396 CHAPTER 6 Trigonometric Functions 125. Challenge Problem Cycling A bicycle has a pedal drive wheel with radius 5.2 inches and a rear cog wheel with radius 1.8 inches. See the figure. How many revolutions will the pedals need to make to move the bicycle 50 feet if the wheels have a diameter of 30 inches? Round to the nearest tenth. 1.8 in. 5.2 in. 124. Challenge Problem Geometry See the figure.The measure of arc BE is 2 .π Find the exact area of the portion of the rectangle ABCD that falls outside of the circle whose center is at A.* A E B C D ED = 7 — ‘Are You Prepared?’ Answers 1. π = C r 2 ; π = A r2 2. ⋅ r t *Courtesy of the Joliet Junior College Mathematics Department Retain Your Knowledge Problems 133–142 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 142. Consider the function f x x x x x x 1 5 9 4 89 2 84 1 5 4 3 2 ( ) = − + + − + (a) Use a graphing utility to graph f. Determine the turning points on the graph of f. (b) The function f x x x x x 9 3 89 84 4 3 2 ( ) ′ = − + + − is called the first derivative of f. Find the zeros of f .′ That is, solve f x 0. ( ) ′ = (c) Compare the x-coordinates of the turning points of f to the zeros of f .′ What do you notice? (d) Use a graphing utility to determine the intervals where f is increasing. (e) The function f is increasing where f x 0. ( ) ′ > Use the derivative to determine the intervals for which f is increasing. Because polynomials are continuous over their domain, all endpoints are included in the intervals describing where f is increasing. However, in general, the numbers at the endpoints must be tested separately to determine if they should be included in the intervals describing where a function is increasing or decreasing. (f) Compare the answers from parts (d) and (e).What do you notice? 133. Find the zero of f x x3 7. ( ) = + 134. Solve: x x 5 2 5 14 2 + = − 135. Write the function that is finally graphed if the following transformations are applied in order to the graph of y x . = 1. Shift left 3 units. 2. Reflect about the x-axis. 3. Shift down 4 units. 136. Find the horizontal and vertical asymptotes of R x x x x 3 12 5 14 . 2 2 ( ) = − − − 137. Find c so the points c 2, ( ) and 1,4 ( ) − are on a line perpendicular to x y 2 5. − = 138. Solve: x 2 3 5 8 − + = 139. Find the domain of h x x x 3 9 . 2 ( ) = − 140. Find the difference quotient of f x x2 5. 3 ( ) = − 141. Multiply: x3 2 3 ( ) − Explaining Concepts 126. Do you prefer to measure angles using degrees or radians? Provide justification and a rationale for your choice. 127. What is 1 radian? What is 1 degree? 128. Which angle has the larger measure: 1 degree or 1 radian? Or are they equal? 129. Explain the difference between linear speed and angular speed. 130. For a circle of radius r, a central angle of θ degrees subtends an arc whose length s is s r 180 . π θ = Discuss whether this statement is true or false. Defend your position. 131. Discuss why ships and airplanes use nautical miles to measure distance. Explain the difference between a nautical mile and a statute mile. 132. Investigate the way that speed bicycles work. In particular, explain the differences and similarities between 5-speed and 9-speed derailleurs. Be sure to include a discussion of linear speed and angular speed.
RkJQdWJsaXNoZXIy NjM5ODQ=