Linear Approximation

Front

What is Linear Approximation?

Back

Linear approximation is a method of estimating the value of a function near a given point using the tangent line at that point.

Front

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

Back

Front

How do you find the slope of the tangent line at a point a for a function f(x)?

Back

The slope of the tangent line at point a is given by the derivative, f'(a).

Front

What is the first step in using linear approximation to estimate f(x) at a point near a?

Back

The first step is to find the value of f(a).

Front

What is the significance of the derivative in linear approximation?

Back

The derivative at a point gives the slope of the tangent line, which is used to create the linear approximation.

Front

If f(4) = 1 and f'(4) = 3, what is the linear approximation of f(4.5)?

Back

Using linear approximation, f(4.5) ≈ f(4) + f'(4)(4.5 - 4) = 1 + 3(0.5) = 2.5.

Front

What is the purpose of linear approximation in calculus?

Back

Linear approximation is used to estimate the values of functions that are difficult to compute directly.

Front

What is the relationship between linear approximation and tangent lines?

Back

Linear approximation uses the tangent line at a point to estimate the value of the function near that point.

Front

Back

You can use the linear approximation of the cube root function near a known value.

Front

What is the second derivative test and how does it relate to linear approximation?

Back

The second derivative test is not needed for linear approximation; it is used to determine concavity and inflection points.