MTH 251 – Week 5 lecture notes - What information is given by derivative functions?
Drawing Functions based upon Derivative Sign Information
Table 1: 4 Basic Curve Shapes and the attendant Derivative Signs
Example
Sketch onto Figure 1 a continuous curve, , that has the following properties.
·
· over and
· over and
· over and
· over
Four different functions are shown in figures W - Z. In each question on this page a property is stated that is true for only one of the functions shown in figures W - Z. For each property, state the figure letter that shows the function with the stated property. Write below each question a brief explanation of how you made your determination.
a. The function whose first derivative is increasing over the interval is shown in Figure
b. The function whose first derivative is negative over the interval is shown in Figure
c. The function whose antiderivative is decreasing over is shown in Figure
d. The function whose first derivative has a local minimum point at is in Figure
e. The function whose antiderivative is concave up over is shown in Figure
f. The function whose antiderivative has a local maximum point at is in Figure
The graph of a first derivative, , is shown in Figure 21. Sketch onto Figure 22 the function given that and .
Figure 2 shows the graph of the second derivative of a function named g. Answer each of the following questions. In each case, explain your reasoning.
· Where is g concave up?
· Where does g have its point(s) of inflection?
· Rank the four numbers in increasing order.
· Suppose that . Is g increasing or decreasing at the point where ?
A function is shown in Figure 3. Suppose that is an antiderivative of . Which is greater, or ? How do you know?
Figures A-F show 6 different functions. The first derivative of one of these functions is . Which one is a graph of ? No work need be shown.
Each of the following sentences is true if one of the words/phrases in Table 1 is inserted into the blank. Find the proper word/phrase for each of the blanks. Read each sentence carefully!!
· If is negative at every point over , then is
over the entire interval.
· If is positive at every point over , then is over the entire interval .
· If is increasing over the entire interval , then is over the entire interval .
· If the slope of is increasing over the entire interval , then is
over the entire interval .
· If the slope of is positive over the entire interval , then is
over the entire interval .
· If is continuous and has a local minimum at the point where ,
then is immediately to the right of .
· If the slope of is increasing over the entire interval , then at any point along the
interval , is .
(This question is continued on page 7.)
· If is differentiable and has a local maximum point at then the tangent line to
at the point is .
· If is decreasing over the entire interval then is .
· If is concave up over the entire interval , then is over the entire interval .
· If the slope of is negative over the entire interval ,
then is .
· If is concave up over the entire interval ,
then is .
· If has a local maximum at the point where , then the slope of the tangent line to
is at .
Suppose that and for all values of t. Over what interval(s) is the function decreasing? Explain!
Sketch a curve that satisfies each of the indicated properties. (Sketch a different curve for each stated property. J)
Water flows at a constant rate into a large conical tank (pointy end down J). Let (in ) and (in ft) be, respectively, the volume of water in the tank and the height of the water in the tank t minutes after the water begins to flow. Suppose that five minutes after the water begins to flow the tank is one quarter full. For each of the following expressions, state the units on the expression and state whether the value is positive, negative, or zero. Explain!
Figure 5 is the graph of a function named f. F is an antiderivative of f. Answer each of the following questions about F.
· On what intervals is F increasing? Explain.
· At what values of x does F have a local maximum or local minimum? Explain.
· On what intervals is F concave up? concave down? Explain.
At what values of x does F have an inflection point? Expla
Page 2 of 10