Understanding Derivatives and Tangent Lines

This video tutorial reviews the connections between derivatives and antiderivatives, focusing on graphical analysis. It covers relationships between first and second derivatives, concavity, and points of inflection. The tutorial includes solving a 2008 exam question and demonstrates using a calculator for derivative problems. It concludes with an application of the second fundamental theorem of calculus.

What is the relationship between a function's increase and its derivative?

• The derivative is positive.

• The derivative is zero.

• The derivative does not exist.

• The derivative is negative.

If a function has a relative maximum, what happens to its derivative?

• The derivative changes from positive to negative.

• The derivative remains constant.

• The derivative changes from negative to positive.

• The derivative becomes undefined.

What is true about the derivative at a critical point?

• The derivative is undefined.

• The derivative is always negative.

• The derivative is always positive.

• The derivative is zero or does not exist.

How does concavity affect the first derivative of a function?

• Concave up means the first derivative is increasing.

• Concave down means the first derivative is increasing.

• Concave down means the first derivative is constant.

• Concave up means the first derivative is decreasing.

In the 2008 exam question, what indicates a point of inflection for G?

• G changes from concave up to concave down.

• G prime remains constant.

• G prime changes from increasing to decreasing.

• G changes from increasing to decreasing.

How do you find the average rate of change of a function over an interval?

• By integrating the function over the interval.

• By dividing the change in function values by the change in x-values.

• By finding the derivative at the midpoint.

• By finding the second derivative.

What does a horizontal tangent line indicate about G prime?

• G prime is undefined.

• G prime is zero.

• G prime is positive.

• G prime is negative.

When writing the equation of a tangent line, what is essential to find first?

• The x-intercept.

• The second derivative.

• The y-intercept.

• The slope of the tangent line.

What does the second fundamental theorem of calculus help you find?

• The slope of a tangent line.

• The derivative of a function.

• The integral of a function.

• The area under a curve.