Calculus and simple geometry

In summary, increasing the perimeter of a rectangle does not necessarily mean the area will also increase. This can be seen through the example of a degenerate rectangle with two sides of length 3a and the other two with length 0, which has a larger perimeter than the initial square but still has an area of 0. This concept can also be demonstrated through a family of rectangles with varying lengths, where the area and perimeter do not necessarily increase at the same rate.
  • #1
circa415
20
0
If the perimeter of a rectangle increases, does the area necessarily increase?

Can anyone explain this using calculus?
 
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  • #2
No it does not, and you don't need calculus to find the answer:
Let's say you've got a square initially with side "a".
Then, you look at the degenerate rectangle with two sides 3a, and the other two length 0.

The perimeter of the degenerate rectangle is 3a+0+3a+0=6a, that is, greater than your original square's 4a, yet the rectangle's area is zero..
 
  • #3
Think of a rectangle with corners at (0,0), (0,1/x), (x,1/x), (x,0). Draw a picture-one corner at the origin the opposite on the graph 1/x.

Now we get an entire family of rectangles parameterized by x>0. What can you say about the area of these guys? What about the perimiter as x varies?
 

What is Calculus?

Calculus is a branch of mathematics that deals with the study of change and motion. It involves the analysis of functions and their rates of change, as well as the use of limits, derivatives, and integrals to solve problems.

What is the difference between differential and integral calculus?

Differential calculus is concerned with the study of rates of change and slopes of curves, while integral calculus is concerned with the study of areas under curves and the accumulation of quantities over a given interval.

What is the relationship between calculus and simple geometry?

Calculus and simple geometry are closely related, as calculus provides a way to understand the behavior of geometric figures and their properties. Calculus can be used to find the slope of a curve, the area under a curve, and the volume of a three-dimensional shape.

What are some real-life applications of calculus and simple geometry?

Calculus and simple geometry have many practical applications in fields such as physics, engineering, economics, and computer science. They can be used to model and solve problems involving motion, optimization, and geometric shapes in the real world.

Is it necessary to have a strong background in algebra to understand calculus and simple geometry?

Yes, a strong understanding of algebra is necessary for understanding and applying calculus and simple geometry. Many concepts in calculus and geometry involve manipulating algebraic equations and solving for unknown variables.

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