Tangent Lines and Derivatives

The video tutorial explains how to find the derivative of a function using the quotient rule and how to determine the equation of the tangent line at a specific point. The function given is a quotient of trigonometric expressions, and the derivative is found by applying the quotient rule. The slope of the tangent line is calculated at x = π/3, using trigonometric identities and simplification. Finally, the equation of the tangent line is derived using the point-slope form, providing both an exact and an approximate solution.

What is the main goal of the problem presented in the video?

• To find the integral of a function

• To determine the derivative and tangent line equation

• To solve a system of equations

• To calculate the area under a curve

Which rule is applied to find the derivative of the given function?

• Power Rule

• Quotient Rule

• Chain Rule

• Product Rule

What is the derivative of 3 cosine x?

• -3 cosine x

• 3 cosine x

• -3 sine x

• 3 sine x

What trigonometric values are used to find the slope of the tangent line at x = π/3?

• Sine π/3 = √3/2, Cosine π/3 = 1/2

• Sine π/3 = 0, Cosine π/3 = 1

• Sine π/3 = 1/2, Cosine π/3 = √3/2

• Sine π/3 = √2/2, Cosine π/3 = √2/2

What is the approximate decimal value of the slope of the tangent line?

• -1.764

• 0.764

• 1.764

• -0.764

How do you find the y-coordinate of the point of tangency?

• By differentiating the function

• By substituting x = π/3 into the original function

• By integrating the function

• By solving a quadratic equation

What is the y-coordinate of the point of tangency when x = π/3?

• 3/(4√3 + 1)

• 3/(2√3 + 1)

• 3/(√3 + 1)

• 3/(4 + √3)

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

• Slope-Intercept Form

• Point-Slope Form

• Standard Form

• Vertex Form

What is the slope of the tangent line in slope-intercept form?

• -48/(48 + 8√3 + 1)

• 48π/(3x + 48)

• 3/(4√3 + 1)

• -12/(4√3 + 1)

What is the y-intercept of the tangent line in slope-intercept form?

• 48π/(3x + 48)

• -48/(48 + 8√3 + 1)

• 3/(4√3 + 1)

• 12/(4√3 + 1)