Tangent Line Approximations and Derivatives

The video tutorial explains how to use a tangent line to approximate the square root of 49.1. It begins by introducing the concept of using a tangent line at x = 49, which is close to 49.1 and simplifies perfectly. The tutorial then details finding the point of tangency and calculating the slope using derivatives. With this information, the equation of the tangent line is derived. Finally, the tangent line is used to approximate the square root of 49.1, resulting in a decimal approximation of 7.0071.

Why is x = 49 chosen for the tangent line approximation?

• Because it is an even number

• Because it is a perfect cube

• Because it is close to 49.1 and its square root is a whole number

• Because it is a prime number

What is the main reason the tangent line is a good approximation near x = 49?

• The function is quadratic

• The tangent line closely follows the function near the point of tangency

• The function is linear

• The tangent line is always accurate

What is the y-coordinate of the point of tangency for x = 49?

• 6

• 7

• 8

• 9

How is the derivative of the square root function expressed?

• x to the power of 2

• 1 divided by 2 square root of x

• 1 divided by 2x

• x to the power of 1/2

What is the slope of the tangent line at x = 49?

• 1/7

• 1/14

• 1/28

• 1/21

Which form is used to find the equation of the tangent line?

• Slope-intercept form

• Point-slope form

• Standard form

• Quadratic form

What is the equation of the tangent line derived in the video?

• y = 1/14x + 7/2

• y = 1/7x - 14/2

• y = 1/7x + 14/2

• y = 1/14x - 7/2

How is 49.1 expressed as an improper fraction?

• 491/10

• 49/100

• 49/10

• 491/100

What is the exact approximation for the square root of 49.1 using the tangent line?

• 981/140

• 980/140

• 983/140

• 982/140

What is the rounded decimal approximation of the square root of 49.1?

• 7.0070

• 7.0072

• 7.0073

• 7.0071