Derivative of a Function: Understanding Slope and Continuity

In summary, the conversation discusses a problem statement and relevant equations for a question related to number 27. The teacher's answer is provided, explaining the connection between the slope and a point on the graph. The person asking the question initially did not understand but later gained understanding. The conversation also includes a request for further clarification on what it means for a function to be continuous. The summary concludes that a function is continuous when there are no breaks or holes, and provides an explanation for how to determine the constants in the given equations.
  • #1
grace77
43
0
Problem statement

ImageUploadedByPhysics Forums1394018429.382539.jpg

My question is for number 27.
Revelant equation
None

Attempt at a solution

I'm not sure where to start.
ImageUploadedByPhysics Forums1394018545.381550.jpg
ImageUploadedByPhysics Forums1394018553.886597.jpg


This is my teachers answer. I understand how the slope is 1 for x greater than -1 and that it is -2 at x greater than -1 and that there is a point at (0,-1) but I don't understand how they connect to form that final pic. I think I'm missing something ,can someone help me?
 
Physics news on Phys.org
  • #2
Never mind I think I understand it now. I think I was just confused by the previous picture leading up to the answer.
 
  • #3
grace77 said:
Never mind I think I understand it now. I think I was just confused by the previous picture leading up to the answer.
Maybe someone could elaborate on this for me? Thanks
 
  • #4
grace77 said:
Maybe someone could elaborate on this for me? Thanks

What does it mean for a function to be continuous ?
 
  • #5
SammyS said:
What does it mean for a function to be continuous ?

That there are no breaks or holes
 
  • #6
If f'(x)= 1, for x< -1, then f(x)= x+ C1 for some constant C1, for x< -1, so the graph is a straight line with slope 1.

If f'(x)= -2, for x> -1, then f(x)= -2x+ C2 for some constant C2, for x> -1, so the graph is a straight line with slope -2.

Since f is continuous, the two lines must meet at x= -1. That means that -1+ C1= -2(-1)+ C2.

That, together with f(0)= C2= -1 is sufficient to determine both C2 and C1.
 

FAQ: Derivative of a Function: Understanding Slope and Continuity

What is the derivative of a function?

The derivative of a function is the rate of change of the function at a specific point. It represents the slope of the tangent line to the function at that point.

How is the derivative of a function calculated?

The derivative of a function is calculated using the limit definition of a derivative, which involves finding the slope of a tangent line as the change in input approaches zero. Alternatively, it can also be calculated using differentiation rules and formulas.

What is the purpose of finding the derivative of a function?

The derivative of a function is useful in many applications, such as in physics, engineering, and economics. It helps to analyze the rate of change of a quantity and predict its behavior over time. It is also used to find maximum and minimum values of a function.

What are the different types of derivatives?

The most common types of derivatives are first-order derivatives, which represent the slope of a function, and second-order derivatives, which represent the concavity of a function. Other types include partial derivatives, which involve multiple variables, and higher-order derivatives, which represent the rate of change of the rate of change of a function.

Can every function have a derivative?

No, not every function has a derivative. A function must be continuous and have a defined slope at every point in order for it to have a derivative. Discontinuous functions or functions with sharp corners or cusps do not have derivatives at those points.

Similar threads

Can you clarify your understanding for parts (a) and (b)?
Replies
11
Views
2K
f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε
Replies
7
Views
873
Can this function still be a constant function?
Replies
11
Views
877
Find Values of Osculating Circle Tangent to a Parabola
Replies
4
Views
838
The relation between the normal and the slope of a cylindrical curve
Replies
2
Views
1K
Determine a function and derivative from a given graph
Replies
8
Views
3K
Study Function: Continuity, Derivatives & Differentiability
Replies
6
Views
1K
Are Two Random Numbers More Likely to Have a Quotient Closer to an Odd Integer?
Replies
23
Views
2K
Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1
Replies
19
Views
973
Find nth derivative of e^m(arcsinx) where x=0
Replies
7
Views
2K

Hot Threads

Recent Insights

Back
Top