Understanding Derivatives and Tangent Lines

The video tutorial explains how to find the derivative of the function f(x) = 4 sin(x) and use it to determine the equation of the tangent line at the point (π/6, 2). The process involves calculating the derivative, finding the slope of the tangent line, and using the point-slope form to derive the equation. The equation is then converted to slope-intercept form for easier interpretation. Finally, the solution is verified graphically to ensure accuracy.

What is the derivative of the function f(x) = 4 sin(x)?

• 4 cos(x)

• 4 sin(x)

• cos(x)

• sin(x)

How do you find the slope of the tangent line at a specific point?

• By using the original function

• By finding the second derivative

• By calculating the area under the curve

• By evaluating the derivative function at that point

What is the value of cos(π/6) on the unit circle?

• √3/2

• 1/2

• 1

• 0

What is the slope of the tangent line at the point (π/6, 2)?

• 2

• 3

• √3

• 2√3

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

• Point-slope form

• Quadratic form

• Slope-intercept form

• Standard form

What is the equation of the tangent line in point-slope form?

• y = x + 2√3

• y = 4x - 2

• y = 2√3x + 2

• y - 2 = 2√3(x - π/6)

How do you convert the equation of a line from point-slope form to slope-intercept form?

• Divide by the x-coordinate

• Multiply by the slope

• Distribute and add the constant to both sides

• Subtract the y-intercept

What is the approximate value of 2√3?

• 4.46

• 2.46

• 3.46

• 1.46

What is the purpose of graphing the function and tangent line?

• To find the derivative

• To verify the tangent line equation

• To calculate the area under the curve

• To determine the maximum value

What is the approximate y-intercept of the tangent line?

• 0.19

• 1.19

• 2.19

• 3.19