Video Example " Section... A pulse laser operates by storing charge on a capacitor and releasing it suddenly when the laser is fired. The data in the table describe the charge Q remaining on the capacitor (measured in coulombs) at time t (measured in seconds after the laser is fired). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t = 0.04. (Note: The slope of the tangent line represents the electric current flowing from the capacitor to the laser (measured in amperes).) O 0.00 10 0.02 8.187 0.04 6.702 0.06 5.487 0.08 4.491 0.10 3.677 Solution In the following figure we plot the data and use it to sketch a curve that approximates the graph of the function. (coulombs) 0.02 0.04 0.06 0.08 0:10 (seconds) Given the points P(0.04, 6.702) and R(0.00, 10) on the graph, we find that the slope of the secant line PR, rounded to two decimal places, is as follows. mPR = 10 - 0.00 - 0.04 The following table shows the results of similar calculations for the slopes of other secant lines. m PR (0.00, 10) -82.45 (0.02, 8.187) -74.25 (0.06, 5.487) -60.75 (0.08, 4.491) -55.28 (0.10, 3.677) -50.42 From this table we would expect the slope of the tangent line at t = 0.04 to lie somewhere between -74.25 and -60.75. In fact, the average of the slopes of the two closest secant lines is as follows. (-74.25 - 60.75) = So , by this method , we estimate the slope of the tangent line , rounded to the nearest integer , to be about Another method is to draw an approximation to the tangent line at P and measure the sides of the triangle ABC, as in the figure below. Q (coulombs )