Derivatives & L'Hôpital's Rule Explained

In summary, the conversation revolves around the topic of finding the derivative using the definition and L'Hopital's rule. The person was unable to make any progress and was prompted to consider other theorems, specifically the Mean Value Theorem. After some back and forth, the person remembered the Mean Value Theorem and was able to solve the problem.
  • #1
MrNotknowinganything
7
0
Homework Statement
there be a function ##f(x)## continues in the section ##[1,2]## and Shearing in the Section ##(1,2)##. Suppose also that takes place ##\frac{f(2)}{f(1)}=2##. prove thet exists a ##c \in(1,2)## so that ##f(c)=c \cdot f_{(c)}^{\prime}##
My best attempt :
\begin{array}{l}
1<c<2,1 \leq f(x) \leq 2 \\
f(c)=c f^{\prime}(c) \rightarrow c=\frac{f(c)}{f^{\prime}(c)} \\
f(2)=2 f(1)
\end{array}
Relevant Equations
there be a function ##f(x)## continues in the section ##[1,2]## and Shearing in the Section ##(1,2)##. Suppose also that takes place ##\frac{f(2)}{f(1)}=2##. prove thet exists a ##c \in(1,2)## so that ##f(c)=c \cdot f_{(c)}^{\prime}##
Tried to use the information to put it in the definition of derivative and lopital but I couldn't get to anything
 
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  • #2
:welcome:

I guess you have translated this question into English? I think I can work out that "shearing" means "differentiable".

According to the Homework guidelines you must post your best attempt to doing this yourself.
 
  • #3
Tried to use the information to put it in the definition of derivative and lopital but I couldn't get to anything. This is my "attempt "
16424315120934266545030016434446.jpg
 
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Likes PeroK
  • #4
Can you think of any theorems you have learned recently?
 
  • #5
PeroK said:
Can you think of any theorems you have learned recently?
Intermediate value test
 
  • #6
MrNotknowinganything said:
Intermediate value test
Any others? Like one that involves a derivative?
 
  • #7
PeroK said:
Any others? Like one that involves a derivative?
Derivative definition
 
  • #8
MrNotknowinganything said:
Derivative definition
That's not a theorem.
 
  • #9
MrNotknowinganything said:
Derivative definition
PeroK said:
Any others? Like one that involves a derivative?
Lopital rule
 
  • #10
That's for limits. What about the Mean Value Theorem?
 
  • #11
PeroK said:
That's for limits. What about the Mean Value Theorem?
I don't recall maybe I forgot , I'll be sure to check up on it now
 
  • #12
PeroK said:
That's for limits. What about the Mean Value Theorem?
Solved it thank you
 

FAQ: Derivatives & L'Hôpital's Rule Explained

What are derivatives and why are they important in mathematics?

Derivatives are a mathematical concept that represents the rate of change of a function at a specific point. They are important because they allow us to analyze and understand the behavior of functions, such as determining maximum and minimum values, and solving optimization problems.

How is the derivative of a function calculated?

The derivative of a function is calculated using the limit definition, which involves finding the slope of a tangent line to the function at a specific point. This can also be done using various rules, such as the power rule, product rule, and chain rule.

What is L'Hôpital's Rule and when is it used?

L'Hôpital's Rule is a mathematical theorem that provides a method for evaluating indeterminate forms, such as 0/0 or ∞/∞. It states that the limit of a quotient of two functions is equal to the limit of their derivatives, under certain conditions. It is often used in calculus to simplify complex limits and solve problems involving derivatives.

Can L'Hôpital's Rule be applied to all indeterminate forms?

No, L'Hôpital's Rule can only be applied to indeterminate forms of the type 0/0 or ∞/∞. It cannot be used for other indeterminate forms, such as 1^∞ or ∞ - ∞, and it also cannot be used if the limit involves a product or quotient of functions.

Are there any limitations or drawbacks to using L'Hôpital's Rule?

While L'Hôpital's Rule can be a useful tool in solving certain problems, it should not be relied on blindly. It can sometimes give incorrect results if used incorrectly, and it is important to understand the underlying concepts and conditions for its application. Additionally, it may not always provide a solution, as there are some limits that cannot be evaluated using L'Hôpital's Rule.

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