4.8 Practice

Front

What is the formula for the linear approximation of a function f(x) at a point (c, f(c))?

Back

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

Front

What is the first step in finding the equation of the tangent line to a curve at a point (a, f(a))?

Back

Find the value of f(a).

Front

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

Back

The slope of the tangent line to the curve at the point (a, f(a)).

Front

How do you estimate f(2.5) using linear approximation at x = 2 for f(x) = x^3 - x?

Back

Calculate f(2) and f'(2), then use L(2.5) = f(2) + f'(2)(2.5 - 2).

Front

What is the significance of the second derivative f''(a) in relation to tangent lines?

Back

It is not necessary for finding the equation of the tangent line.

Front

If f(5) = 3 and f'(5) = 4, how do you estimate f(4.8) using local linear approximation?

Back

Use f(4.8) ≈ f(5) + f'(5)(4.8 - 5) = 3 + 4(-0.2) = 2.2.

Front

What is the purpose of linear approximation in calculus?

Back

To estimate the value of a function near a point using the tangent line.

Front

What is the relationship between the function value and its derivative at a point for linear approximation?

Back

The function value gives the y-coordinate, and the derivative gives the slope of the tangent line.

Front

How do you write the equation of a tangent line in point-slope form?

Back

T(x) = f(a) + f'(a)(x - a).

Front

What is the linear approximation of f(x) = x^2 at x = 1?

Back

L(x) = f(1) + f'(1)(x - 1) = 1 + 2(x - 1).