Understanding Tangent Lines

The video tutorial explains the concept of tangent lines to functions, focusing on their properties and significance. A tangent line at a point on a curve has the same slope as the curve at that point, known as the point of tangency. The tutorial illustrates how tangent lines serve as linear approximations of functions and discusses their role in representing the instantaneous rate of change. It also covers the conditions under which a function may not have a tangent line, such as at points of discontinuity or sharp corners. Graphical examples, including quadratic and cubic functions, demonstrate how the slope of tangent lines relates to the increasing or decreasing nature of functions. The video concludes by highlighting interactive tools like Desmos for further exploration.

What is a tangent line to a function?

• A line that touches the curve at a point with the same slope

• A line that intersects the curve at multiple points

• A line that is parallel to the x-axis

• A line that is perpendicular to the curve

What is the point of tangency?

• The point where the tangent line is horizontal

• The point where the tangent line is vertical

• The point where the tangent line and curve have the same slope

• The point where the tangent line crosses the x-axis

Why is a tangent line considered a linear approximation?

• Because it is a straight line

• Because it approximates the curve at the point of tangency

• Because it intersects the curve at multiple points

• Because it is parallel to the y-axis

What does the slope of a tangent line represent?

• The total change of the function over an interval

• The average rate of change of the function

• The maximum value of the function

• The instantaneous rate of change of the function

When does a function not have a tangent line?

• At points where the function is increasing

• At points of continuity

• At points of discontinuity or sharp corners

• At points where the function is decreasing

What happens when the slope of the tangent line is negative?

• The function is decreasing

• The function is increasing

• The function has a maximum

• The function is constant

What indicates a relative minimum in a function?

• When the slope of the tangent line changes from negative to positive

• When the slope of the tangent line is positive

• When the slope of the tangent line is zero

• When the slope of the tangent line changes from positive to negative

What indicates a relative maximum in a function?

• When the slope of the tangent line changes from negative to positive

• When the slope of the tangent line is negative

• When the slope of the tangent line changes from positive to negative

• When the slope of the tangent line is zero

What is a key feature of a tangent line at the point of tangency?

• It is always horizontal

• It has the same slope as the curve at that point

• It is always vertical

• It intersects the curve at multiple points

Where can you find a tool to animate tangent lines?

• In a math textbook

• On the Desmos website

• On a graphing calculator

• In a calculus classroom