MHF4U Cycle 4B Notes - P3 PDF

MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Period: Horizontal distance of one full cycle/revolution LESSON 4: APPLICATIONS OF TRIGONOMETRIC FUNCTIONS Amplitude: |a| = vertical distance between the Ymax/Ymin...

MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Period: Horizontal distance of one full cycle/revolution LESSON 4: APPLICATIONS OF TRIGONOMETRIC FUNCTIONS Amplitude: |a| = vertical distance between the Ymax/Ymin and the Yeoa. Half of the vertical distance of the sinusoidal wave. Start at lowest point (neg cosine) Let t represent the time in seconds. Let h(t) represent the height in metres. The height of the rider at 52 seconds is approximately 7.96 metres. e) When is the rider above (higher than) 20 metres? The rider is above 20 metres in between 16.28 seconds and 43.72 seconds. Add/subtract the period until you reach one full revolution (that is what the question wants). We don't need to do it here because it will exceed one full revolution. The rider is 20 metres above the ground at about 16.28 seconds and 43.72 seconds. Page 1 of 13 MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Amplitude is the vertical distance at still water (axis of the curve) to the peak (max) or trough (min) of the wave. In other words, it is half the vertical distance of the peak to the trough of the wave. Page 2 of 13 MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Let's make a positive cosine equation: d = 0 and a > 0 t represents the number of hours after midnight. Add/subtract the period of 8 hours until you reach The water is 5 metres high at about 2.32, 5.68, 10.32 hours. Page 3 of 13 MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions I will model the height of point P using a positive cosine function where d = 0 and a > 0. Page 4 of 13 MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Horizontal distance of one full cycle- from one minimum [or maximum] to the next minimum [or maximum]. Horizontal line that cuts the sinusoidal in half. Half of the vertical distance of the sinusoidal. I will model the pendulum with a positive cosine function using d = 0 according to the graph. Positive cosine starts at the maximum and decreases after. Page 5 of 13 MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions LESSON 5: APPLICATIONS OF EXPONENTIAL & LOGARITHMIC EQUATIONS pH Scale (Acidity or Alkalinity) Example 1: (in mol/L). Page 6 of 13  Find the hydrogen ion concentration of each acid, and then compare (ratio). MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions The acid with pH 1.6 is about 7.94 times stronger than an acid with pH 2.5. Richter Scale (Magnitude of Earthquakes) Biggest Earthquakes ever recorded: Valdivia, Chile 22 May 1960 (magnitude 9.5) Prince William Sound, Alaska 28 Example 2: If the average earthquake measures 4.5 on the Richter scale, how March 1964 (magnitude 9.2) much more intense is an earthquake that measures 8? Find the intensity of each earthquake, then compare. Sumatra, Indonesia 26 December 2004 (magnitude 9.1) Sendai, Japan 11 March 2011 (magnitude 9.0) Kamchatka, Russia 4 November 1952 (magnitude 9.0) The earthquake that measure 8 on a Richter Scale is about 3162 times more intense than an earthquake of 4.5 on a Richter scale. Decibel Scale (Sound Intensity) Example 3: How many times more intense is the sound of a rock concert than the sound of a subway? The rock concert is 1000 times more intense than the sound of a subway. Page 7 of 13  MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Exponential Growth/Decay Exponential Growth: Exponential Decay: 𝑟 𝑛𝑡 𝑥 𝑥 Compound Interest: 𝐴 = 𝑃 (1 + 𝑛) ( ) ( ) 𝐴(𝑥) = 𝐴𝑜 (1 + 𝑔) 𝑡 𝐴(𝑥) = 𝐴𝑜 (1 − 𝑑) 𝑡 𝐴 = 𝐹𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑟 𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 ($) 𝑥 𝑥 𝑃 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑟 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 ($) Doubling Time: 𝐴(𝑥) = 𝐴0 (2) 𝑡 Half-Life: 𝐴(𝑥) = 𝐴0 ( ) 1 𝑡 t = half-life (is the time 𝑟 = 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟 (𝑑𝑒𝑐𝑖𝑚𝑎𝑙) 𝑥 it takes for 2 something to 𝑡 = 𝑡𝑖𝑚𝑒 (𝑦𝑒𝑎𝑟𝑠) Tripling Time: 𝐴(𝑥) = 𝐴0 (3) 𝑡 cut in half). 𝑛 = 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑥 Quadrupling Time: 𝐴(𝑥) = 𝐴0 (4) 𝑡 𝐴𝑜 = initial value 𝑔 & 𝑑 are the growth or decay rates in decimals t = time it takes to grow/decay 𝑥 = time It takes 8 hours for the radioactive substance to reduce to half of its amount. Example 4: An investment of $2500 grows at a rate of Example 5: The half-life of a radioactive substance is 8 h. 4.8% per year, compounded semi-annually. How long will How long will it take for a 300 grams sample to decay to it take for the investment to be worth $4000? 20 grams? It would take About 98 years for the investment to grow to $4000. It will take approximately 31.26 hours for the sample to decay to 20 grams. Example 6: RJ has an ant colony, whose population Example 7: Blue jeans fade when washed due to the loss doubles every month. He begins with 100 ants. How long of blue dye from the fabric. If each washing removes will it take until the colony reaches a population of 3200 about 2.2% of the original dye from the fabric, how many ants? washings are required to give a pair of jeans a well-worn look of 30% of the original dye? It will take about 54 washes for the jeans to have 30% It will take 5 months for the ant of its original dye. Page 8 of 13 colony reach a population of 3200. MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions LESSON 6: COMBINATION OF FUNCTIONS Page 9 of 13  MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions This equation is already in factored form Log function TIMES the sine function The profit will be greater than or equal to 0 when we sell in between Page 10 of 13 and including two hundred to about seven-and-a-half hundred units. MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions LESSON 7: AVERAGE & INSTANTANEOUS RATE OF CHANGE OF ALL FUNCTIONS The unit of any rate of change is _________ per 1 unit of _________. Polynomial Function Example 1: For the function, , a) Calculate the average rate of change on the b) Estimate the instantaneous rate of change interval 𝑥 ∈ [1, 3]. at x = 2. You will learn how to calculate the exact instantaneous rate of change in Gr. 12 Calculus & Vectors. For now, we will use the average rate of change on the interval of a point really close to our observed x-value and the observed x-value to estimate the instantaneous rate of change. Page 11 of 13 MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Rational (Reciprocal) Function Example 2: The demand function for snack cakes at a large bakery is given by the Marginal Revenue The instantaneous rate of change at a function,. The x-units are given in thousands point on a revenue function is called the of cakes, and the price per snack cake, p(x), is in dollars. Estimate the marginal revenue. It is a measure of the marginal revenue for x = 0.75. I needed to find estimated additional revenue from selling revenue function one more item. first! dollars per thousands of cake The marginal revenue is about $0.28 per snack cake for every thousands of cake. Shoutout to Kaylyn for catching my mistake! Exponential & Logarithmic Functions Example 3: For 500 g of a radioactive substance with a Example 4: As a tornado moves, its speed increases. The half-life of 5.2 h, the amount remaining is given by the function, , relates the speed of the formula, , where M is the mass wind, S, in miles per hour, near the centre of a tornado remaining and t is the time in hours. Estimate the to the distance that the tornado has travelled, d, in instantaneous rate of change in mass at 1 day. miles. Calculate the average rate of change for the speed of the wind at the centre of a tornado from mile 10 to Shoutout to Emily for fixing my mistake. mile 100. Great attention to detail! The average rate of change for the speed of the wind at the centre of a tornado from mile 10 to mile 100 is grams per hour 31/30 mph per mile. The instantaneous rate of change is The instantaneous rate about -58.33 grams per hour. This of change is about -2.72 means that the radioactive substance is grams per hour. This decaying at a rate of 58.33 grams per means that the hour. radioactive substance is decaying at a rate of 2.72 grams per hour. Page 12 of 13 MHF4U | Cycle 4B: More Problem Solving & Combinations of Functions Trigonometric Functions Example 5: For the function, , Example 6: For the following calculate the average rate of change on the interval graph of a function, state Thanks to Arya for correcting my two intervals in which the mistake. My calculator was in DEGREE function has an average rate. mode. Should be in RADIANS!!!!!!!!!!!!!!!!!!!!!!!!! of change in f(x) that is… a) Zero b) A negative value c) A positive value Horizontal line Slope = 0 Increasing line Decreasing line Slope = positive Slope = negative Slope of the Secant & Tangent Lines Use AROC to calculate the slope of the secant line. Use IROC to estimate the slope of the tangent line. Example 7: Page 13 of 13