When we estimate distances from velocity data, it is sometimes necessary to use times to, t1, t2, t3, .. . that are not equally spaced. We can still estimate distances using the time periods At; = t; - t;- 1- For example, a space shuttle was launched on a mission, the purpose of which was to install a new motor in a satellite. The table provided gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height, h, above Earth's surface of the space shuttle, 62 seconds . after liftoff. Event Time (s) Velocity (ft/s) Launch Begin roll maneuver 10 185 End roll maneuver 15 319 Throttle to 89% 20 447 Throttle to 67% 32 742 Throttle to 104% 59 1325 Maximum dynamic pressure 62 1445 Solid rocket booster separation 125 4151 Step 1 We can obtain an upper estimate for the height by using the final velocity for each time interval.· The first interval begins at t = 0 and ends at t = 10, so its width is Ati %D

1. Find the equation of the line tangent to the curve y = Vx at the point (4,2). Hint: Write vx = x1/2 and apply the power rule. Include a screenshot or sketch of the graph of the curve and the tangent line together on the same graph. Desmos.com is a nice, free graphing tool. %3D

Determine if the following functions are one-to-one by inspecting their graphs using the Vertical line test (VLT) and the Horizontal Line Test (HLT). You do not need to sketch the graphs.1.) Y=5x-2VLT:HLT:One-to-One: 2.) Y= 1/xVLT:HLT:One-to-One:

3. Find the total shaded area, given that r = 1.4 inches, 6 = 288°, and the small circles (Pac- dots) have radius 0.22 in. Express your answers both as a function of a and as a number rounded to 2 decimal places. Include units with both answers.