What is the equation of the line tangent to # f(x)=sqrt(x^2+3x+6) # at # x=-1 #?

Answer 1

#y-2=1/4(x+1)#

We can find the point of tangency:

#f(-1)=sqrt(1-3+6)=sqrt4=2#
The function passes through the point #(-1,2)#.
Now, all we need to do is find the slope of the tangent line, which equals #f'(-1)#.
To find #f'(x)#, write the function using exponential powers.
#f(x)=(x^2+3x+6)^(1/2)#

Now, differentiate using the chain rule.

#f'(x)=1/2(x^2+3x+6)^(-1/2)*d/dx(x^2+3x+6)#
#f'(x)=1/(2(x^2+3x+6)^(1/2))*(2x+3)#
#f'(x)=(2x+3)/(2sqrt(x^2+3x+6))#

The slope of the tangent line is

#f'(-1)=(-2+3)/(2sqrt(1-3+6))=1/(2sqrt4)=1/4#
The equation of the line passing through #(-1,2)# with a slope of #1/4#, written in point-slope form, is
#y-2=1/4(x+1)#
Graphed are the function and its tangent line at #x=-1#:

graph{(y-sqrt(x^2+3x+6))(y-2-1/4(x+1))=0 [-10.625, 9.375, -1.72, 8.28]}

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Answer 2

To find the equation of the line tangent to the function f(x) = √(x^2 + 3x + 6) at x = -1, we need to find the derivative of the function and evaluate it at x = -1.

The derivative of f(x) = √(x^2 + 3x + 6) can be found using the chain rule.

f'(x) = (1/2) * (x^2 + 3x + 6)^(-1/2) * (2x + 3)

Evaluating f'(x) at x = -1:

f'(-1) = (1/2) * ((-1)^2 + 3(-1) + 6)^(-1/2) * (2(-1) + 3)

Simplifying:

f'(-1) = (1/2) * (4)^(-1/2) * (1)

f'(-1) = 1/4

Therefore, the slope of the tangent line at x = -1 is 1/4.

Using the point-slope form of a line, where the slope is 1/4 and the point is (-1, f(-1)), we can find the equation of the tangent line.

Substituting x = -1 into the original function:

f(-1) = √((-1)^2 + 3(-1) + 6)

f(-1) = √(1 - 3 + 6)

f(-1) = √4

f(-1) = 2

Therefore, the point on the tangent line is (-1, 2).

Using the point-slope form:

y - y1 = m(x - x1)

y - 2 = (1/4)(x - (-1))

Simplifying:

y - 2 = (1/4)(x + 1)

y - 2 = (1/4)x + 1/4

y = (1/4)x + 9/4

Hence, the equation of the line tangent to f(x) = √(x^2 + 3x + 6) at x = -1 is y = (1/4)x + 9/4.

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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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