Gr 12 Core Maths Rates of Change

Summary

This video explains the concept of rates of change in differential calculus, equating gradients, first derivatives, and rates of change. Using examples like modeling water volume in a dam and calculating the maximum income from sales, it demonstrates how to compute the first derivative, understand the behavior of functions, and apply these principles in real-world scenarios. Key topics include finding the rate of change, determining when a function starts increasing or decreasing, and calculating maximum values. The video emphasizes the importance of understanding units and applying derivative concepts to solve optimization problems.

Takeaways

• 😀 The first derivative of a gradient function is called the rate of change, which is essentially the same as the gradient or first derivative.
• 😀 When you encounter the term 'rate of change', you should immediately recognize that it refers to the first derivative of the function.
• 😀 In the provided example, the rate of change of the volume of water in a dam is found by taking the first derivative of the volume function with respect to time.
• 😀 Units are important! The volume in the problem is given in thousands of cubic meters, and time is measured in hours, so the rate of change units are 'thousands of cubic meters per hour'.
• 😀 When asked about increasing or decreasing volume, you need to solve for when the rate of change becomes positive (increasing) or negative (decreasing).
• 😀 To find when the volume of water starts decreasing, solve when the first derivative (rate of change) becomes less than zero.
• 😀 The first derivative of the volume function shows that the rate of change is not constant—it changes over time.
• 😀 The maximum volume of water in the dam occurs when the rate of change is zero, i.e., the first derivative equals zero.
• 😀 Optimization problems, like finding the maximum or minimum, require you to set the first derivative equal to zero and solve for the variable.
• 😀 In the business example, the number of watches sold each year decreases by one for every four rand increase in price, so a formula for the income from selling watches over time can be derived using this rate of change concept.

Q & A

What is the relationship between gradients, derivatives, and rates of change?

Gradients, derivatives, and rates of change are all essentially the same concept. A gradient is the rate at which a function changes, and this is what the derivative represents. In differential calculus, the first derivative of a function gives the rate of change, so gradients and derivatives are interchangeable terms for rate of change.

What does the first derivative represent in the context of rates of change?

The first derivative of a function represents the instantaneous rate of change of one variable with respect to another. In other words, it describes how the value of a function changes at a specific point in relation to its input.

In the example with the dam, what units are used to measure the volume of water?

The volume of water is measured in thousands of cubic meters. This is important to remember to avoid confusion when calculating or interpreting results.

How do you calculate the rate at which the volume of water changes in the dam example?

To calculate the rate at which the volume of water changes, you need to find the first derivative of the given volume function. For the function V = 2500 + 500T - 1/80 T^2, the first derivative with respect to T is DV/DT = 500 - 1/4 T.

What does the rate of change of the volume being equal to 375 mean in the dam example?

When the rate of change of the volume is 375, it means that the volume of water in the dam is increasing at a rate of 375 thousand cubic meters per hour at that specific point in time.

How do you determine when the volume of water starts to decrease?

To determine when the volume of water starts to decrease, you need to find when the rate of change (the first derivative) becomes negative. In the example, this happens when T exceeds 2000 hours, as calculated from the equation 500 - 1/4 T < 0.

What is the significance of the first derivative being equal to zero when calculating maximum or minimum values?

The first derivative being equal to zero is significant because it represents critical points where a function might have a maximum or minimum value. For the maximum volume of water in the dam, the first derivative equals zero at T = 2000 hours, indicating the maximum point.

What is the maximum volume of water in the dam, according to the given function?

The maximum volume of water in the dam occurs at T = 2000 hours. Substituting this time into the original volume function gives a maximum volume of 502,500 thousand cubic meters, or 502.5 million cubic meters.

In the second example, how is the number of watches sold per year modeled?

The number of watches sold per year is modeled by the equation 40 - X, where X is the number of years since the starting point. For every year that passes, one less watch is sold, as the price increases by 4 units per year.

How do you calculate the annual income from the sale of watches in the second example?

The annual income is calculated by multiplying the number of watches sold (40 - X) by the price per watch. The price increases by 4 units each year, so the price after X years is 144 + 4X. The total income is therefore (40 - X) * (144 + 4X).