Understanding Calculus

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  Preface
  1. Why Study
  Calculus
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  Motion
  8. Understanding
  Derivative
  9. Derivative
  Approximations
  10. Integration
  Theory
  11. Understanding
  Integration
  12. Differentials

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

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CHAPTER 6 - Applications of the Derivative

Chapter 6 - Applications of the Derivative

Section 6.2 - Instantaneous Velocity and Acceleration

When an objectís distance changes with time, its velocity is the rate at which the distance is changing with respect to time, while its acceleration is the rate at which the velocity is changing with respect to time. As our time interval goes to zero, the velocity and acceleration of an object take on instantaneous values at a certain moment. These instantaneous rate of changes represent the derivatives with respect to time.

To understand how the derivative relates to a moving object, consider a Porsche that accelerates from rest at a constant rate of15 km/hr/s from a starting point d=0. It continues at this acceleration until its velocity is 160 km/hr after which it stops accelerating and maintains its velocity.

If we freeze the moment when 4 seconds have past then its speedometer will read a velocity of exactly 60 km/hr at that instant only. However at t= 4.1 seconds the velocity will be slightly higher since the car is accelerating. This is why we use Calculus to analyze how these accelerations give rise to a changing velocity that results in a changing distance that is covered.

The graph of its distance from the starting point as a function of time is:

From the graph we can see that at t = 4 seconds the car has covered 33 meters. Since the speedometer at that moment reads 60 km/hr, then we can say that at t= 4 seconds, its velocity is a constant 60 km/hr. At t = 4.1 seconds, the velocity has changed due to the car's acceleration. So the speedometer now reads 61 km/hr.

We can assume that from t= 4.0 seconds to t=4.1 seconds the velocity of the car is a constant 60 km/hr. By definition velocity is the distance covered divide by the time taken to cover the distance or Thus,the distance covered by the car in this small time interval, divided by the time, .1 seconds, will give us 60 km/hr.

Since the velocity of the car is increasing, due to its constant rate of acceleration,the velocity of the Porsche at any instant, t,will be whatever the speedometer reads at that moment.

If we were given the relationship for distance covered as a function of time t, then velocity of the car at any time t can be found by calculating the distance covered over a time interval, :

Since the car is accelerating, its velocity is not constant over the interval . We can assume the velocity is constant over an infinitely small time interval,

Therefore we have to take the limit as goes to zero to find the instantaneous rate of change of distance with respect to time. The instantaneous rate of change of distance will correspond exactly to what the speedometer reads at time t.

From the definition of the derivative:

This leads to the extremely important result:

The velocity at any time t is the instantaneous rate of change of the distance function at a time t. By definition the derivative is the instantaneous rate of change of a function over an infinitely small interval. thus the derivative of the distance function, with respect to time is the velocity function for the object

We now need to derive an expression for acceleration as function of time. In the same way that velocity is the rate of change of distance with respect to time, acceleration is the rate of change of velocity with respect to time.

To find the instantaneous acceleration at any time t, we need to take the limit as goes to zero. Without taking the limit, is a discrete value such that the calculated acceleration is the average acceleration is for that interval. By taking the limit as , we are assuming the acceleration is constant over that time interval.

This proves that acceleration is the derivative of the velocity function with respect to time. Since velocity is the first derivative of the distance function with respect to time, the acceleration function is the second derivative of the distance function. In other words the acceleration function is obtained by differentiating the distance function twice. Our results can be summarized as follows:

Next section -> Section 7.1 - What is a Force?

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