# How Fast is the Area of an Equilateral Triangle Changing?

• Peppy
In summary, the average rate of change is the overall change in a quantity over a given interval, while the instantaneous rate of change is the rate of change at a specific point in time. To find the average rate of change, you divide the change in the output by the change in the input over a given interval. The derivative of a function represents the instantaneous rate of change at a specific point and is often used to calculate rates of change in mathematical models. To find the derivative of a function, you can use the limit definition of the derivative or apply derivative rules based on the type of function. Rates of change are used in many real-life applications, such as calculating the speed of a moving object or predicting changes in stock prices.
Peppy
I need help with the question: The height h of an equilateral triangle is increasing at a rate of 3cm/min. How fast is the area changing when h is 5 cm.

What's the area?What's the height?Can u write the former as a function of the latter?

Daniel.

You want $dA/dt$. Use the chain rule:

$$\frac{dA}{dt} = \frac{dA}{dh} \frac{dh}{dt}$$

Of course, you'll need to work out what the two derivatives on the right are.

## 1. What is the difference between average rate of change and instantaneous rate of change?

The average rate of change is the overall change in a quantity over a given interval, while the instantaneous rate of change is the rate of change at a specific point in time. It is like the difference between average speed and instantaneous speed in physics.

## 2. How do you find the average rate of change of a function?

To find the average rate of change of a function, you divide the change in the output (y) by the change in the input (x) over a given interval. This can be represented by the formula: average rate of change = (f(b) - f(a)) / (b - a), where a and b are the input values at the beginning and end of the interval, and f(x) is the function.

## 3. What is the relationship between the derivative and rate of change?

The derivative of a function represents the instantaneous rate of change of that function at a specific point. In other words, it shows how the output is changing with respect to the input at that point. This is why the derivative is often used to calculate rates of change in mathematical models.

## 4. How do you find the derivative of a function?

To find the derivative of a function, you can use the limit definition of the derivative or apply derivative rules based on the type of function. For example, the derivative of a polynomial function can be found by applying the power rule, while the derivative of a trigonometric function can be found using the chain rule.

## 5. How are rates of change used in real life?

Rates of change are used in many real-life applications, such as calculating the speed of a moving object, determining the rate of growth in a population, or predicting changes in stock prices. In engineering and physics, rates of change are essential in understanding the behavior of systems and designing efficient solutions.

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