Module 11 - The Relationship between a Function and Its First and Second Derivatives

Module 11 - Answers

Lesson 1

Answer 1

11.1.1

The coefficient of x in the equation of the tangent line is the value of the derivative at the point of tangency because the derivative is the slope of the tangent line.

Answer 2

11.1.2

Answer 3

11.1.3

The derivative is negative on the interval where the function is decreasing.
The derivative is positive on the intervals where the function is increasing.

Answer 4

11.1.4

Lesson 2

Answer 1

11.2.1

The slopes steadily increase until about x = 0.7 after which the slopes decrease. The screen below is the last screen displayed after running tanimate. The points on the dotted curve represent the slopes of the tangent lines at the corresponding x-values.

Answer 2

11.2.2 The graph changes from concave upward to concave downward at approximately (0.7, -10.863).

Answer 3

11.2.3 The second derivative is positive where the original function is concave up and is negative where the function is concave down. Since f " is the derivative of f ', when f " is positive, that means f ' is increasing and f is therefore concave upward. Similarly, when f " is negative, f ' is decreasing and f is concave downward.

Answer 4

11.2.4