How do you find an equation of the tangent line to the curve #y = arcsin(x/2)# at the point where #x = −sqrt2#?

Answer 1

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

Let #f(x)=arcsin(x/2)#. By the Chain Rule,
#f'(x)=1/sqrt(1-(x/2)^2)*1/2=1/sqrt{4-x^2}#

Thus,

#f(-sqrt(2))=arcsin(-sqrt(2)/2)=-\pi/4# and
#f'(-sqrt(2))=1/sqrt{4-2}=1/sqrt(2)#

The equation of the tangent line is

#y=f(-sqrt(2))+f'(-sqrt(2))*(x-(-sqrt(2)))#
#=-pi/4+1/sqrt(2)(x+sqrt(2))#.
This simplifies to: #y=x/sqrt(2)+1-pi/4#
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Answer 2

To find the equation of the tangent line to the curve y = arcsin(x/2) at the point where x = −sqrt2, we need to find the derivative of the function and evaluate it at x = −sqrt2.

The derivative of y = arcsin(x/2) can be found using the chain rule.

dy/dx = (1/√(1 - (x/2)^2)) * (1/2)

Evaluating the derivative at x = −sqrt2:

dy/dx = (1/√(1 - ((-sqrt2)/2)^2)) * (1/2) = (1/√(1 - 1/2)) * (1/2) = (1/√(1/2)) * (1/2) = (1/√2) * (1/2) = 1/2√2

So, the slope of the tangent line at x = −sqrt2 is 1/2√2.

Using the point-slope form of a line, where (x1, y1) is the given point, and m is the slope:

y - y1 = m(x - x1)

Substituting x1 = −sqrt2, y1 = arcsin((-sqrt2)/2), and m = 1/2√2:

y - arcsin((-sqrt2)/2) = (1/2√2)(x - (-sqrt2))

Simplifying:

y - arcsin((-sqrt2)/2) = (1/2√2)(x + sqrt2)

This is the equation of the tangent line to the curve y = arcsin(x/2) at the point where x = −sqrt2.

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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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