# Applications of Differentiation

• BoogieL80
In summary, the conversation discusses finding critical numbers for the function g(t) = sqrt (4-t) on the interval t < 3. The steps involved in finding critical numbers are explained, including squaring both sides of the equation to eliminate the square root. The terms "LHS" and "RHS" are defined and the concept of performing algebraic manipulations on both sides of an equation is emphasized. The conversation also mentions the importance of defining the interval for the problem and clarifies that g(t) becomes complex for t > 4. The conversation ends with the individual realizing their mistake and thanking the other person for their help.

## Homework Statement

Find any critical numbers of the function

## Homework Equations

g(t) = sqrt (4-t) , t < 3

## The Attempt at a Solution

I'm not really sure how to find a critical number when square roots are involved. I do know typically when finding the critical number you find the derivative of the equation and then set that equal to 0. However I ended up with the derivative dx = 4 sqrt (4-t) / 4-t. I tried setting that to 0 but got no where.

Hint: If SQRT(X) = 0, then X = 0 too, right? What can you do to the RHS to get rid of the square root? What do you get on the LHS when you do the same thing to it?

I'm sorry, what does RHS and LHS stand for?

No, that's my bad. I normally would have defined them with their first use if I wasn't sure you knew already.

LHS = left hand side (of the equation)
RHS = right hand side.

When working with any equation, you must do the same thing to the LHS as the RHS as you perform algebraic manipulations.

Okay, I tried squaring both sides and I got 16/(4-t) = 0. Is that t suppose to still be in the problem? I'm guessing that I have to eventually solve for t since the answer is t = 8/3. I'm assuming maybe I've done this problem incorrectly...

BoogieL80 said:
Okay, I tried squaring both sides and I got 16/(4-t) = 0. Is that t suppose to still be in the problem? I'm guessing that I have to eventually solve for t since the answer is t = 8/3. I'm assuming maybe I've done this problem incorrectly...

More likely, I'm not understanding the problem statement. Sorry, exactly how is the term "critical number" defined? I'd started off thinking maxima and minima, but that doesn't appear to be the case. BTW, they should be defining what interval of t they intend for the problem statement...

$$g(t) = \sqrt {4-t}$$

g(t) goes complex for t>4, I believe.

I think I forgot to factor in my process solving. I think I've figured it out. Thank you for your help :)

Hmm... you sure that's the right equation?.
The function $$g(t) = \sqrt {4-t}$$ is monotone decreasing on I:-infinity<t<3 and its derivative is defined and nonzero everywhere on the interval...

## What is differentiation?

Differentiation is a mathematical concept that involves finding the rate of change of a function. It is used to analyze and model relationships between variables in various fields such as physics, economics, and engineering.

## What are the applications of differentiation?

Differentiation has many applications in real-world situations. Some common examples include finding maximum and minimum values, determining the slope of a curve, and calculating rates of change and velocity.

## How is differentiation used in physics?

In physics, differentiation is used to analyze the motion of objects. By differentiating the position function with respect to time, we can find the velocity and acceleration of an object at any given point. This is essential for understanding the behavior of objects in motion.

## What are the different types of differentiation?

The two main types of differentiation are the derivative and the partial derivative. The derivative is used for functions with one independent variable, while the partial derivative is used for functions with multiple independent variables.

## What are some real-world examples of differentiation?

Differentiation is used in many real-world applications, such as determining the maximum profit for a company, optimizing production processes, and modeling population growth. It is also used in fields such as economics, biology, and medicine to analyze and understand various phenomena.

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