Antidifferentiation Applications

In this topic:

    (1) Finding the demand function given the marginal revenue. A manufacturer estimates that the marginal revenue of a certain commodity is when units are produced. Find the demand function
    
    (2) State the Area as an Antiderivative theorem.
    
    (3) Find the area under the parabola over the interval
    
    (4) A ball is thrown upward with a speed of 48 ft/s from the edge of a cliff 432 feet above the ground. Find its height above the ground seconds later. When does it reach its maximum height? When does it hit the ground?
    

Example (Antidifferentiation Application) Finding the demand function given the marginal revenue. A manufacturer estimates that the marginal revenue of a certain commodity is when units are produced. Find the demand function

    Solution. Since,
    


and because where is the demand function, we must have so that yielding and

Theorem (Area as an Antiderivative)  If is a continuous function such that for all on the closed interval then the area bounded by the curve the -axis, and the vertical lines and viewed as a function of is an antiderivative of on

Example (Area as an Antiderivative) Find the area under the parabola over the interval

    Solution. The area function is given by and we can determine using and so which means Therefore, the area under the curve from is

Example (Antidifferentiation Application) A ball is thrown upward with a speed of 48 ft/s from the edge of a cliff 432 feet above the ground. Find its height above the ground seconds later. When does it reach its maximum height? When does it hit the ground?

    Solution. The motion is vertical and we choose the positive direction to be upward. At time the distance above the ground is and the velocity is decreasing. Therefore the acceleration must be negative Taking the antiderivative, To determine we use the given information of Thus, and so and Therefore, the ball reaches it maximum height at which means Since and using we find that and so Therefore the height function is and so the ball hits the ground when meaning by using the quadratic formula.

• print this page
• pages linking here
• related pages