# I think linear approximation? (square root, tangent, e^x)

• meredith
In summary, the conversation discussed finding the value of f(x) = sqrt(e^x+3) at x=0.08 using linear approximation and the derivative at x=0. The correct answer was found to be 2.04.
meredith

## Homework Statement

the value of f(x) = (sqrrt e^x +3) at x=0.08 obtained from the tangent to the graph at x=0 is...?

## The Attempt at a Solution

i used linear approximation.
(sqrrt e^o +3) + (1/2(sqrrt3+e^0)(0.08)
i got an answer but i know its wrong. i got like 1.72 or something.
did i do it all wrong?

Your use of parentheses doesn't make sense. Do you mean

$$f(x) = \sqrt{e^x+3}$$

$$f(x) = \sqrt{e^x} + 3$$

$$f(x) = \left(\sqrt e\right)^{\;x + 3}$$

$$f(x) =\left(\sqrt e\right)^{\;x} + 3$$

Or even yet something else?

D H said:
Your use of parentheses doesn't make sense. Do you mean

$$f(x) = \sqrt{e^x+3}$$

$$f(x) = \sqrt{e^x} + 3$$

$$f(x) = {\sqrt e}^{x + 3}$$

$$f(x) = {\sqrt e}^x + 3$$

Or even yet something else?

yes i meant the first one sorry i don't know how to do that stuff!

You could have written it as f(x)=sqrt(e^x+3) and that would have been fine.
f(x)=(sqrt e^x+3) was pretty much meaningless.

What is the derivative of f(x) at x=0?

D H said:
You could have written it as f(x)=sqrt(e^x+3) and that would have been fine.
f(x)=(sqrt e^x+3) was pretty much meaningless.

What is the derivative of f(x) at x=0?

would that be 1/2sqrrt(1+3) = 1/4?
so then i multiply that by 0.08
ok i got it thanks!

## 1. What is linear approximation?

Linear approximation is a method used in calculus to approximate the value of a function at a certain point by using the tangent line at that point. It is a useful tool for estimating values of functions that are difficult to calculate directly.

## 2. How is linear approximation used for square root functions?

For square root functions, linear approximation involves finding the equation of the tangent line at a certain point on the function and using that line to approximate the value of the function at that point. This is helpful when finding the square root of a number with a non-perfect square, as it provides a close estimate.

## 3. Can linear approximation be used for tangent functions?

Yes, linear approximation can be used for tangent functions. In this case, the tangent line is used to approximate the value of the function at a certain point. This is particularly useful for finding the value of tangent functions at angles that are not easily calculable.

## 4. How is linear approximation used for e^x functions?

For e^x functions, linear approximation involves using the tangent line at a point to approximate the value of the function at that point. This is helpful when trying to find the value of e^x at a non-integer value of x, as it provides a close estimate.

## 5. What are the limitations of linear approximation?

Linear approximation is only accurate for small intervals around the point of approximation. The further away the point is from the point of approximation, the less accurate the estimate will be. Additionally, it can only be used for functions that are differentiable at the point of approximation.

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