How do you find the Linear Approximation at x=0 of #y=sqrt(3+3x)#?

Answer 1

I got: #y=sqrt(3)/2x+sqrt(3)#

The Linear Approximation should be a line that can substitute (in a narrow interval) your curve.
To find the equation of this line we need to find the slope #m# in the specific point of coordinates #(x_0,y_0)# and use the general expression for a line as:

#y-y_0=m(x-x_0)#

The slope can be found deriving our function and evaluating the derivative at #x=0#;
#y'=1/(2sqrt(3+3x))*3=3/(2sqrt(3+3x)#

At #x=0#

#y'(0)=3/(2sqrt(3))=3/(2sqrt(3))*(sqrt(3))/(sqrt(3))=sqrt(3)/2#

This will be the slope #m# of our line approximating the original curve at #x=0#.
The coordinate #y# of the specific point can be found substituting #x=0# into our original function writing:

#y(0)=sqrt(3+0)=sqrt(3)#

So, the line through our point and having slope #m# will then be:

#y-sqrt(3)=sqrt(3)/2(x-0)#
Or
#y=sqrt(3)/2x+sqrt(3)#

Graphically:

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Answer 2

The linear approximation at ( x = 0 ) of ( y = \sqrt{3 + 3x} ) is given by ( L(x) = f'(0)(x - 0) + f(0) ), where ( f(x) = \sqrt{3 + 3x} ).

First, find ( f'(x) ): [ f'(x) = \frac{d}{dx} \sqrt{3 + 3x} ] [ = \frac{1}{2\sqrt{3 + 3x}} \cdot 3 ] [ = \frac{3}{2\sqrt{3 + 3x}} ]

Next, evaluate ( f'(0) ): [ f'(0) = \frac{3}{2\sqrt{3}} ] [ = \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} ] [ = \frac{3\sqrt{3}}{6} ] [ = \frac{\sqrt{3}}{2} ]

Now, find ( f(0) ): [ f(0) = \sqrt{3 + 3(0)} ] [ = \sqrt{3} ]

Therefore, the linear approximation at ( x = 0 ) of ( y = \sqrt{3 + 3x} ) is: [ L(x) = \frac{\sqrt{3}}{2} \cdot x + \sqrt{3} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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