(5) 6. Place the expression you obtained in #5 in Yl in your graphing calculator. Complete the following table: X Yl (5) 7. Using your observations from the table, complete this sentence on your paper: As x gets smaller and smaller, I'1 - - gets closer to (5) 8. Go back to your graph and estimate the value of the slope of the tangent line at (0,0). 9. Compare your answer in #7 with your result from #8. Using your observations from the table, complete this sentence on your paper: (5) The slope of the tangent line to ) = sinx at x =0 is (5) Since you have the slope, you can find the equation of the tangent line at (0.0). The equation of the tangent line to j' = sin x at x = 0 is1 Let's consider the problem that has given rise to the branch of calculus called differential calculus: the tangent problem. This problem relates to finding the slope of the tangent line to a curve at a given point. To understand how this is done we are going to consider the point (0,0) on the graph of (x) = sin x . 1. On graph paper, sketch the graph of y = sin x and draw a tangent line at (0,0). Your scale should allow you to graph points very close to (0,0). Is the slope of this line positive? Negative? Explain your reasoning. 2. On graph paper, sketch the graph of y = sinx . Your scale should allow you to graph points very close to (0,0). Draw the secant line joining (0, 0) and (. sin =3)- Find the slope of this line as a decimal. Is the slope of the tangent line at (0, 0) greater than or less than the slope of this secant line? 3. In the same graph as #2 draw the secant line joining (0, 0) and (2 . sin ) Find the slope of this line as a decimal. Is the slope of the tangent line at (0, 0) greater than or less than the slope of this secant line? 4. In the same graph as #2 draw the secant line joining (0, 0) and sin _ Find the slope of this line as a decimal. Is the slope of the tangent line at (0, 0) greater than or less than the slope of this secant line? 5. Find the slope of the line joining (x, f(x)) and (x +h, f(x+ A)) in terms of the nonzero number h. Taking x =0, verify that /=~ ~ yield the slopes you calculated in part #2 - #4 above