# What is the local linearization of #y = sin^-1x # at a=1/4?

We will only keep term of degree 0 and 1.

By signing up, you agree to our Terms of Service and Privacy Policy

To find the local linearization of ( y = \sin^{-1}(x) ) at ( a = \frac{1}{4} ), we'll follow these steps:

- Find the derivative of ( y = \sin^{-1}(x) ) using the chain rule: ( \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}} ).
- Evaluate the derivative at ( x = \frac{1}{4} ) to find the slope of the tangent line: ( \frac{1}{\sqrt{1 - (\frac{1}{4})^2}} = \frac{4}{\sqrt{15}} ).
- Use the point-slope form of a line to find the equation of the tangent line: ( y - y_1 = m(x - x_1) ), where ( (x_1, y_1) = \left(\frac{1}{4}, \sin^{-1}\left(\frac{1}{4}\right)\right) ).
- Substitute the values into the equation to find the local linearization.

The local linearization of ( y = \sin^{-1}(x) ) at ( a = \frac{1}{4} ) is given by:

[ y - \sin^{-1}\left(\frac{1}{4}\right) = \frac{4}{\sqrt{15}}\left(x - \frac{1}{4}\right) ]

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?
- How do you maximize and minimize #f(x,y)=e^-x+e^(-3y)-xy# subject to #x+2y<7#?
- How do you use the linear approximation to #f(x, y)=(5x^2)/(y^2+12)# at (4 ,10) to estimate f(4.1, 9.8)?
- How do you find the linear approximation of the function #g(x)=root5(1+x)# at a=0?
- How do you minimize and maximize #f(x,y)=x-y/(x-y/(x-y))# constrained to #1<yx^2+xy^2<16#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7