I Questions for Further Exploration 1. Use a graphing utility to estimate Limit 1 in part (c) cosine functions are each other's derivatives? If not, above graphically and numerically. which is the derivative of the other? , sin h 1111 2 l g hm I; . . 4. Amusement Park Ride An amusement park ride is constructed such that its height h in feet above ground in terms of the horizontal distance x in feet From the starting point can be modeled by he): 50 +45 sin 3-5 0515300. 2. Use a graphing utility to estimate Limit 2 in part (c) 150' above graphically and numericalbl. (a) The formula for the derivative of fix) = a + b sin or biscos ex. Use this to nd the derivative of Mac). (b) Find the value of the derivative of 11(1) when x = 50, 150, 200, and 250. . cos it : 1 11m = 0 1350 h 3. Follow the steps used in parts (a), (b), and (c) above to develop a formula for the derivative of the function gtx) '2 cos x. [s it tme hat he sine ind Mathematics ( Precalculus and Calculus) Class : 12 B Chapter 12 Project . Tangent Lines to Sine Curves In this project you will develop a formula for the slope of the tangent line to the sine curve at an arbitrary point (x, y). In other words, you will derive a formula for the derivative of the function f(x) = sin x. The first approach will be graphi- y = sin x cal, the second will be numerical, and the third will be algebraic. a. Graphical Approach Use a graphing utility to graph the function f (x) = - 6 6 sin x as shown at the right. On a separate sheet of paper, plot your esti-mates of the slope of this curve at various x-values. For instance, some values of the slope underneath the sine curve have been plotted at the right. The slope is approximately 1 at x = 0, 0.8 at x = 0.5, and 0 at x = 1.5. After you have plotted 15 or 20 points, connect them with a continuous curve. Do you recog- nize this curve? b. Numerical Approach The slope of the tangent line is given by the formula m = lim A( x + h) - f(x) h The difference quotient [f(x + h) - f(x)]/h is a good approximation of the slope when h is small. If you choose h = 0.01, you have the following approx- imation for the slope of the sine curve at x. N m= sin(x + 0.01) - sin x 0.01 Use the table feature of a graphing utility to complete the table below. Plot these points and compare your results with part (a). -2 x 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 m c. Algebraic Approach To calculate the slope of the tangent line to f (x) = sin x algebraically, you need two trigonometric limits. (See Questions 1 and 2 below.) sin h Limit 2: lim cosh - 1 Limit 1: 10% h - = 0 h Use these limits to find a formula for the slope of the sine curve