Exploring the Definition and Calculation of Derivatives

What does the derivative of a function tell you?

• The maximum value of the function

• The slope of the tangent line at any point

• The area under the curve

• The average rate of change

What is a tangent line?

• A line that is parallel to the curve

• A line that intersects the curve at two points

• A line that is perpendicular to the curve

• A line that touches the curve at only one point

Why is it challenging to find the slope of a tangent line at a single point?

• Because the tangent line does not exist

• Because the slope is always zero

• Because we need two points to find the slope

• Because the curve is not continuous

What is the difference quotient?

• The sum of the differences in y-values

• The product of the differences in x-values

• The quotient of differences in y-values over differences in x-values

• The difference between the maximum and minimum values

How do we make the slope of the secant line more accurate?

• By decreasing the distance between the points

• By ignoring the second point

• By increasing the distance between the points

• By using a different formula

What is the definition of the derivative?

• The limit of the secant slope as the distance between points approaches zero

• The maximum slope on the curve

• The average slope over a given interval

• The sum of all slopes on the curve

What does f'(x) represent?

• The integral of the function

• The average value of the function

• The derivative of the function

• The original function

In the example provided, what is the derivative of the function 3x^2 + 12?

• 6x

• 3x

• 12x

• 9x

Why is the limit process considered tedious in practice?

• Because it is not a valid method

• Because it is not accurate

• Because it only works for linear functions

• Because it requires a lot of extra algebra

What is a faster and simpler way to find derivatives in practice?

• Using the integral rules

• Using the derivative rules

• Using the average rate of change

• Using the limit process