Linear Approximations

Front

What is the definition of linear approximation?

Back

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

Front

What is the formula for linear approximation?

Back

The formula for linear approximation is: \( f(x) \approx f(a) + f'(a)(x - a) \) where \( a \) is the point of tangency.

Front

How do you find the derivative of a function?

Back

To find the derivative of a function, apply the rules of differentiation (power rule, product rule, quotient rule, chain rule) to obtain \( f'(x) \).

Front

What does it mean for a function to be differentiable at a point?

Back

A function is differentiable at a point if it has a defined derivative at that point, meaning the function is smooth and has no sharp corners or discontinuities.

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 approximate the function's value near that point.

Front

Estimate \( f(4.8) \) for \( f(x) \) given \( f(5) = 3 \) and \( f'(5) = 4 \).

Back

Using linear approximation: \( f(4.8) \approx f(5) + f'(5)(4.8 - 5) = 3 + 4(-0.2) = 3 - 0.8 = 2.2 \).

Front

Use linear approximation to estimate \( (30)^{-2/5} \) given \( f(x) = x^{-2/5} \) at \( x = 32 \).

Back

Using linear approximation: \( f(30) \approx f(32) + f'(32)(30 - 32) \) gives an estimate of \( \frac{41}{160} \).

Front

Estimate \( f(2.5) \) for \( f(x) = x^3 - x \) using linear approximation at \( x = 2 \).

Back

Using linear approximation: \( f(2.5) \approx f(2) + f'(2)(2.5 - 2) = 11.5 \).

Front

What is the tangent line approximation?

Back

The tangent line approximation is the linear function that best approximates a function near a given point, represented as \( y = f(a) + f'(a)(x - a) \).

Front

How do you apply linear approximation to estimate cube roots?

Back

To estimate cube roots using linear approximation, find the derivative of \( f(x) = x^{1/3} \) and use the tangent line at a nearby point.