# Exploring Variable Slopes in Advanced Functions

• MartynaJ
In summary, the conversation discusses the concept of average and instantaneous rate of change in a function. The speaker attempts to solve a problem using a straight line with a slope of 5 for an interval, but struggles to understand how the same function can have a different slope for a different interval. The other speaker explains that average rate of change is the average slope over a specified interval and gives an example of how the average slope can be 5 for an interval with a line that is flat on one part and has a slope of 10 on another part. The concept of instantaneous rate of change is also briefly discussed.
MartynaJ
Homework Statement
Sketch a possible graph for the function that satisfies all of the following criteria and justify
your sketch by clearly labelling each of the given information.
□ Average rate of change = 5 for t belongs to [1, 5]
□ Average rate of change = -20 for t belongs to [1,10]
□ Average rate of change = 0 for t belongs to [2 , 9]
□ Instantaneous rate of change = -10 at t = 2
□ Instantaneous rate of change = 0 at t =3
Relevant Equations
above please
So I attempted this problem and to satisfy the first condition (for t in the range of [1, 5]), I drew the straight line that has a slope of 5 (i.e. f(x)=5x). I just don't understand how I can have the same function with a different slope (average rate of change) for the interval [1,10] or for [2 , 9]... Any help please!

As for the first statement it means, say f(t)
$$f(5)-f(1)=5(5-1)=20$$ and so on.

WWGD, Delta2 and DaveE
By "average rate of change" they mean the average slope over the specified interval. So, for example, you can have an average slope of 5 over [1, 5] with a line that is flat on [1, 3.5] and has a slope of 10 on [3.5, 5].

Frankly I'm not sure what "instantaneous rate of change" means. I would guess it means "constant rate of change" or "rate of change at each point".

DaveE said:
you can have an average slope of 5 over [1, 5] with a line that is flat on [1, 3.5] and has a slope of 10 on [3.5, 5].
Wouldn't that give an overall change of 15 instead of 20?
DaveE said:
what "instantaneous rate of change" means
It just means the slope at that point.

haruspex said:
Wouldn't that give an overall change of 15 instead of 20?
Oops! Yes. I guess I'm better at math than arithmetic, LOL. I didn't split the interval in half as I intended.

haruspex said:
It just means the slope at that point.
Yes, I missed that it was defined at a point not the whole interval.

## 1. What are advanced functions in graphing?

Advanced functions in graphing refer to mathematical equations or functions that involve more complex operations and concepts, such as logarithms, trigonometry, and exponential functions. These functions often have more intricate and varied graphs compared to basic linear or quadratic functions.

## 2. How do I graph advanced functions?

To graph advanced functions, you will need to use a graphing calculator or a computer program that can plot the function's points and create a visual representation of the graph. You can also manually plot points by substituting different values for the independent variable and then connecting the points to create a curve or line.

## 3. What is the importance of graphing advanced functions?

Graphing advanced functions allows us to visualize and understand complex mathematical concepts and relationships. It also helps in solving equations and making predictions based on the behavior of the function's graph.

## 4. What are some common advanced functions used in science?

In science, some common advanced functions used in graphing include logarithmic functions, exponential functions, and trigonometric functions. These functions are often used to model real-world phenomena and make predictions about their behavior.

## 5. How can I determine the domain and range of an advanced function?

The domain of an advanced function is the set of all possible input values (independent variable) for which the function is defined. The range is the set of all possible output values (dependent variable) that the function can produce. To determine the domain and range, you can analyze the behavior of the function's graph and identify any restrictions or patterns in the input and output values.

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