Tangent Lines and Linear Approximations

The video tutorial explains how to find the linear approximation of the function f(x) = √x at x = 4. It involves determining the equation of the tangent line at this point and using it to estimate f(4.1). The process includes finding the point of tangency, calculating the slope of the tangent line using the derivative, and formulating the tangent line equation in slope-intercept form. Finally, the linear approximation is used to estimate the value of the function at x = 4.1, demonstrating its accuracy compared to the exact value.

What is the purpose of finding a linear approximation in this context?

• To find the exact value of the function

• To estimate the function value near a point

• To determine the maximum value of the function

• To calculate the area under the curve

What is the y-coordinate of the point of tangency for f(x) = √x at x = 4?

• 3

• 2

• 1

• 4

Which rule is applied to find the derivative of f(x) = √x?

• Product Rule

• Quotient Rule

• Power Rule

• Chain Rule

What is the slope of the tangent line to f(x) = √x at x = 4?

• 1/5

• 1/4

• 1/3

• 1/2

In point-slope form, what is the equation of the tangent line at x = 4?

• y - 2 = 1/5(x - 4)

• y - 2 = 1/4(x - 4)

• y - 2 = 1/2(x - 4)

• y - 2 = 1/3(x - 4)

What is the slope-intercept form of the tangent line equation derived?

• y = 1/4x + 1

• y = 1/2x + 1

• y = 1/3x + 1

• y = 1/5x + 1

How do you use the linear equation to approximate f(4.1)?

• Substitute x = 4.1 into the linear equation

• Substitute x = 4.1 into the inverse function

• Substitute x = 4.1 into the derivative

• Substitute x = 4.1 into the original function

What is the approximate value of √4.1 using the linear approximation?

• 2.035

• 2.015

• 2.025

• 2.045

Why is the linear approximation considered a good estimate for √4.1?

• It is very close to the true value

• It is greater than the true value

• It is less than the true value

• It is exactly equal to the true value

What is the main advantage of using linear approximation?

• It simplifies complex calculations

• It provides exact solutions

• It increases the function's complexity

• It eliminates the need for derivatives