Sample Problems, 2.10

Sample Problems, Lesson 2.10

2.10, #12, Given Problem.

The graph of the derivative of a continuous function f is shown.	2.10, #12, Given Problem.
Intervals where f(x) is increasing: (1,6) , (8,9) Intervals where f(x) is decreasing: (0,1) , (6,8)	Part (a). We observe the graph of the derivative and look for any intervals where the derivative is positive. (Remember, a positive derivative indicates that the curve is increasing) Note that we are unsure of what happens after x = 9.
Local maxima or minima: x = 1, 6, and 8	Part (b). Again this is done by observation of the graph. Here we are looking for where the derivative is zero.
Concave up: (-2,2) , (3,5) , (7,9) Concave down: (2,3) , (5,7)	Part (c). To find the intervals where the function is concave up we are looking for when the slope of the frist derivative curve is positive. To find the intervals where the function is concave down we are looking for when the slope of the frist derivative curve is negative.
Points of inflection: x = 2, 3, 5, 7	Part (d). We know that we will have a point of inflection when the second derivative equals zero.. This will happen on our first derivative curve whenever the slope of the tangent line equals zero.
	Part (e). This sketch incorporates all of the analyses above. We use the increasing and decreasing information to get a sense for the general shape of the graph. The points of inflection are shown as points on the answer graph at left. We also use the local maximum and minimum information.

2.10, #14, Given Problem.

2.10, #14, Given Problem.

There are many correct answers here. The graph at the left is correct because:

1) its slope is always negative (this corresponds to the directive in the problem to have the first derivative always be negative, and

2) it is concave up. (this corresponds to the directive in the problem to have the second derivative always be positive)

2.10, #26, Given Problem.

The graph of a function f is shown. Which graph is an antiderivative of f and why?	2.10, #26, Given Problem.
We first rule out curve c. We are down to the choices a, and b.	The slope of our antiderivative must be zero where the f curve crosses the x-axis.
Curve a is the antiderivative of f.	The slope of our antiderivative must be positive before the x-axis crossing and negative thereafter.