Question

Use linear approximation, i.e. the tangent line, to approximate 3.83 as follows: Let f(x) = x³. The equation of the tangent line to f(x) at x = 4 can be written in the form y = mx + b - where m is: and where b is: Using this, we find our approximation for 3.8³ is 0

Answer 1

In linear approximation, the tangent line is used to approximate the value of a function nearby a specific point. The equation of the tangent line to f(x) = x³ at x = 4 is y = 48x - 128. Using this equation, the approximation for 3.83³ is 39.04.

Step-by-step explanation:

In linear approximation, the tangent line to the function at a specific point is used to approximate the value of the function nearby. In this case, the function is f(x) = x³ and we want to approximate 3.83.

First, we find the equation of the tangent line to f(x) at x = 4. The derivative of f(x) is 3x², so at x = 4, the slope of the tangent line (m) is 3*4² = 48.

The equation of the tangent line is y = 48(x-4) + f(4), where f(4) = 4³ = 64. Simplifying the equation, we get y = 48x - 128.

Using this equation, the approximation for 3.83³ is obtained by plugging in x = 3.83 into the equation: y = 48(3.83) - 128 = 39.04.