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## What is Calculus?

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**What is Calculus?**• Calculus involves mathematics that deals with rates of change that are not constant. • In Algebra, you work with constant rates of change. • In the formula (rate)(time) = (distance), the rate is a constant rate. There is only one problem . . .**A few things move at a constant velocity, but many don’t.**. .**Here, this bungee jumper experiences acceleration, followed**by deceleration**Differential Equations: The Study of Rates of Change**• In the study of differential equations, we are able to take a rate equation and “solve it”. • To “solve” a differential equation means to write the equation in the form that does not contain rates. • Have you ever heard of “exponential growth”?**Exponential Growth**• Exponential Growth occurs when the rate of growth of some “thing”is directly proportional to the amount of that “thing” present. • Example: Plants in a garden grow exponentially. • A possible equation describing this growth could be • dy/dt = 0.10y where y = the mass of the plant after “t” days and the plant increases in mass approximately 10% (0.10) each day. • What is dy/dt?**What is dy/dt?**• dy/dt is the “Rate of Growth” of the plant measured in mass units per time. • For example if y = 20grams, dy/dt = 0.1(20) = 2 grams of new growth per day. But the next day, the mass of the plant is about 22 grams so, dy/dt = 0.1(22) = 2.2 grams of growth and so on . . . • The growth rate keeps growing! What is the solution?**The Solution**• Using “techniques” from Calculus, we may “solve” the differential equation dy/dt=0.10y to get the equation • y = 10e0.095t where we use y=10 for day 0. • A “picture” is worth a lot here!**How Bad Shocks Affects Car Ride Quality**• We can use Calculus and Differential Equations to actually simulate the ride of a car with bad shock absorbers! • In the suspension system of a car, there are two major components: 1) Springs to cushion the ride. 2) Shocks to “dampen” the bounce.**Vertical Acceleration in a Car Ride When Hitting a Bump**• When you hit a bump while driving a car, there is a lot of “up and down” change in position and acceleration and deceleration occurring. We use dy/dt to represent the change in position with respect to time and we use d2y/dt2 to represent acceleration.**Equation For This System**• The second order differential equation is m* d2y/dt2 + kd*dy/dt + ks*y = 0 where m = mass of rear end of vehicle kd = the damping coefficient due to the shocks ks = the spring coefficient • Note that “damping” is primarily achieved by the shock absorber but additional damping occurs due to frictional heat losses. • After looking up a value for “m” and experimentally determining kd and ks, the equation obtained is 24.2*d2y/dt2 + 400*dy/dt + 2400*y = 0**Comparisonof Solutions Graphed**Good Shocks 24.2*d2y/dt2 + 400*dy/dt + 2400*y = 0 Bad Shocks 24.2*d2y/dt2 + 200*dy/dt + 2400*y = 0 Very Bad Shocks 24.2*d2y/dt2 + 100*dy/dt + 2400*y = 0 Note: For this problem, the “solutions” are obtained “graphically”.**1986 Toyota CelicaSuspension System**Rear Shocks in Good Condition Vertical Displacement of Rear in Feet is Plotted Against Time in Seconds**1986 Toyota CelicaSuspension System**Rear Shocks in BAD Condition Viscosity is 1/2 as Much Vertical Displacement of Rear in Feet is Plotted Against Time in Seconds**1986 Toyota CelicaSuspension System**Rear Shocks in VERY BAD Condition Viscosity is 1/4 as Much Vertical Displacement of Rear in Feet is Plotted Against Time in Seconds