Linear Approximation Concepts

This lecture covers linear approximation, focusing on determining whether it results in an overestimate or underestimate. The instructor uses a specific example involving the function f(x) = sqrt(x + 3) to demonstrate the process. The lecture explains how to set up the problem, calculate the linearization, and analyze the results. The key takeaway is understanding how linear approximation works and how to determine the accuracy of the approximation.

What is the primary goal of linear approximation?

• To calculate integrals

• To solve differential equations

• To estimate values using tangent lines

• To find the exact value of a function

In the example problem, what is the function f(x) used?

• f(x) = x^2 + 3

• f(x) = 3x

• f(x) = x + 3

• f(x) = sqrt(x + 3)

What is the value of 'a' in the example problem?

• 2

• 1

• 3.98

• 0.98

What is the derivative of f(x) = sqrt(x + 3)?

• 1/2 * (x + 3)^(-1/2)

• 1/2 * (x + 3)^(1/2)

• 2 * (x + 3)^(-1/2)

• 2 * (x + 3)^(1/2)

How do you determine if the linear approximation is an overestimate?

• If the tangent line is parallel to the curve

• If the tangent line is below the curve

• If the tangent line intersects the curve

• If the tangent line is above the curve

What is the calculated linear approximation value for f(0.98)?

• 1.99

• 2.00

• 1.98

• 1.995

Why is it important to recognize the form of the function in linear approximation?

• To determine the correct point of tangency

• All of the above

• To ensure the correct derivative is used

• To simplify the calculation process