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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 . . .
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A few things move at a constant velocity, but many don’t. . .
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Here is a case of a sharp decrease in velocity.
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Here, this bungee jumper experiences acceleration, followed by deceleration
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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”?
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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?
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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?
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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!
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Mass of a Garden PlantGraph of y = 10e0.095t
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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.
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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.
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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
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Comparison of 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”.
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1986 Toyota CelicaSuspension System
Rear Shocks in Good Condition
Vertical Displacement of Rear in Feet is Plotted Against Time in Seconds
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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
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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
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