Even Differentiable Functions and Linearization

In summary, the conversation includes discussing the special properties of even differentiable functions and finding the linearization of a function with a definite integral. The first question is asking for reasons behind the statement that every even differentiable function ends with an odd function, while the second question involves understanding the concept of linearization with a definite integral. The conversation also briefly touches on the topic of natural logarithms and proving that the function f(x)=x-lnx is increasing for x>1. The conversation ends with a request for help and a response that new questions should be posted as a separate post.
  • #1
n310
4
0
Is there anything special about even differentiable function of x? Give reasons behind your answer.
and
Find the linearization of
g(x)= 3+ ∫sec(t-1)dt at x=-1
It is a definite intergral going from 1 to x^2.. a=1 b=x^2

I understand how to do regular linearization problems but with this having a definite integral I am having a hard time.
With the first question the most I can get out of it is that every even differentiable function ends with an odd function. In order words its symmetric to the y-axis. However, I need more reasons.
I would appreciate the help as soon as possible.
Thank You
 
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  • #2
I have no idea what you mean by "every even differentiable ends with an odd function". In what sense does one function "end" with another?

It is true that the graph of an even function is symmetric to the y-axis. What does that tell you about its derivative (the slope of the tangent line to the graph) at x= 0?

The "linearization" of a function, f(x), at x= a, is f(a)+ f'(a)(x- a). Here they are referring to the linearization at x= -1 which is f(-1)+ f'(-1)x.
 
  • #3
I know the formula for Linearization. I just did not know how to solve for it considering it had a definite integral.
I am confused with the first question that's why I asked for more input.
Thank You for the reply :)
 
  • #4
ok, well g is a function of x, so u will be linearising w.r.t. x, and x appears in the upper limit of the definite integral

can you try and differentiate the integral?

though a little sloppy, it may help to assume you know the indefinite integral, call it F(t), then apply the limits and differentiate
 
  • #5
Natural Logarithms

a) Prove that f(x)=x-ln x is increasing from x>1.
b) Using part (a), show that lnx < x if x>1

I don't know how to go about this to start it off, I would appreciate the help, and a few steps to get me started asap. Thank You
 
  • #6
thank you, I got a bit of help from my classmates, took a while but I got there..
 
  • #7
no worries, if you have new questions, please make a new post
 

1. What is the definition of an even differentiable function?

An even differentiable function is a function that is symmetric about the y-axis, meaning that the value of the function at x is equal to the value of the function at -x. Additionally, an even differentiable function must have a continuous derivative at all points.

2. How is a linearization of a function defined?

A linearization of a function is a linear approximation of the function near a given point. It is the best straight-line approximation to the function at that point, and can be found using the tangent line to the function at that point.

3. What is the purpose of linearization in calculus?

The purpose of linearization in calculus is to approximate a complicated function with a simpler linear function, making it easier to analyze and calculate values of the function at specific points. This can also be useful in finding approximate solutions to equations involving the function.

4. Can all functions be linearized?

No, not all functions can be linearized. A function can only be linearized if it is differentiable at the point of interest. If a function is not differentiable at a point, it cannot be approximated by a linear function at that point.

5. How can linearization be used in real-world applications?

Linearization can be used to approximate and analyze complex functions in various fields such as physics, engineering, and economics. For example, in physics, linearization can be used to approximate the motion of a pendulum or a spring. In economics, it can be used to analyze the relationship between variables such as supply and demand.

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