Question

1. Find linearization for each of the following functions (equation of the tangent line) 2. Then use it for approximating the value of the functions at x=1.1.

Answer 1

The question asks for the linearization of a given function to approximate its value at x=1.1. Linearization is found by calculating the function's derivative to find the slope of the tangent line at the given point and then forming the linear equation. This linear approximation is then applied to estimate the function's value near the point.

Step-by-step explanation:

The student's question involves finding the linearization of a function, which is the equation of the tangent line at a given point. This linear approximation is then used to estimate the value of the function near that point, in this case at x=1.1.

To find the linearization, we need to know the function being linearized and its derivative at the point of interest. The general process includes calculating the derivative (slope of the tangent line) of the function at the given point and then using the point-slope form of the linear equation to write the tangent line's equation.

Here are the steps to achieve this:

1. Identify the function and the point where the linearization is needed.
2. Calculate the derivative of the function at that point. This gives the slope m.
3. Use the point-slope form y - y1 = m(x - x1) where (x1, y1) is the point on the function and m is the slope from step 2.
4. Rewrite the equation in slope-intercept form y = mx + b, rounding coefficients to four decimal places.
5. Use this linear equation to approximate the value of the function at x=1.1, which is the linearization's main application.

To demonstrate this, let's assume the function is f(x) and at x=1, f(x) is known and f'(x) is evaluated to be the slope of the tangent. For a function f(x) = x2, the linearization at x=1 would be calculated using f'(x)=2x, and since f'(1)=2, the tangent line at x=1 would have a slope of 2.

The point (1, f(1)) is (1, 1), hence the tangent line equation in point-slope form is y - 1 = 2(x - 1). This simplifies to y = 2x - 1, and to estimate the value at x=1.1, we plug into the linear equation: y ≈ 2(1.1) - 1 = 1.2.

The complete question is: How do you find the linearization of f(x)=x^3 at the point x=2?