Understanding Tangent Lines and Slopes

The video tutorial explains the concept of tangent lines on a graph, focusing on their slopes. It covers how tangent lines can have negative, zero, or positive slopes depending on the graph's behavior. The tutorial uses a parabola to illustrate these concepts, showing that slopes are negative when the function decreases, zero at high or low points, and positive when the function increases. It also introduces interval notation and inequalities to describe these intervals. The importance of tangent lines is highlighted as they provide the instantaneous rate of change, linking to the derivative function.

What is a tangent line?

• A line that crosses a curve at multiple points

• A line that is parallel to a curve

• A line that is perpendicular to a curve

• A line that touches a curve at a single point without crossing it

When does a tangent line have a negative slope?

• When the curve is at a high point

• When the curve is increasing

• When the curve is horizontal

• When the curve is decreasing

What happens to the slope of the tangent line at the vertex of a parabola?

• It becomes undefined

• It becomes zero

• It becomes negative

• It becomes positive

In which interval is the function increasing?

• From -5 to positive infinity

• From 2 to positive infinity

• From -5 to 2

• From negative infinity to -5

How is the interval where the function is decreasing expressed in interval notation?

• (-∞, ∞)

• (2, ∞)

• (-5, 2)

• (-∞, -5)

At which points are the tangent lines horizontal?

• x = -3 and x = 0

• x = 0 and x = 3

• x = -2 and x = 5

• x = -5 and x = 2

What does a horizontal tangent line indicate about the slope?

• The slope is zero

• The slope is positive

• The slope is negative

• The slope is undefined

What is the significance of tangent lines in calculus?

• They determine the area under a curve

• They provide the instantaneous rate of change

• They indicate the maximum value of a function

• They show the average rate of change

What is the derivative in relation to a tangent line?

• The length of the tangent line

• The slope of the tangent line

• The angle of the tangent line

• The distance from the origin to the tangent line

How can the intervals of positive slope be expressed using inequalities?

• x > 5 and x < -2

• x < 2 or x > -5

• x < -5 or x > 2

• x > -5 and x < 2