Limit value using definition of derivative

In summary, the conversation discusses a limit problem involving a function and its derivative. The correct approach is to find the derivative of the function and substitute the given value to find the limit. The final answer is pi/4 * (√2)/2.
  • #1
karkas
132
1

Homework Statement


I have encountered a limit that I wasn't really sure whether I was solving right, it's limit (a). I thought that I could find it considering a function f(x) (b) , find the derivative of that function and substitute x->1 so that I can find the value of the limit. Is this correct thinking?

Homework Equations



(a) lim(x->1) [ sin(x*pi/4) - sin(pi/4) ] / x-1

(b) f(x) = sin(x *pi/4) and so f ' (x) = pi/4 cos(x* pi/4)

The Attempt at a Solution


That would mean that (a) = f ' (1) = pi/4 * (√2)/2 , right?
 
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  • #2
yes that is right
 
  • #3
Ok, excellent, thanks!
 

What is the definition of derivative?

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

How do you find the limit value using the definition of derivative?

To find the limit value using the definition of derivative, we first find the slope of the secant line between two points on the function. Then, we take the limit of this slope as the two points get closer and closer together. This limit value is equal to the derivative at that point.

What is the significance of the limit value in the definition of derivative?

The limit value in the definition of derivative represents the instantaneous rate of change of a function at a specific point. It allows us to calculate the slope of a curve at any given point, which is useful in many real-world applications.

Can the limit value of a derivative be undefined?

Yes, the limit value of a derivative can be undefined. This can happen when the function is not continuous at that point or when there is a sharp corner or vertical tangent at that point.

How is the concept of limit value used in real-world applications?

The concept of limit value is used in many real-world applications, such as calculating the speed of an object at a specific time, determining the rate of change in a business's profits, and finding the maximum or minimum values of a function. It is also used in physics, engineering, and economics to model and analyze various phenomena.

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