Understanding Tangent Lines and Derivatives

The video tutorial explains how to find points on the graph of f(x) = 2/x where the tangent line is parallel to the line y = 8/3x + 4. It involves setting the derivative of the function equal to the slope of the given line, solving for x, and then evaluating the function at these x values to find the corresponding y coordinates. The process includes rewriting the function using negative exponents, finding the derivative, and rationalizing the denominator.

What is the slope of the line y = 8/3x + 4?

• 1

• 8/3

• 2

• 4

What is the function f(x) rewritten using negative exponents?

• 2x^1

• 2x^-1

• 2/x^2

• x^2/2

What is the derivative of f(x) = 2/x?

• 2/x^2

• x^2/2

• 2x

• -2/x^2

What equation do we solve to find the x-coordinates where the tangent line is parallel to the given line?

• 2x = 8/3

• x^2 = 8/3

• 2/x = 8/3

• -2/x^2 = 8/3

What are the x-coordinates where the tangent line is parallel to the line y = 8/3x + 4?

• ±2/√3

• ±√3/2

• ±3/4

• ±4/3

How do you find the y-coordinate for x = √3/2?

• Divide x by 2

• Multiply x by 2

• Add 4 to x

• Evaluate f(x) at x = √3/2

What is the y-coordinate for x = √3/2?

• 4/√3

• 2/√3

• √3/4

• 3/4

What is the process called when you multiply the numerator and denominator by the same value to remove a square root from the denominator?

• Expanding

• Factoring

• Simplifying

• Rationalizing

What is the graphical representation of the function and the line used for?

• To solve equations

• To calculate derivatives

• To find the slope

• To show where tangent lines are parallel

What color is used to represent the given line y = 8/3x + 4 in the graph?

• Blue

• Yellow

• Green

• Red