MAT186 Tutorial 1

MAT186H1F - Calculus I Fall 2023 During the first 25 minutes of your (first) tutorial next week, you will complete a two-problem tutorial worksheet (possibly with parts) drawn from these problems. You may discuss the problems in groups but the write-up and submission of your worksheet must be completed individually. During the remaining 25 minutes of your tutorial, you will discuss in groups any of the problems included on this sheet. They are meant for your own practice. You do not need to complete them all before your tutorial but please bring some of your draft work and a willingness to share your ideas or listen intently. You may also use the remaining time of your tutorial to discuss Assignment 1. Throughout your tutorial, TAs will circulate throughout the rooms as you work in small groups providing hints and feedback on your preliminary ideas. I. Building Blocks 1.1 Vocabulary 1. Recall the definitions for the following terms to determine if the statements below are TRUE or FALSE. Provide a sentence or two that either correct the statement (if false ) or justify the statement (if true ). (a) sin θ is the x -coordinate of the point on the unit circle that is reached after traveling the distance and direction indicated by the angle θ , from the point (1 , 0). (b) sin - 1 ( x ) yields radian measures between 0 and π 2 (inclusive) or 3 π 2 and 2 π (inclusive) for which sine equals x . (d) Given a transformed sine function, g ( x ) = c · sin[ a ( x + b )]+ d , the amplitude, | c | , can be determined by calculating half of the distance between the maximum and minimum values of the function. (i.e. | c | = max - min 2 ). (e) The function f ( x ) = sin x + 1 is neither even nor odd. 1.2 Fundamentals and Basic Procedural Exercises 2. Determine if the following statements are TRUE or FALSE with a short justification. (a) Given an invertible function f and its inverse f - 1 , f - 1 ( f ( x )) = x for all x ∈ R . (b) √ sin 2 x = sin x for all x ∈ R . (c) The function f ( x ) = 2 cos(3 x - 6) + 1 has an amplitude of 2, vertical shift of 1, a period of 2 π , and a horizontal shift 6 units to the right. (d) e ln x +1 = x + 1 for all x ∈ R . (e) The domain of ln( x 2 ) is ( -∞ , ∞ ). (f) If g ( x ) = 5 x and h ( x ) = 8 2 x , then f ( x ) = g ( x ) h ( x ) is an exponential function. 3. Show that the only values of x that satisfy x log 10 x = 100 x are x = 100 or x = 1 10 . 1 of 3

II. Core Exercises 4. Newton's Law of Cooling states that an object cools at a rate that is proportional to the difference in temperature between the object and its surrounding environment. The calculus that we will learn later on will allow us to determine the following model for the object's temperature at time t : T ( t ) = T s + D 0 e - kt where k is a positive constant that depends on the type of object, and D 0 is the initial difference between the object's temperature and the temperature of its surrounding environment ( T s ). (a) Suppose that a cup of coffee with a temperature of 90 ◦ C is placed in a room with a temperature of 30 ◦ C. If the coffee's temperature decreases to 60 ◦ C after 10 minutes, determine the function T that models the coffee's temperature at time t. (b) Describe what T - 1 represents in the context given in Part (a). (c) Determine a function for T - 1 . 5. When you board a ferris wheel, your feet are 1 meter off the ground. At the highest point of the ride, your feet are 99 meters above the ground. It takes 30 seconds for the ride to complete one full clockwise revolution. Model your height above the ground at t seconds after the ride starts using a sine function. 6. When you board a ferris wheel, your feet are 1 meter off the ground. At the highest point of the ride, your feet are 99 meters above the ground. It takes 30 seconds for the ride to complete one full clockwise revolution. Model your height above the ground at t seconds after the ride starts using a cosine function. 7. Let P ( t ) = 10000 1 + Ae - t denote the population of deer in West Grey Bruce County, Ontario at time t ≥ 0 measured in years. (a) Let P 0 > 0. Find A such that the initial population of deer is P 0 . (b) If 0 < P 0 < 10000 can we find a time T such that P ( T ) = 10001. Explain. (c) Suppose extra deer are imported from Alaska at t = 0 such that the initial population of deer in West Grey Bruce County is 20000. What happens to the deer population in West Grey Bruce County after a very long time. Explain your answer without the use of limits. (d) Let the initial population of deer in West Grey Bruce County be 1000. At what time T is the population of deer in West Grey Bruce County 5000? 8. Recall from the A3 PCE that ln n ( x ) denotes the n -times composition of ln( x ). More explicitly, ln n ( x ) = ln(ln( · · · ln(ln( x )) · · · )) | {z } composed n times . For all x > 0, define the function I ( x ) to be the smallest positive integer n such that ln n ( x ) ≤ 1. In other words, for a given x > 0, what is the minimum number of compositions of ln x that you need until the value is less than or equal to 1. For example, I (10) = 2 because ln(10) ≈ 2 . 302 > 1 and ln 2 (10) = ln(ln(10)) ≈ 0 . 83 ≤ 1. (a) What is the range of I ( x )? 2 of 3