What is the equation of the line tangent to # f(x)=sqrt(x^2+3x+6) # at # x=-1 #?
We can find the point of tangency:
Now, differentiate using the chain rule.
The slope of the tangent line is
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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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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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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