Gradients and areas under graphs are essential concepts in mathematics that are often used in calculus, physics, engineering, and economics. These concepts are significant in understanding the behavior and relationship between two variables.
Read on to find out what gradients and areas under graphs are, why they matter, and how they are used in practical applications. If you wonder how to find the gradient of a line with an equation, I wrote a whole article that I encourage you to check out.
What Is Gradient?
Gradients refer to the slope of a curve at a particular point. The slope of a line is constant, but the slope of a curve changes at different points along the curve. To calculate the gradient of a curve, we use the formula dy/dx, where dy represents the change in the y-coordinate, and dx represents the change in the x-coordinate.
The gradient at a specific point on a curve represents the rate of change of the curve at that point. The concept of gradient is applicable in physics, where it’s used to determine the velocity of an object at a specific point, among other applications.
To explore more gradient, I encourage you to head over to Khan Academy or watch this video or the video below.
What Is The Area Under a Graph?
The area under a graph refers to the region between the curve and the x-axis. Calculating the area under a graph is useful in determining the total quantity of a function. In physics, the area under a speed-time graph is used to calculate the distance traveled by an object.
In calculus, the area under a curve visually represents an integral. You can interpret the area under a curve as an accumulated amount of whatever the function is modeling. Integrals are used to find the area between a curve and a given axis.
Whether it is finding the area under a velocity-time graph to determine displacement or the area under a population growth graph, areas under graphs play an essential role in physics, economics, and other fields. The area under a curve may also represent the total change in a quantity over a specific time period.
If you prefer to watch a video, I encourage you to watch the video below.
Is The Area Under The Graph Same As the Gradient?
The gradient of the graph is acceleration or the rate of change of speed, whereas the area under the graph represents the distance. In other words, gradients can help us optimize functions and describe changes in quantities over time, while areas under graphs can represent the total change in a quantity over a specific time period.
Gradients and areas under graphs are also applicable in economics and other fields. The gradient can be used to determine the elasticity of demand, which refers to the responsiveness of the quantity demanded of a commodity to a change in the price of the commodity. Furthermore, the area under a demand curve represents the total revenue generated by the commodity.
Practical Applications Of Gradients And Areas Under Graphs
A common application of gradients and areas under graphs is optimization. In optimization, we look for the maximum or minimum value of a function. The gradient of the function represents the rate of change of the function, and we can use it to determine the direction of the steepest ascent or descent. The area under a curve can be used to find the maximum or minimum value of a function by calculating the total quantity of the function.
Another application of gradient and the areas under the graph is in economics; the slope of a demand curve represents the willingness of consumers to pay a marginal price for goods or services. At the same time, the area under that same demand curve represents the consumer surplus of that product or service.
Furthermore, gradients, or the rate of change of a function, are essential in calculus. Particularly, gradients relate to derivatives, which measure the instantaneous rate of change of a function at a specific point. This concept is useful in many areas of science and engineering, such as physics, where gradients are used to describe the rate of change of temperature or pressure.
Similarly, in physics, the force acting on an object can be determined by finding the slope of a position-time graph, while the work done by that force is represented by the area under the same graph.
What to read next:
- Distance Time Graph: Definition, interpretation, and Benefits!
- What is m in y=mx+c: Understanding the role of ‘m’ in linear equations!
- The Easiest Way to Find the Equation of a Straight Line Given Two Points.
Wrapping Up
Gradients and areas under graphs are essential concepts applicable in various fields, including physics, economics, engineering, finance, and optimization.
For instance, in finance, gradients can help determine the optimal investment strategy to maximize profit while minimizing risk.