Linear Approximation

Front

What is the purpose of linear approximation in calculus?

Back

Linear approximation is used to estimate 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

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

Front

What are the steps to find the equation of the tangent line to a curve at a point (a, f(a))?

Back

1. Find f(a). 2. Find f'(a). 3. Use the point-slope form of the line: y - f(a) = f'(a)(x - a).

Front

What does it mean if a function is concave up in relation to linear approximation?

Back

If a function is concave up, linear approximations will underestimate the actual function values.

Front

What does it mean if a function is concave down in relation to linear approximation?

Back

If a function is concave down, linear approximations will overestimate the actual function values.

Front

How do you determine if a linear approximation will overestimate or underestimate a function?

Back

Check the concavity of the function: if it's concave up, it underestimates; if it's concave down, it overestimates.

Front

What is the second derivative test used for in the context of linear approximation?

Back

The second derivative test helps determine the concavity of the function, which affects whether linear approximations will overestimate or underestimate.

Front

Given f(x) = 3x^2, what is the linear approximation at x = 1?

Back

L(x) = 6(x - 1) + 3 = 6x - 3.

Front

What is the significance of the point (c, f(c)) in linear approximation?

Back

It is the point at which the linear approximation is centered and where the tangent line touches the curve.

Front

What is the relationship between the first derivative and the slope of the tangent line?

Back

The first derivative f'(c) at a point c gives the slope of the tangent line to the curve at that point.