 𝟏𝟐𝟗 𝒚𝒆𝒂𝒓𝒔 Back to the Board. The half life of a certain radioactive substance is 12 hours. There are 8 grams present initially. a) Express the amount of substance remaining as a function of time. b) When will there be 1 gram remaining? And the Answer Is: a) Amount = 𝟖 𝟏 𝟐 b) After 36 hours Back to the Board. 𝒕 𝟏𝟐 The population of Glenbrook is 375,999 and is increasing at the rate of 2.25% per year. Predict when the population will be 1 million. And the Answer Is: After about 44.081 years Back to the Board. Write an equation for the lines parallel and perpendicular to the line 𝟐𝒙 + 𝒚 = 𝟒 and contains the point 𝑷(−𝟐, 𝟐). And the Answer Is: 𝑷𝒂𝒓𝒂𝒍𝒍𝒆𝒍: 𝒚 = −𝟐𝒙 − 𝟐 𝟏 𝑷𝒆𝒓𝒑𝒆𝒏𝒅𝒊𝒄𝒖𝒍𝒂𝒓: 𝒚 = 𝒙 + 𝟑 𝟐 Back to the Board. Find the domain and range of the following function: 𝟏 𝒚= +𝟑 𝒙 And the Answer Is: 𝑫𝒐𝒎𝒂𝒊𝒏: 𝒙 ≠ 𝟎 𝑹𝒂𝒏𝒈𝒆: 𝒚 ≥ 𝟎 Back to the Board. Determine if the following function is even or odd: 𝟑 𝒙 𝒚= 𝟐 𝒙 −𝟏 And the Answer Is: Odd function Back to the Board. Find the formula of the piecewise function displayed on the following graph: And the Answer Is: 𝟑 − 𝒙, 𝒇 𝒙 = 𝟐𝒙, Back to the Board. 𝒙≤𝟏 𝟏<𝒙 Find the composition of functions 𝒇(𝒇 𝒙 ), 𝒇(𝒈 𝒙 ), 𝒈(𝒈 𝟓 ), and 𝒈(𝒇 −𝟏 ) when 𝒇 𝒙 = 𝟏 𝒙𝟐 and 𝒈 𝒙 = 𝒙−𝟏 And the Answer Is: 𝒇 𝒇 𝒙 𝟒 =𝒙 𝟏 𝒇 𝒈 𝒙 = 𝒙−𝟏 𝒈 𝒈 𝟓 =𝟏 𝒈 𝒇 −𝟏 = 𝟎 Back to the Board. Is the function 𝐲 = 𝟏 𝟐 𝒙 +𝒙 one-to-one? Explain why or why not. And the Answer Is: No because not every output has only one input. (Does not pass the horizontal line test.) Back to the Board. Does the function 𝟐𝒙 𝒇(𝒙) = 𝒆 − 𝟐 have an inverse? 1 If yes, find f If not, explain why. Back to the Board. And the Answer Is: Yes, 𝐥𝐧 𝒙 + 𝟐 1 f ( x) = 𝟐 Back to the Board. Find the inverse of the following function and verify that −𝟏 −𝟏 𝒇 𝒇 𝒙 = 𝒇 𝒇 𝒙 =𝒙: 𝟐 𝒇 𝒙 =𝒙 +𝟏 𝒙≥𝟎 And the Answer Is: 𝒇 −𝟏 𝒇 𝒇 𝒙 −𝟏 = 𝒇−𝟏 𝒇 𝒙 Back to the Board. 𝒙 = 𝒙−𝟏 𝒙−𝟏 = 𝟐 +𝟏= 𝒙−𝟏 +𝟏=𝒙 𝒙𝟐 + 𝟏 − 𝟏 = 𝒙𝟐 = 𝒙 Solve the following equation algebraically and support your answer graphically. 𝒙 −𝒙 𝟐 +𝟐 =𝟓 And the Answer Is: 𝒙 = 𝐥𝐨𝐠 𝟐 𝟓 ± 𝟐𝟏 𝟐 ≈ −𝟐. 𝟐𝟔 𝒐𝒓 𝟐. 𝟐𝟔 Back to the Board. −𝟏 Graph 𝒇 𝒙 , 𝒇 𝒙 and 𝒚 = 𝒙 on the same screen. 𝒇 𝒙 = 𝐥𝐨𝐠 𝒙 What do you notice? And the Answer Is: 𝒇−𝟏 𝒙 𝒊𝒔 𝒂 𝒓𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒇 𝒙 𝒐𝒗𝒆𝒓 𝒕𝒉𝒆 𝒍𝒊𝒏𝒆 𝒚 = 𝒙 Back to the Board. Determine the period and amplitude and draw the graph of the following function: 𝒚 = 𝟑 𝐜𝐬𝐜 𝟑𝒙 + 𝝅 − 𝟐 And the Answer Is: 𝟐𝝅 𝟑 Period: Amplitude: 3 Back to the Board. Find the value of the six trigonometric functions at 𝜽 given that 𝜽 = 𝐜𝐨𝐬 −𝟏 𝟗 𝟒𝟏 And the Answer Is: 𝟒𝟎 𝐬𝐢𝐧 𝜽 = 𝟒𝟏 𝟒𝟏 𝐜𝐬𝐜 𝜽 = 𝟒𝟎 𝟗 𝐜𝐨𝐬 𝜽 = 𝟒𝟏 𝟒𝟏 𝐬𝐞𝐜 𝜽 = 𝟗 𝟒𝟎 𝐭𝐚𝐧 𝜽 = 𝟗 𝟗 𝐜𝐨𝐭 𝜽 = 𝟒𝟎 Back to the Board. Evaluate the following expression: 𝐬𝐢𝐧 𝐜𝐨𝐭 −𝟏 𝟏𝟐 𝟓 And the Answer Is: 𝐬𝐢𝐧 𝐜𝐨𝐭 Back to the Board. −𝟏 𝟏𝟐 𝟓 𝟓 = 𝟏𝟑 Show that 𝐜𝐬𝐜 𝒙 is an odd function of x. Using this, show that the reciprocal of an odd function is also odd. And the Answer Is: 𝒓 𝒓 𝐜𝐬𝐜(−𝜽) = =− = − 𝐜𝐬𝐜 𝜽 −𝒚 𝒚 The reciprocal of cosecant is the sine function: 𝟏 −𝒚 𝒚 = 𝐬𝐢𝐧 −𝜽 = =− = −𝐬𝐢𝐧(𝜽) 𝐜𝐬𝐜 −𝜽 𝒓 𝒓 Back to the Board. Solve the following equation in the specified interval: 𝐜𝐬𝐜 𝒙 = 𝟐 𝟎 < 𝒙 < 𝟐𝝅 And the Answer Is: 𝝅 𝟓𝝅 𝒙 = 𝐚𝐧𝐝 𝟔 𝟔 Back to the Board.