Analyzing Tangent Line Slopes

The video tutorial explains how to graphically determine the derivative of the sine function. It begins by identifying points where the derivative is zero, such as maxima and minima, and then estimates the derivative values between these points. By connecting these points, the graph of the derivative is formed, revealing that the derivative of the sine function is the cosine function.

What is the initial function used in the graphical example?

• f(x) = cos(x)

• f(x) = tan(x)

• f(x) = sin(x)

• f(x) = x^2

What is the significance of maximum and minimum points in estimating derivatives graphically?

• They are points where the function is decreasing.

• They indicate where the function is undefined.

• They show where the slope of the tangent is zero.

• They are points where the function is increasing.

At which x-values is the derivative of the sine function zero?

• x = -2π, 0, 2π

• x = 0, π, 2π

• x = -π/2, π/2, 3π/2

• x = -3π/2, -π/2, π/2, 3π/2

What is the slope of the tangent line at x = π/2?

• Approximately 0

• Approximately 1

• Approximately -1

• Approximately 2

What is the slope of the tangent line at x = -π/2?

• Approximately 0

• Approximately 1

• Approximately -1

• Approximately 2

What is the slope of the tangent line at x = 3π/2?

• Approximately -1

• Approximately 2

• Approximately 1

• Approximately 0

What is the slope of the tangent line at x = -3π/2?

• Approximately 2

• Approximately -1

• Approximately 0

• Approximately 1

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

• Approximately 0

• Approximately 2

• Approximately 1

• Approximately -1

What does the test technique help determine about the derivative?

• Where the derivative is positive

• Where the derivative is zero

• Where the derivative is negative

• Where the derivative is undefined

Between which points is the derivative expected to be negative?

• Between 0 and π/2

• Between π/2 and 3π/2

• Between -3π/2 and -π/2

• Between -π and π