# Finding Second and Third Derivative from Graph

• a1234
In summary, the conversation discusses finding g'(2), g''(2), and g'''(4) for a given function f(x). It is determined that g'(2) = f(2) = -2, g''(2) = f'(2) = DNE due to a sharp curve, and g'''(4) = f''(4) = 0 due to the first derivative being a constant and the second derivative being 0.
a1234

## Homework Statement

The problem asks to find g'(2), g''(2), and g'''(4).

## Homework Equations

and attempt at solution[/B]

The derivative of g(x) is just the function f(x). So g'(2) = f(2) = -2.

I'm not sure how to find g''(2) and g'''(4).
I understand that g''(2) is f'(2), but how do I find the derivative of f from this graph? Same question for g'''(4) = f''(4).

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Why don't you simply compute ##g(x)\,##? Just integrate for ##1 \leq x \leq 2## then for ##2 \leq x \leq 3\, , \,3\leq x \leq 4## and ##4\leq x \leq 5\,.## After this you could draw this function and see whether and where it is differentiable and what the values are.

a1234 said:

## Homework Statement

The problem asks to find g'(2), g''(2), and g'''(4).

## Homework Equations

and attempt at solution[/B]

The derivative of g(x) is just the function f(x). So g'(2) = f(2) = -2.

I'm not sure how to find g''(2) and g'''(4).
I understand that g''(2) is f'(2), but how do I find the derivative of f from this graph? Same question for g'''(4) = f''(4).

The graph gives a complete and exact description of the function ##f(x)##. You just need to translate the given information into formulas.

I think I'm getting somewhere...
g''(2) = f'(2) is DNE because of the sharp curve, where the tangent line is a vertical line.
g'''(4) = f''(4) is 0 because the first derivative is a constant and the second is 0.

## 1. What are derivatives?

Derivatives are mathematical concepts that describe the rate of change of a function with respect to its independent variable. They are used to measure how much a function is changing at a specific point.

## 2. How do you find the second and third derivative from a graph?

To find the second derivative, you will need to take the derivative of the first derivative. Similarly, to find the third derivative, you will need to take the derivative of the second derivative. This process can be done using calculus rules and techniques.

## 3. Why is it important to find the second and third derivative from a graph?

The second and third derivatives provide valuable information about the behavior of a function. They can help determine the concavity, inflection points, and local extrema of a function, which are important in understanding the behavior and properties of the function.

## 4. Can you find the second and third derivative from any type of graph?

Yes, it is possible to find the second and third derivative from any type of graph as long as the function is continuous and differentiable. However, some graphs may require more advanced techniques and may not have a well-defined second or third derivative at certain points.

## 5. What are some real-life applications of finding the second and third derivative from a graph?

Finding the second and third derivative can be used in various fields, such as physics, economics, and engineering. For example, in physics, the second derivative of position with respect to time is acceleration, and the third derivative is jerk. In economics, the second derivative of a profit function can help determine the optimal level of production. In engineering, the second derivative can be used to analyze the stability of a system.

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