4.5 Linearization

Front

What is linearization in calculus?

Back

Linearization is the process of approximating a function near a given point using the tangent line at that point.

Front

What is the formula for the linearization of a function y=f(x) near x=a?

Back

L(x) = f(a) + f'(a)(x - a)

Front

How do you find the derivative of a function?

Back

The derivative of a function measures the rate at which the function's value changes as its input changes.

Front

What does the derivative f'(a) represent in the context of linearization?

Back

f'(a) represents the slope of the tangent line to the graph of f(x) at the point x=a.

Front

If f(x) = x^3, what is the linearization of f(x) at x=2?

Back

L(x) = f(2) + f'(2)(x - 2) = 8 + 12(x - 2) = 12x - 16.

Front

What is the purpose of using linearization?

Back

Linearization is used to approximate complex functions with simpler linear functions for easier calculations.

Front

How do you find the equation of the tangent line to a function at a given point?

Back

To find the tangent line, calculate the derivative at the point to get the slope, then use the point-slope form of the line.

Front

What is the differential dy of a function y=f(x)?

Back

The differential dy is given by dy = f'(x)dx, where dx is a small change in x.

Front

If y = 4x^2 + x + 3, what is dy?

Back

dy = (8x + 1)dx.

Front

What is the significance of the point of tangency in linearization?

Back

The point of tangency is where the linear approximation is most accurate, as it is the point where the function and the tangent line touch.