Whether the problem involves a baseball, a rocket or a wayward car, you've been given an equation for velocity and asked to find its maximum -- and perhaps the time at which that maximum occurs. If you've taken calculus, you'll find the solution to this problem quickly and simply. But even if algebra is as much math as you've taken, you can still find the answer through your calculator.

## Calculus

Take the derivative of the velocity equation with respect to time. This derivative is the equation for acceleration. For example, if the equation for velocity were v=3sin(t), where t is time, the equation for acceleration would be a=3cos(t).

Set the acceleration equation equal to zero and solve for time. More than one solution may exist; that's OK. Remember acceleration is the slope of the velocity equation; the derivative is just the slope of the original line. When the slope is equal to zero, the line is horizontal. This occurs at an extremum: a maximum or a minimum. In the example, a=3cos(t)=0 when t=pi/2 and t=(3pi)/2.

Test each solution to determine whether it is a maximum or a minimum. Choose a point just to the left of the extremum and just to the right. If acceleration is negative to the left and positive to the right, the point is a minimum velocity. If acceleration is positive to the left and negative to the right, the point is a maximum velocity. In the example, a=3cos(t) is positive just before t=pi/2 and negative just after, so it is a maximum; however, (3pi)/2 is a minimum because a=3cos(t) is negative just before (3pi)/2 and positive just after.

If you find more than one maximum, simply plug in times to the original velocity equation to compare the velocities at those extrema. Whichever velocity is larger is the absolute maximum.

## Calculator

Press the "Y=" button and enter the velocity equation.

Graph the function. Just by looking at the graph, you should be able to estimate where the maximum is.

Press "2nd," "Calc," "Max." Use the arrow buttons to move along the graph just to the left of the maximum and press enter. Arrow just to the right of the maximum, and again press "Enter." Arrow between those points and enter your best guess of the position of the maximum.

Record the time (x-value) and velocity (y-value) of the calculator's more precise solution of the maximum.

If the original velocity equation involves a sine or cosine, watch out for times that the calculator reports that involve many decimal places. Your real answer for time may likely involve pi. Divide the decimal time by pi. If the quotient is close to a fraction, it likely is that fraction, rounded to a decimal by the calculator. Go back to the graph, press "Trace," and enter the exact fraction -- including the pi button on your calculator. If you get the same maximum that the calculator found originally, then the maximum does indeed occur at the fractional multiple of pi.